SP2004F Lecture07-01 Digital Signal Processingberlin.csie.ntnu.edu.tw/Courses/2004F-SpeechRecognition/Slides... · 2004 SP- Berlin Chen 3 Two Main Approaches to Digital Signal Processing

Digital Signal Processing

References1 A V Oppenheim and R W Schafer Discrete-time Signal Processing 19992 X Huang et al Spoken Language Processing Chapters 5 63 J R Deller et al Discrete-Time Processing of Speech Signals Chapters 4-64 J W Picone ldquoSignal modeling techniques in speech recognitionrdquo proceedings of the IEEE

September 1993 pp 1215-1247

Berlin Chen 2004

2004 SP- Berlin Chen 2


bull Digital Signalndash Discrete-time signal with discrete amplitude

bull Digital Signal Processingndash Manipulate digital signals in a digital computer

[ ]nx


Two Main Approaches to Digital Signal Processing

bull Filtering

bull Parameter Extraction

FilterSignal in Signal out

[ ] nx [ ] ny

ParameterExtraction

Signal in Parameter out

[ ] nx

1

12

11

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

mc

cc

2

22

21

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

mc

cc

2

1

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

Lm

L

L

c

cc

eg1 Spectrum Estimation2 Parameters for Recognition

Amplify or attenuate some frequencycomponents of [ ] nx


Sinusoid Signals

ndash amplitude (振幅)ndash angular frequency (角頻率) ndash phase (相角)

bull Eg speech signals can be decomposed as sums of sinusoids

[ ] ( )φω += nAnx cos

ωφ

A

Tf ππω 22 ==

[ ] ⎟⎠⎞

⎜⎝⎛ minus=

2cos πωnAnx

samples 25=T

10frequency normalized

lele ff

Period represented by number of samples

[ ] [ ]

( )2122

2)(cos

==rArr=rArr

⎟⎠⎞

⎜⎝⎛ minus+=+=

kN

kkN

NnANnxnx

πωπω

πω


Sinusoid Signals (cont)

bull is periodic with a period of N (samples)

bull Examples (sinusoid signals)

ndash is periodic with period N=8ndash is periodic with period N=16ndash is not periodic

[ ]nx[ ] [ ]nxNnx =+

( ) ( )φωφω +=++ nANnA cos)(cos)21( 2 kkN == πω

Nπω 2

=

[ ] ( )4cos1 nnx π=[ ] ( )83cos2 nnx π=

[ ] ( )nnx cos3 =


Sinusoid Signals (cont)[ ] ( )

( )

8

integers positive are and 824

44cos)(

4cos

4cos

4cos

1

11

11

1

=there4

sdotrArrsdot=rArr

⎟⎠⎞

⎜⎝⎛ +=⎟

⎠⎞

⎜⎝⎛ +=⎟

⎠⎞

⎜⎝⎛=

=

N

kNkkN

NnNnn

nnx

ππ

πππππ

[ ] ( )

( )

( )

16

numbers positive are and 3

1628

3

83

83cos

83cos

83cos

83cos

2

222

22

2

=there4

=rArrsdot=sdotrArr

⎟⎠⎞

⎜⎝⎛ sdot+sdot=⎟

⎠⎞

⎜⎝⎛ +=⎟

⎠⎞

⎜⎝⎛ sdot=

=

N

kNkNkN

NnNnn

nnx

ππ

πππππ

[ ] ( )( ) ( )( ) ( )

exist t doesn integers positive are and

2 cos1cos1cos

cos

3

3

3

33

3

NkN

kNNnNnn

nnx

there4

sdot=rArr+=+sdot=sdot=

=

Q

π



bull Complex Exponential Signalndash Use Eulerrsquos relation to express complex numbers

( )φφφ sincos

jAAez

jyxzj +==rArr

+=

( )number real a is A

Re

Im

φφ

sin cos

AyAx

==

22 yx + 22 yx

x

+22 yx

y

+

Imaginary part

real part



bull A Sinusoid Signal

[ ] ( )( )

φω

φω

φω

jnj

nj

eAeAe

nAnx

Re Re

cos

=

=

+=+

real part



bull Sum of two complex exponential signals with same frequency

ndash When only the real part is considered

ndash The sum of N sinusoids of the same frequency is another sinusoid of the same frequency

( ) ( )

( )

( )φω

φω

φφω

φωφω

+

++

=

=

+=

+

nj

jnj

jjnj

njnj

Ae

Aee

eAeAe

eAeA10

10

10

10

( ) ( ) ( )φωφωφω +=+++ nAnAnA coscoscos 1100

numbers real are and 10 AAA



bull Trigonometric Identities

1100

1100coscossinsintan

φφφφ

φAAAA

++

=

( ) ( )( )

( )10102

120

1010102

120

2

21100

21100

2

cos2

sinsincoscos2

sinsincoscos

φφ

φφφφ

φφφφ

minus++=

+++=

+++=

AAAA

AAAAA

AAAAA


Some Digital Signals



bull Any signal sequence can be represented as a sum of shift and scaled unit impulse sequences (signals)

[ ]nx

[ ] [ ] [ ]knkxnxk

minussum=infin

minusinfin=δ

scaleweightedTime-shifted unit impulse sequence

[ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ]31211121221

33221101122

3

2

minus+minusminus+minus+++minus++=minus+minus+minus+++minus++minus=

minus=minus= sumsumminus=

infin

minusinfin=

nnnnnnnxnxnxnxnxnx

knkxknkxnxkk

δδδδδδδδδδδδ

δδ

321012 minusminus=n


Digital Systems

bull A digital system T is a system that given an input signal x[n] generates an output signal y[n]

[ ] [ ] nxTny =

[ ]nx bullT [ ]ny


Properties of Digital Systems

bull Linearndash Linear combination of inputs maps to linear

combination of outputs

bull Time-invariant (Time-shift)ndash A time shift of in the input by m samples give a shift in

the output by m samples

[ ] [ ] [ ] [ ] nxbTnxaTnbxnaxT 2121 +=+

[ ] [ ] mmnxTmny forallplusmn=plusmn

time sift [ ][ ] samples shift left )0 (if

samples shift right )0 (if mmmnx

mmmnxrArrgt+rArrgtminus


Properties of Digital Systems (cont)

bull Linear time-invariant (LTI)ndash The system output can be expressed as a

convolution (迴旋積分) of the input x[n] and the impulse response h[n]

ndash The system can be characterized by the systemrsquos impulse response h[n] which also is a signal sequence

bull If the input x[n] is impulse the output is h[n][ ]nδ

DigitalSystem

[ ]nδ [ ]nh



bull Linear time-invariant (LTI)ndash Explanation

[ ] [ ] [ ]knkxnxk

minus= suminfin

minusinfin=δ

[ ] [ ] [ ] [ ] [ ]

[ ] [ ][ ] [ ]nhnx

knhkx

knTkx

knkxTnxT

k

k

k

lowast=

minus=

minus=

minus=rArr

sum

sum

sum

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

δ

δ

scale

Time-shifted unit impulse sequence

Time invariant

DigitalSystem

[ ]nδ [ ]nh

linear

Time-invariant

[ ] [ ][ ] [ ]knhkn

nhnT

T

minus⎯rarr⎯minus

⎯rarr⎯

δ

δ

Impulse response

convolution



bull Linear time-invariant (LTI)ndash Convolution

bull Example

[ ]nδ 0 12

13

-2[ ]nh

LTI[ ]nx

0 1 2

23

1

0

3

0 12

39

-6

1

2

1 23

26

-4

LTI

2

1

2 34

1 3

-2

Length=M=3

Length=L=3

Length=L+M-1

[ ]nδsdot3

[ ]12 minussdot nδ

[ ]21 minussdot nδ

0

3

1

11

2

13

-1

4

-2

Sum up

[ ]ny

[ ]nhsdot3

[ ]12 minussdot nh

[ ]2minusnh



bull Linear time-invariant (LTI)ndash Convolution Generalization

bull Reflect h[k] about the origin (rarr h[-k]) bull Slide (h[-k] rarr h[-k+n] or h[-(k-n)] ) multiply it with

x[k]bull Sum up [ ]kx

[ ]kh

[ ]kh minus

Reflect Multiply

slide

Sum up

n

[ ] [ ] [ ]

[ ] ( )[ ]nkhkx

knhkxny

k

k

minusminus=

minus=

sum

suminfin

minusinfin=

infin

minusinfin=


0 12

13

-2

0-1-2

13

-2

0 1 2

23

1

0

3

10-1

13

-21

11

210

13

-2

1

2

321

13

-2 -1

3

432

13

-2 -2

4

0

3

1

11

2

13

-1

4

-2

Sum up

[ ]kh

[ ]kx

[ ]kh minus

[ ] 0 =nny

[ ]1+minuskh

[ ]2+minuskh

[ ]3+minuskh

[ ]4+minuskh

[ ] 1 =nny

[ ] 2 =nny

[ ] 3 =nny

[ ] 4 =nny

[ ]ny

Reflect [ ] [ ] [ ][ ] [ ]knhkx

nhnxny

kminus=

lowast=

suminfin

minusinfin=

Convolution



bull Linear time-invariant (LTI)ndash Convolution is commutative and distributive

[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=

[ ]nh 1 [ ]nh 2 [ ]nh 2 [ ]nh 1

[ ]nh 2

[ ]nh 1

[ ] [ ]nhnh 21 +

ndash An impulse response has finite durationraquo Finite-Impulse Response (FIR)

ndash An impulse response has infinite durationraquo Infinite-Impulse Response (IIR)

[ ] [ ] [ ][ ] [ ] [ ]nhnhnx

nhnhnx

12

21

=

[ ] [ ] [ ]( )[ ] [ ] [ ] [ ]nhnxnhnx

nhnhnx

21

21

+=+

Commutation

Distribution



bull Prove convolution is commutative

[ ] [ ] [ ]

[ ] [ ]

[ ] [ ] ( )

[ ] [ ]nxnh

knmhmnx

knhkx

nhnxny

m

k

mlet

=

minus=sum minus=

minussum=

=

infin

minusinfin=

infin

minusinfin=



bull Linear time-varying Systemndash Eg is an amplitude modulator [ ] [ ] nnxny 0cos ω=

[ ] [ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] ( )0000

00011

0

101

cosBut

coscos

suppose

nnωnnxnny

nnnxnnxny

nnynynnxnx

minusminus=minus

minus==

minus=rArrminus=

ωω



bull Bounded Input and Bounded Output (BIBO) stable

ndash A LTI system is BIBO if only if h[n] is absolutely summable

[ ] infinlesuminfin

minusinfin=kkh

[ ][ ] nBny

nBnx

y

x

forallinfinltle

forallinfinltle



bull Causalityndash A system is ldquocasualrdquo if for every choice of n0 the output

sequence value at indexing n=n0 depends on only the input sequence value for nlen0

ndash Any noncausal FIR can be made causal by adding sufficient long delay

[ ] [ ] [ ]sum sum= =

minus+minus=K

k

M

mkkk mnxknyny

1000 βα

z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα


Discrete-Time Fourier Transform (DTFT)

bull Frequency Responsendash Defined as the discrete-time Fourier Transform of

ndash is continuous and is periodic with period=

ndash is a complex function of

[ ]nh( )ωjeH

π2( )ωjeH

ω( ) ( ) ( )

( ) ( )ωω

ωωω

jeHjj

ji

jr

j

eeH

ejHeHeHang=

+=

magnitudephase

( )ωjeH

proportional to twotimes of the samplingfrequency


Discrete-Time Fourier Transform (cont)

bull Representation of Sequences by Fourier Transform

ndash A sufficient condition for the existence of Fourier transform

[ ] infinltsuminfin

minusinfin=nnh

[ ]( )

( )

[ ]mnnmnm

mnmn

emnj

de

mnj

mnj

minus=⎩⎨⎧

ne=

=

minusminus

=

minus= minus

minus

minusminus

int

δ

πππ

ωπ

ππ

ω

ππ

ω

0 1

sin)(2

121

)(

)(

( ) [ ]

[ ] ( ) [ ]

[ ] [ ] [ ] [ ]nhmnmhdemh

deemhdeeHnh

enheH

m

mnj

m

nj

m

mjnjj

n

njj

=minus==

==

=

sumintsum

int sumint

sum

infin

minusinfin=minus

minusinfin

minusinfin=

minus

infin

minusinfin=

minusminus

infin

minusinfin=

minus

δωπ

ωπ

ωπ

ππ

ω

ππ

ωωππ

ωω

ωω

21

21

21

invertible is ansformFourier tr

)(

( ) [ ]

[ ] ( )int

sum

minus

infin

minusinfin=

minus

=

=

ππ

ωω

ωω

ωπ

deeHnh

enheH

njj

n

njj

2

1

absolutely summable

DTFT

Inverse DTFT


Discrete-Time Fourier Transform (cont)bull Convolution Property

( ) [ ]

[ ] [ ] [ ]

( ) [ ] [ ]

[ ] [ ]

( ) ( )

[ ] ( ) ( )ωω

ωω

ωω

ωω

ωω

jj

jj

nj

nk

kj

nj

n k

j

k

n

njj

eHeXnhnx

eHeX

enhekx

eknhkxeY

knhkxnhnxny

enheH

hArrlowastthere4

=

⎟⎠

⎞⎜⎝

⎛=

minus=

minus=lowast=

=

minusinfin

minusinfin=

infin

minusinfin=

minus

minusinfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus

sumsum

sum sum

sum

sum

][

][][

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )ωωω

ωωω

ωωω

jjj

jjj

jjj

eHeXeY

eHeXeY

eHeXeY

ang+ang=angrArr

=rArr

=

knnknnknn

minusminus=minusrArr+=rArrminus=



bull Parsevalrsquos Theorem

ndash Define the autocorrelation of signal

[ ] ( )intsum minus

infin

minusinfin== π

πω ω

πdeXnx j

n

22 21

power spectrum

[ ]nx[ ]

( ) ][][][][

][][

nxnxlnxlx

mxnmxnR

l

mxx

minuslowast=minusminus=

+=

lowastinfin

minusinfin=

infin

minusinfin=

sum

sum

( ) ( ) ( ) ( ) 2 ωωωω XXXS xx ==

hArr

[ ] ( ) ( )intint minusminus==

π

π

ωπ

π

ω ωωπ

ωωπ

deXdeSnR njnjxxxx

2

21

21

[ ] [ ] [ ] [ ] ( )sum intsuminfin

minusinfin=minus

infin

minusinfin=

===

=

mmxx dXmxmxmxR

nπ

πωω

π22

210

0Set

The total energy of a signalcan be given in either the time or frequency domain

)(

lnnlmnml

minusminus=minus=rArr+=

jyxz jyxz minus=rArr+=

conjugatecomplex



bull A LTI system with impulse responsendash What is the output for

[ ]nh[ ]ny [ ] ( )φω += nAnx cos

[ ] ( )[ ] [ ] [ ] ( )

[ ] ( ) [ ]

[ ] ( )( )

( ) [ ]( ) ( )( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( )( )[ ] ( ) ( ) ( )( )ωω

ωωωω

φωω

ωφω

ωφω

ωφω

φω

φω

φω

φω

φωφω

φω

ω

ω

jj

jjjj

eHjnjj

eHjjnj

jnj

kj

k

nj

knj

k

nj

nj

eHneHAny

eHneHjAeHneHA

eeHA

eeHAe

eHAe

ekhAe

Aekh

nhAeny

Aenxnxnx

njAnx

j

j

ang++=rArr

ang+++ang++=

=

=

=

sum=

sum=

=

=+=

+=

ang++

ang+

+

minusinfin

minusinfin=

+

+minusinfin

minusinfin=

+

+

cos

sincos

sin

0

10

1

[ ]ny [ ]ny1

( )( ) attenuate 1

amplify 1

lt

gt

ω

ω

j

j

eH

eH

Systemrsquos frequency response




Z-Transform

bull z-transform is a generalization of (Discrete-Time) Fourier transform

ndash z-transform of is defined as

bull Where a complex-variablebull For Fourier transform

ndash z-transform evaluated on the unit circle

( ) [ ]suminfin

minusinfin=

minus=n

nznhzH

[ ] ( )zHnh

[ ] ( )ωjeHnh

[ ]nh

ωjrez =

( ) ( ) ωω

jezj zHeH

==

)1( == zez jω

complex plane

unit circle

Im

Re

ω

ωjez =


Z-Transform (cont)

bull Fourier transform vs z-transformndash Fourier transform used to plot the frequency response of a filterndash z-transform used to analyze more general filter characteristics eg

stability

bull ROC (Region of Converge)ndash Is the set of z for which z-transform exists (converges)

ndash In general ROC is a ring-shaped region and the Fourier transform exists if ROC includes the unit circle ( )

[ ] infinltsuminfin

minusinfin=

minus

n

nznh

complex plan

R1

R2

Re

Im

absolutely summable

1=z


Z-Transform (cont)

bull An LTI system is defined to be causal if its impulse response is a causal signal ie

ndash Similarly anti-causal can be defined as

bull An LTI system is defined to be stable if for every bounded input it produces a bounded outputndash Necessary condition

bull That is Fourier transform exists and therefore z-transform include the unit circle in its region of converge

[ ] 0for 0 lt= nnh

[ ] 0for 0 gt= nnh

Right-sided sequence

Left-sided sequence

[ ] infinltsuminfin

minusinfin=nnh

[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


Z-Transform (cont)

bull Right-Sided Sequencendash Eg the exponential signal

[ ] [ ] [ ]⎩⎨⎧

ltge

==0for 00for 1

ere wh 1 1 nn

nunuanh n

( ) ( ) 11

1 11

minus

infin

minusinfin=

minusminusinfin

minusinfin= minus=== sumsum

azazzazH

n

nn

n

n

If 11 ltminusaz

azROC gtthere4 is 1

Re

Im

a[ ] 1 if exists

ofansformFourier tr

1 ltanhtimes

have a pole at(Pole z-transform goes to infinity)

az =

the unit cycle

0


Z-Transform (cont)

bull Left-Sided Sequencendash Eg

[ ] [ ] )-0-1-2(n 1 2 2 infin=minusminusminus= nuanh n

( ) [ ]

( ) 11

1

10

1

1

12

11

11111

1

minusminus

minus

minus

infin

=

minusinfin

=

minus

minusminus

minusinfin=

minusinfin

minusinfin=

minus=

minusminus

minus=minus

minus=minus=minus=

minus=minusminusminus=

sumsum

sumsum

azzaza

zazaza

zaznuazH

n

n

n

nn

n

n

nn

n

n If 11 ltminus za

azROC ltthere4 is 2

Re

Im

a[ ] [ ]

minusinfinrarr

lt

nnhnh

a

aslly exponentia go will because existt doesn

of ansformFourier tr the1when

22times

the unit cycle

0


Z-Transform (cont)

bull Two-Sided Sequencendash Eg

[ ] [ ] [ ]121

31 3 3 minusminus⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus= nununh

nn

[ ]

[ ]21

211

1 121

31

311

1 31

1

1

ltminus

⎯rarrlarrminusminus⎟⎠⎞

⎜⎝⎛minus

gt+

⎯rarrlarr⎟⎠⎞

⎜⎝⎛ minus

minus

minus

zz

nu

zz

nu

zn

zn

Re

Im

times

the unit cycle

times21

31

minus [ ]circleunit theincludet doesn because

existt doesn of ansformFourier tr

3

3

ROCnh

31 and

21 is 3 gtltthere4 zzROC

( )⎟⎠⎞

⎜⎝⎛ minus⎟⎠⎞

⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ minus

=minus

++

=minusminus

21

31

1212

211

1

311

1 11

3

zz

zz

zzzH

0


Z-Transform (cont)

bull Finite-length Sequencendash Eg

[ ]⎩⎨⎧ minuslele

=others 0

10 3 4

Nnanh

n

( ) ( ) ( )azaz

zazazazzazH

NN

N

NN

n

nN

n

nn

minusminus

=minus

minus=== minusminus

minusminus

=

minusminus

=

minus sumsum 11

11

0

11

04

11

1

0except plane- entire is 4 =there4 zzROC

13221 minusminusminusminus ++++ NNNN azaazz

Im

times

the unit cycle

31

minus

7 poles at zero A pole and zero at is cancelled az =

4π

( ) 11 2

minus== N kaez Nkj

k

π

If N=8

0 N-1

Re


Z-Transform (cont)

bull Properties of z-transform1 If is right-sided sequence ie and if ROC

is the exterior of some circle the all finite for which will be in ROC

bull If ROC will include

2 If is left-sided sequence ie the ROC is the interior of some circle

bull If ROC will include 3 If is two-sided sequence the ROC is a ring4 The ROC canrsquot contain any poles

[ ]nh [ ] 1 0 nnnh le=

01 gen

0rz gtz

infin=z

[ ]nh [ ] 2 0 nnnh ge=

02 ltn 0=z[ ]nh

A causal sequence is right-sided with ROC is the exterior of circle including

01 geninfin=zthere4


Summary of the Fourier and z-transforms


LTI Systems in the Frequency Domain

bull Example 1 A complex exponential sequencendash System impulse response

ndash Therefore a complex exponential input to an LTI systemresults in the same complex exponential at the output but modified by

bull The complex exponential is an eigenfunction of an LTI system and is the associated eigenvalue

[ ]nh

[ ] [ ] [ ] [ ]

[ ]

( ) njj

kjnj

knj

eeH

ekhe

ekhnhnxny

ωω

ωω

ω

=

=

=lowast=

minusinfin

infin=

minusinfin

infin=

sum

sum

-k

)(

-k( )

responsefrequency system theas toreferredoften isIt

response impulse system the of ansformFourier tr theωjeH

[ ] njenx ω=[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=

[ ] ( ) [ ]nxeHnxT jω=

( )jωeH

( )jωeH

scalar


[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjωnjjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

ang++=

+=

+=

+=

+minusangminus+ang

+minus+

minusminusminus

φω

φωφω

φωφω

ωφωφ

LTI Systems in the Frequency Domain (cont)

bull Example 2 A sinusoidal sequence

ndash System impulse response

[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0

ωφωφ

φω

minusminus+=

+=

[ ]nh

( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

minus

minus

+=rArr

minus=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω

jωjω

eeHeH

eHeHangminus

minus

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos

sincos

jje

jejyxz

jejyxz

j

j

j

minus=minus+minus=

minus=rArrminus=

+=rArr+=

minus

magnitude response phase response



bull Example 3 A sum of sinusoidal sequences

ndash A similar expression is obtained for an input consisting of a sum of complex exponentials

[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1

cos ωω φωmagnitude response phase response



bull Example 4 Convolution Theorem

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ] [ ] 1 lt= suminfin

minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minus= sum

infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1

[ ] [ ] ( ) ( )ωω jj eHeXnhnx hArrlowast

has a nonzero value when

πω2

Pk =



bull Example 5 Windowing Theorem [ ] [ ] ( ) ( )ωω

πjj eXeWnwnx lowasthArr

21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ minusminus=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ minuslowast=

⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=


Difference Equation Realization for a Digital Filter

bull The relation between the output and input of a digital filter can be expressed by

[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus

rarrlinearity and delay properties

A rational transfer functionCausalRightsided the ROC outside the outmost poleStableThe ROC includes the unit circleCausal and Stableall poles must fall inside the unit circle (not including zeros)

z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα


Difference Equation Realization for a Digital Filter (cont)


Magnitude-Phase Relationship

bull Minimum phase systemndash The z-transform of a system impulse response sequence ( a

rational transfer function) has all zeros as well as poles inside the unit cycle

ndash Poles and zeros called ldquominimum phase componentsrdquondash Maximum phase all zeros (or poles) outside the unit cycle

bull All-pass systemndash Consist a cascade of factor of the form

ndash Characterized by a frequency responsewith unit (or flat) magnitude for all frequencies

1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a

Poles and zeros occur at conjugate reciprocal locations

111

1 =minus minusaz

z-a


Magnitude-Phase Relationship (cont)

bull Any digital filter can be represented by the cascade of a minimum-phase system and an all-pass system

( ) ( ) ( )zHzHzH apmin=

( )( )

( ) ( )( ) ( )

( )( ) ( )( )

( )( )( )( ) filter pass-all a is 11

filter phase minimum a also is 1

where

111

filter) phase minimum a is ( 1

as expressed becan circleunit theoutside

1)( 1 zero oneonly has that Suppose

1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters

bull FIR (Finite Impulse Response)ndash The impulse response of an FIR filter has finite durationndash Have no denominator in the rational function

bull No feedback in the difference equation

ndash Can be implemented with simple a train of delay multiple and add operations

( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β


First-Order FIR Filters

bull A special case of FIR filters

[ ] [ ] [ ]1minus+= nxnxny α ( ) 11 minus+= zzH α

( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH

pre-emphasis filter0ltα


Discrete Fourier Transform (DFT)

bull The Fourier transform of a discrete-time sequence is a continuous function of frequencyndash We need to sample the Fourier transform finely enough to be

able to recover the sequencendash For a sequence of finite length N sampling yields the new

transform referred to as discrete Fourier transform (DFT)

( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π

Inverse DFT Synthesis

DFT Analysis


Discrete Fourier Transform (cont)

[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall

minussdotminusminussdotminusminus

minussdotminussdotminus

minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π



bull Orthogonality of Complex Exponentials ( )

⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=



bull Parsevalrsquos theorem

[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density


Analog Signal to Digital Signal

[ ] ( ) period sampling TnTxnx a=

rate sampling 1TFs =

Analog Signal

Discrete-time Signal or Digital Signal

nTt =

sampling period=125μs=gtsampling rate=8kHz

Digital SignalDiscrete-time

signal with discreteamplitude


Analog Signal to Digital Signal (cont)

Impulse TrainTo

Sequence [ ] ( ))( ˆ nTxnx a=

Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a

Continuous-TimeSignal

Discrete-TimeSignal

Digital Signal

Discrete-time signal with discrete

amplitude

[ ] nx

Continuous-Time to Discrete-Time Conversion

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

Periodic Impulse Train ( ) [ ]nxtxs by specifieduniquely becan

switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T



bull A continuous signal sampled at different periods

T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa

frequency) (sampling22ss F

Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin

minusinfin=ΩminusΩ=Ω

ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧

ΩgtΩrArrΩgtΩminusΩ

2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧

ΩltΩrArrΩltΩminusΩ

2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ

=ΩΩ otherwise 0 2 sT

jRs

( ) ( ) ( )ΩΩ=Ω Ω jXjRjX pa s

( ) ( )ΩΩ jXjX pa from recovered bet can

Low-pass filter

high frequency componentsgot superimposed onlow frequency components

bull Spectra



bull To avoid aliasing (overlapping fold over)ndash The sampling frequency should be greater than two times of

frequency of the signal to be sampled rarrndash (Nyquist) sampling theorem

bull To reconstruct the original continuous signalndash Filtered with a low pass filter with band limit

bull Convolved in time domain sΩ

( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2

ltlt ASCII85EncodePages false AllowTransparency false AutoPositionEPSFiles true AutoRotatePages All Binding Left CalGrayProfile (Dot Gain 20) CalRGBProfile (sRGB IEC61966-21) CalCMYKProfile (US Web Coated 050SWOP051 v2) sRGBProfile (sRGB IEC61966-21) CannotEmbedFontPolicy Warning CompatibilityLevel 14 CompressObjects Tags CompressPages true ConvertImagesToIndexed true PassThroughJPEGImages true CreateJDFFile false CreateJobTicket false DefaultRenderingIntent Default DetectBlends true ColorConversionStrategy LeaveColorUnchanged DoThumbnails false EmbedAllFonts true EmbedJobOptions true DSCReportingLevel 0 EmitDSCWarnings false EndPage -1 ImageMemory 1048576 LockDistillerParams false MaxSubsetPct 100 Optimize true OPM 1 ParseDSCComments true ParseDSCCommentsForDocInfo true PreserveCopyPage true PreserveEPSInfo true PreserveHalftoneInfo false PreserveOPIComments false PreserveOverprintSettings true StartPage 1 SubsetFonts true TransferFunctionInfo Apply UCRandBGInfo Preserve UsePrologue false ColorSettingsFile () AlwaysEmbed [ true ] NeverEmbed [ true ] AntiAliasColorImages false DownsampleColorImages true ColorImageDownsampleType Bicubic ColorImageResolution 300 ColorImageDepth -1 ColorImageDownsampleThreshold 150000 EncodeColorImages true ColorImageFilter DCTEncode AutoFilterColorImages true ColorImageAutoFilterStrategy JPEG ColorACSImageDict ltlt QFactor 015 HSamples [1 1 1 1] VSamples [1 1 1 1] gtgt ColorImageDict ltlt QFactor 015 HSamples [1 1 1 1] VSamples [1 1 1 1] gtgt JPEG2000ColorACSImageDict ltlt TileWidth 256 TileHeight 256 Quality 30 gtgt JPEG2000ColorImageDict ltlt TileWidth 256 TileHeight 256 Quality 30 gtgt AntiAliasGrayImages false DownsampleGrayImages true GrayImageDownsampleType Bicubic GrayImageResolution 300 GrayImageDepth -1 GrayImageDownsampleThreshold 150000 EncodeGrayImages true GrayImageFilter DCTEncode AutoFilterGrayImages true GrayImageAutoFilterStrategy JPEG GrayACSImageDict ltlt QFactor 015 HSamples [1 1 1 1] VSamples [1 1 1 1] gtgt GrayImageDict ltlt QFactor 015 HSamples [1 1 1 1] VSamples [1 1 1 1] gtgt JPEG2000GrayACSImageDict ltlt TileWidth 256 TileHeight 256 Quality 30 gtgt JPEG2000GrayImageDict ltlt TileWidth 256 TileHeight 256 Quality 30 gtgt AntiAliasMonoImages false DownsampleMonoImages true MonoImageDownsampleType Bicubic MonoImageResolution 1200 MonoImageDepth -1 MonoImageDownsampleThreshold 150000 EncodeMonoImages true MonoImageFilter CCITTFaxEncode MonoImageDict ltlt K -1 gtgt AllowPSXObjects false PDFX1aCheck false PDFX3Check false PDFXCompliantPDFOnly false PDFXNoTrimBoxError true PDFXTrimBoxToMediaBoxOffset [ 000000 000000 000000 000000 ] PDFXSetBleedBoxToMediaBox true PDFXBleedBoxToTrimBoxOffset [ 000000 000000 000000 000000 ] PDFXOutputIntentProfile () PDFXOutputCondition () PDFXRegistryName (httpwwwcolororg) PDFXTrapped Unknown Description ltlt FRA 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 ENU (Use these settings to create PDF documents with higher image resolution for improved printing quality The PDF documents can be opened with Acrobat and Reader 50 and later) JPN ltFEFF3053306e8a2d5b9a306f30019ad889e350cf5ea6753b50cf3092542b308000200050004400460020658766f830924f5c62103059308b3068304d306b4f7f75283057307e30593002537052376642306e753b8cea3092670059279650306b4fdd306430533068304c3067304d307e305930023053306e8a2d5b9a30674f5c62103057305f00200050004400460020658766f8306f0020004100630072006f0062006100740020304a30883073002000520065006100640065007200200035002e003000204ee5964d30678868793a3067304d307e30593002gt DEU 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 PTB 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 DAN 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 NLD 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 ESP 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 SUO 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 ITA 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 NOR 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 SVE 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 KOR ltFEFFd5a5c0c1b41c0020c778c1c40020d488c9c8c7440020c5bbae300020c704d5740020ace0d574c0c1b3c4c7580020c774bbf8c9c0b97c0020c0acc6a9d558c5ec00200050004400460020bb38c11cb97c0020b9ccb4e4b824ba740020c7740020c124c815c7440020c0acc6a9d558c2edc2dcc624002e0020c7740020c124c815c7440020c0acc6a9d558c5ec0020b9ccb4e000200050004400460020bb38c11cb2940020004100630072006f0062006100740020bc0f002000520065006100640065007200200035002e00300020c774c0c1c5d0c11c0020c5f40020c2180020c788c2b5b2c8b2e4002egt CHS ltFEFF4f7f75288fd94e9b8bbe7f6e521b5efa76840020005000440046002065876863ff0c5c065305542b66f49ad8768456fe50cf52068fa87387ff0c4ee563d09ad8625353708d2891cf30028be5002000500044004600206587686353ef4ee54f7f752800200020004100630072006f00620061007400204e0e002000520065006100640065007200200035002e00300020548c66f49ad87248672c62535f003002gt CHT ltFEFF4f7f752890194e9b8a2d5b9a5efa7acb76840020005000440046002065874ef65305542b8f039ad876845f7150cf89e367905ea6ff0c4fbf65bc63d066075217537054c18cea3002005000440046002065874ef653ef4ee54f7f75280020004100630072006f0062006100740020548c002000520065006100640065007200200035002e0030002053ca66f465b07248672c4f86958b555f3002gt gtgtgtgt setdistillerparamsltlt HWResolution [2400 2400] PageSize [612000 792000]gtgt setpagedevice



bull Digital Signalndash Discrete-time signal with discrete amplitude

bull Digital Signal Processingndash Manipulate digital signals in a digital computer

[ ]nx



bull Filtering



[ ] nx [ ] ny

ParameterExtraction


[ ] nx

1

12

11

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

mc

cc

2

22

21

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

mc

cc

2

1

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

Lm

L

L

c

cc




Sinusoid Signals




ωφ

A

Tf ππω 22 ==

[ ] ⎟⎠⎞

⎜⎝⎛ minus=

2cos πωnAnx

samples 25=T


lele ff


[ ] [ ]

( )2122

2)(cos

==rArr=rArr

⎟⎠⎞

⎜⎝⎛ minus+=+=

kN

kkN

NnANnxnx

πωπω

πω






[ ]nx[ ] [ ]nxNnx =+


Nπω 2

=

[ ] ( )4cos1 nnx π=[ ] ( )83cos2 nnx π=

[ ] ( )nnx cos3 =



( )

8


44cos)(

4cos

4cos

4cos

1

11

11

1

=there4

sdotrArrsdot=rArr

⎟⎠⎞

⎜⎝⎛ +=⎟

⎠⎞

⎜⎝⎛ +=⎟

⎠⎞

⎜⎝⎛=

=

N

kNkkN

NnNnn

nnx

ππ

πππππ

[ ] ( )

( )

( )

16


1628

3

83

83cos

83cos

83cos

83cos

2

222

22

2

=there4

=rArrsdot=sdotrArr

⎟⎠⎞


⎠⎞

⎜⎝⎛ +=⎟

⎠⎞

⎜⎝⎛ sdot=

=

N

kNkNkN

NnNnn

nnx

ππ

πππππ

[ ] ( )( ) ( )( ) ( )


2 cos1cos1cos

cos

3

3

3

33

3

NkN

kNNnNnn

nnx

there4


=

Q

π




( )φφφ sincos

jAAez

jyxzj +==rArr

+=


Re

Im

φφ

sin cos

AyAx

==

22 yx + 22 yx

x

+22 yx

y

+

Imaginary part

real part




[ ] ( )( )

φω

φω

φω

jnj

nj

eAeAe

nAnx

Re Re

cos

=

=

+=+

real part






( ) ( )

( )

( )φω

φω

φφω

φωφω

+

++

=

=

+=

+

nj

jnj

jjnj

njnj

Ae

Aee

eAeAe

eAeA10

10

10

10






1100

1100coscossinsintan

φφφφ

φAAAA

++

=

( ) ( )( )

( )10102

120

1010102

120

2

21100

21100

2

cos2

sinsincoscos2

sinsincoscos

φφ

φφφφ

φφφφ

minus++=

+++=

+++=

AAAA

AAAAA

AAAAA






[ ]nx

[ ] [ ] [ ]knkxnxk

minussum=infin

minusinfin=δ


[ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ]31211121221

33221101122

3

2



infin

minusinfin=

nnnnnnnxnxnxnxnxnx

knkxknkxnxkk


δδ

321012 minusminus=n


Digital Systems


[ ] [ ] nxTny =

[ ]nx bullT [ ]ny


















DigitalSystem

[ ]nδ [ ]nh




[ ] [ ] [ ]knkxnxk

minus= suminfin

minusinfin=δ

[ ] [ ] [ ] [ ] [ ]

[ ] [ ][ ] [ ]nhnx

knhkx

knTkx

knkxTnxT

k

k

k

lowast=

minus=

minus=

minus=rArr

sum

sum

sum

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

δ

δ

scale


Time invariant

DigitalSystem

[ ]nδ [ ]nh

linear

Time-invariant

[ ] [ ][ ] [ ]knhkn

nhnT

T


⎯rarr⎯

δ

δ

Impulse response

convolution




bull Example

[ ]nδ 0 12

13

-2[ ]nh

LTI[ ]nx

0 1 2

23

1

0

3

0 12

39

-6

1

2

1 23

26

-4

LTI

2

1

2 34

1 3

-2

Length=M=3

Length=L=3

Length=L+M-1

[ ]nδsdot3

[ ]12 minussdot nδ

[ ]21 minussdot nδ

0

3

1

11

2

13

-1

4

-2

Sum up

[ ]ny

[ ]nhsdot3

[ ]12 minussdot nh

[ ]2minusnh






[ ]kh

[ ]kh minus

Reflect Multiply

slide

Sum up

n

[ ] [ ] [ ]

[ ] ( )[ ]nkhkx

knhkxny

k

k

minusminus=

minus=

sum

suminfin

minusinfin=

infin

minusinfin=


0 12

13

-2

0-1-2

13

-2

0 1 2

23

1

0

3

10-1

13

-21

11

210

13

-2

1

2

321

13

-2 -1

3

432

13

-2 -2

4

0

3

1

11

2

13

-1

4

-2

Sum up

[ ]kh

[ ]kx

[ ]kh minus

[ ] 0 =nny

[ ]1+minuskh

[ ]2+minuskh

[ ]3+minuskh

[ ]4+minuskh

[ ] 1 =nny

[ ] 2 =nny

[ ] 3 =nny

[ ] 4 =nny

[ ]ny

Reflect [ ] [ ] [ ][ ] [ ]knhkx

nhnxny

kminus=

lowast=

suminfin

minusinfin=

Convolution




[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=

[ ]nh 1 [ ]nh 2 [ ]nh 2 [ ]nh 1

[ ]nh 2

[ ]nh 1

[ ] [ ]nhnh 21 +



[ ] [ ] [ ][ ] [ ] [ ]nhnhnx

nhnhnx

12

21

=

[ ] [ ] [ ]( )[ ] [ ] [ ] [ ]nhnxnhnx

nhnhnx

21

21

+=+

Commutation

Distribution




[ ] [ ] [ ]

[ ] [ ]

[ ] [ ] ( )

[ ] [ ]nxnh

knmhmnx

knhkx

nhnxny

m

k

mlet

=

minus=sum minus=

minussum=

=

infin

minusinfin=

infin

minusinfin=




[ ] [ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] ( )0000

00011

0

101

cosBut

coscos

suppose

nnωnnxnny

nnnxnnxny

nnynynnxnx

minusminus=minus

minus==

minus=rArrminus=

ωω





[ ] infinlesuminfin

minusinfin=kkh

[ ][ ] nBny

nBnx

y

x

forallinfinltle

forallinfinltle






[ ] [ ] [ ]sum sum= =

minus+minus=K

k

M

mkkk mnxknyny

1000 βα

z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα






[ ]nh( )ωjeH

π2( )ωjeH

ω( ) ( ) ( )

( ) ( )ωω

ωωω

jeHjj

ji

jr

j

eeH

ejHeHeHang=

+=

magnitudephase

( )ωjeH






[ ] infinltsuminfin

minusinfin=nnh

[ ]( )

( )

[ ]mnnmnm

mnmn

emnj

de

mnj

mnj

minus=⎩⎨⎧

ne=

=

minusminus

=

minus= minus

minus

minusminus

int

δ

πππ

ωπ

ππ

ω

ππ

ω

0 1

sin)(2

121

)(

)(

( ) [ ]

[ ] ( ) [ ]


deemhdeeHnh

enheH

m

mnj

m

nj

m

mjnjj

n

njj

=minus==

==

=

sumintsum

int sumint

sum

infin

minusinfin=minus

minusinfin

minusinfin=

minus

infin

minusinfin=

minusminus

infin

minusinfin=

minus

δωπ

ωπ

ωπ

ππ

ω

ππ

ωωππ

ωω

ωω

21

21

21


)(

( ) [ ]

[ ] ( )int

sum

minus

infin

minusinfin=

minus

=

=

ππ

ωω

ωω

ωπ

deeHnh

enheH

njj

n

njj

2

1

absolutely summable

DTFT

Inverse DTFT



( ) [ ]

[ ] [ ] [ ]

( ) [ ] [ ]

[ ] [ ]

( ) ( )

[ ] ( ) ( )ωω

ωω

ωω

ωω

ωω

jj

jj

nj

nk

kj

nj

n k

j

k

n

njj

eHeXnhnx

eHeX

enhekx

eknhkxeY

knhkxnhnxny

enheH

hArrlowastthere4

=

⎟⎠

⎞⎜⎝

⎛=

minus=

minus=lowast=

=

minusinfin

minusinfin=

infin

minusinfin=

minus

minusinfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus

sumsum

sum sum

sum

sum

][

][][

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )ωωω

ωωω

ωωω

jjj

jjj

jjj

eHeXeY

eHeXeY

eHeXeY

ang+ang=angrArr

=rArr

=

knnknnknn






[ ] ( )intsum minus

infin

minusinfin== π

πω ω

πdeXnx j

n

22 21

power spectrum

[ ]nx[ ]

( ) ][][][][

][][

nxnxlnxlx

mxnmxnR

l

mxx


+=

lowastinfin

minusinfin=

infin

minusinfin=

sum

sum

( ) ( ) ( ) ( ) 2 ωωωω XXXS xx ==

hArr


π

π

ωπ

π

ω ωωπ

ωωπ

deXdeSnR njnjxxxx

2

21

21


minusinfin=minus

infin

minusinfin=

===

=

mmxx dXmxmxmxR

nπ

πωω

π22

210

0Set


)(

lnnlmnml



conjugatecomplex





[ ] ( )[ ] [ ] [ ] ( )

[ ] ( ) [ ]

[ ] ( )( )

( ) [ ]( ) ( )( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( )( )[ ] ( ) ( ) ( )( )ωω

ωωωω

φωω

ωφω

ωφω

ωφω

φω

φω

φω

φω

φωφω

φω

ω

ω

jj

jjjj

eHjnjj

eHjjnj

jnj

kj

k

nj

knj

k

nj

nj

eHneHAny

eHneHjAeHneHA

eeHA

eeHAe

eHAe

ekhAe

Aekh

nhAeny

Aenxnxnx

njAnx

j

j

ang++=rArr

ang+++ang++=

=

=

=

sum=

sum=

=

=+=

+=

ang++

ang+

+

minusinfin

minusinfin=

+

+minusinfin

minusinfin=

+

+

cos

sincos

sin

0

10

1

[ ]ny [ ]ny1

( )( ) attenuate 1

amplify 1

lt

gt

ω

ω

j

j

eH

eH





Z-Transform





( ) [ ]suminfin

minusinfin=

minus=n

nznhzH

[ ] ( )zHnh

[ ] ( )ωjeHnh

[ ]nh

ωjrez =

( ) ( ) ωω

jezj zHeH

==

)1( == zez jω

complex plane

unit circle

Im

Re

ω

ωjez =


Z-Transform (cont)


stability



[ ] infinltsuminfin

minusinfin=

minus

n

nznh

complex plan

R1

R2

Re

Im

absolutely summable

1=z


Z-Transform (cont)





[ ] 0for 0 lt= nnh

[ ] 0for 0 gt= nnh


Left-sided sequence

[ ] infinltsuminfin

minusinfin=nnh

[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


Z-Transform (cont)


[ ] [ ] [ ]⎩⎨⎧

ltge

==0for 00for 1

ere wh 1 1 nn

nunuanh n

( ) ( ) 11

1 11

minus

infin

minusinfin=

minusminusinfin


azazzazH

n

nn

n

n

If 11 ltminusaz

azROC gtthere4 is 1

Re

Im

a[ ] 1 if exists

ofansformFourier tr

1 ltanhtimes


az =

the unit cycle

0


Z-Transform (cont)



( ) [ ]

( ) 11

1

10

1

1

12

11

11111

1

minusminus

minus

minus

infin

=

minusinfin

=

minus

minusminus

minusinfin=

minusinfin

minusinfin=

minus=

minusminus

minus=minus

minus=minus=minus=


sumsum

sumsum

azzaza

zazaza

zaznuazH

n

n

n

nn

n

n

nn

n

n If 11 ltminus za

azROC ltthere4 is 2

Re

Im

a[ ] [ ]

minusinfinrarr

lt

nnhnh

a



22times

the unit cycle

0


Z-Transform (cont)


[ ] [ ] [ ]121


⎠⎞

⎜⎝⎛minus⎟

⎠⎞


nn

[ ]

[ ]21

211

1 121

31

311

1 31

1

1

ltminus


⎜⎝⎛minus

gt+


⎜⎝⎛ minus

minus

minus

zz

nu

zz

nu

zn

zn

Re

Im

times

the unit cycle

times21

31



3

3

ROCnh

31 and


( )⎟⎠⎞


⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ minus

=minus

++

=minusminus

21

31

1212

211

1

311

1 11

3

zz

zz

zzzH

0


Z-Transform (cont)



=others 0

10 3 4

Nnanh

n

( ) ( ) ( )azaz

zazazazzazH

NN

N

NN

n

nN

n

nn

minusminus

=minus

minus=== minusminus

minusminus

=

minusminus

=

minus sumsum 11

11

0

11

04

11

1



Im

times

the unit cycle

31

minus


4π

( ) 11 2

minus== N kaez Nkj

k

π

If N=8

0 N-1

Re


Z-Transform (cont)






[ ]nh [ ] 1 0 nnnh le=

01 gen

0rz gtz

infin=z

[ ]nh [ ] 2 0 nnnh ge=

02 ltn 0=z[ ]nh


01 geninfin=zthere4








[ ]nh

[ ] [ ] [ ] [ ]

[ ]

( ) njj

kjnj

knj

eeH

ekhe

ekhnhnxny

ωω

ωω

ω

=

=

=lowast=

minusinfin

infin=

minusinfin

infin=

sum

sum

-k

)(

-k( )



[ ] njenx ω=[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


( )jωeH

( )jωeH

scalar


[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjωnjjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

ang++=

+=

+=

+=

+minusangminus+ang

+minus+

minusminusminus

φω

φωφω

φωφω

ωφωφ




[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0

ωφωφ

φω

minusminus+=

+=

[ ]nh

( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

minus

minus

+=rArr

minus=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω

jωjω

eeHeH

eHeHangminus

minus

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos

sincos

jje

jejyxz

jejyxz

j

j

j

minus=minus+minus=

minus=rArrminus=

+=rArr+=

minus






[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2




bull Filtering



[ ] nx [ ] ny

ParameterExtraction


[ ] nx

1

12

11

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

mc

cc

2

22

21

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

mc

cc

2

1

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

Lm

L

L

c

cc




Sinusoid Signals




ωφ

A

Tf ππω 22 ==

[ ] ⎟⎠⎞

⎜⎝⎛ minus=

2cos πωnAnx

samples 25=T


lele ff


[ ] [ ]

( )2122

2)(cos

==rArr=rArr

⎟⎠⎞

⎜⎝⎛ minus+=+=

kN

kkN

NnANnxnx

πωπω

πω






[ ]nx[ ] [ ]nxNnx =+


Nπω 2

=

[ ] ( )4cos1 nnx π=[ ] ( )83cos2 nnx π=

[ ] ( )nnx cos3 =



( )

8


44cos)(

4cos

4cos

4cos

1

11

11

1

=there4

sdotrArrsdot=rArr

⎟⎠⎞

⎜⎝⎛ +=⎟

⎠⎞

⎜⎝⎛ +=⎟

⎠⎞

⎜⎝⎛=

=

N

kNkkN

NnNnn

nnx

ππ

πππππ

[ ] ( )

( )

( )

16


1628

3

83

83cos

83cos

83cos

83cos

2

222

22

2

=there4

=rArrsdot=sdotrArr

⎟⎠⎞


⎠⎞

⎜⎝⎛ +=⎟

⎠⎞

⎜⎝⎛ sdot=

=

N

kNkNkN

NnNnn

nnx

ππ

πππππ

[ ] ( )( ) ( )( ) ( )


2 cos1cos1cos

cos

3

3

3

33

3

NkN

kNNnNnn

nnx

there4


=

Q

π




( )φφφ sincos

jAAez

jyxzj +==rArr

+=


Re

Im

φφ

sin cos

AyAx

==

22 yx + 22 yx

x

+22 yx

y

+

Imaginary part

real part




[ ] ( )( )

φω

φω

φω

jnj

nj

eAeAe

nAnx

Re Re

cos

=

=

+=+

real part






( ) ( )

( )

( )φω

φω

φφω

φωφω

+

++

=

=

+=

+

nj

jnj

jjnj

njnj

Ae

Aee

eAeAe

eAeA10

10

10

10






1100

1100coscossinsintan

φφφφ

φAAAA

++

=

( ) ( )( )

( )10102

120

1010102

120

2

21100

21100

2

cos2

sinsincoscos2

sinsincoscos

φφ

φφφφ

φφφφ

minus++=

+++=

+++=

AAAA

AAAAA

AAAAA






[ ]nx

[ ] [ ] [ ]knkxnxk

minussum=infin

minusinfin=δ


[ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ]31211121221

33221101122

3

2



infin

minusinfin=

nnnnnnnxnxnxnxnxnx

knkxknkxnxkk


δδ

321012 minusminus=n


Digital Systems


[ ] [ ] nxTny =

[ ]nx bullT [ ]ny


















DigitalSystem

[ ]nδ [ ]nh




[ ] [ ] [ ]knkxnxk

minus= suminfin

minusinfin=δ

[ ] [ ] [ ] [ ] [ ]

[ ] [ ][ ] [ ]nhnx

knhkx

knTkx

knkxTnxT

k

k

k

lowast=

minus=

minus=

minus=rArr

sum

sum

sum

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

δ

δ

scale


Time invariant

DigitalSystem

[ ]nδ [ ]nh

linear

Time-invariant

[ ] [ ][ ] [ ]knhkn

nhnT

T


⎯rarr⎯

δ

δ

Impulse response

convolution




bull Example

[ ]nδ 0 12

13

-2[ ]nh

LTI[ ]nx

0 1 2

23

1

0

3

0 12

39

-6

1

2

1 23

26

-4

LTI

2

1

2 34

1 3

-2

Length=M=3

Length=L=3

Length=L+M-1

[ ]nδsdot3

[ ]12 minussdot nδ

[ ]21 minussdot nδ

0

3

1

11

2

13

-1

4

-2

Sum up

[ ]ny

[ ]nhsdot3

[ ]12 minussdot nh

[ ]2minusnh






[ ]kh

[ ]kh minus

Reflect Multiply

slide

Sum up

n

[ ] [ ] [ ]

[ ] ( )[ ]nkhkx

knhkxny

k

k

minusminus=

minus=

sum

suminfin

minusinfin=

infin

minusinfin=


0 12

13

-2

0-1-2

13

-2

0 1 2

23

1

0

3

10-1

13

-21

11

210

13

-2

1

2

321

13

-2 -1

3

432

13

-2 -2

4

0

3

1

11

2

13

-1

4

-2

Sum up

[ ]kh

[ ]kx

[ ]kh minus

[ ] 0 =nny

[ ]1+minuskh

[ ]2+minuskh

[ ]3+minuskh

[ ]4+minuskh

[ ] 1 =nny

[ ] 2 =nny

[ ] 3 =nny

[ ] 4 =nny

[ ]ny

Reflect [ ] [ ] [ ][ ] [ ]knhkx

nhnxny

kminus=

lowast=

suminfin

minusinfin=

Convolution




[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=

[ ]nh 1 [ ]nh 2 [ ]nh 2 [ ]nh 1

[ ]nh 2

[ ]nh 1

[ ] [ ]nhnh 21 +



[ ] [ ] [ ][ ] [ ] [ ]nhnhnx

nhnhnx

12

21

=

[ ] [ ] [ ]( )[ ] [ ] [ ] [ ]nhnxnhnx

nhnhnx

21

21

+=+

Commutation

Distribution




[ ] [ ] [ ]

[ ] [ ]

[ ] [ ] ( )

[ ] [ ]nxnh

knmhmnx

knhkx

nhnxny

m

k

mlet

=

minus=sum minus=

minussum=

=

infin

minusinfin=

infin

minusinfin=




[ ] [ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] ( )0000

00011

0

101

cosBut

coscos

suppose

nnωnnxnny

nnnxnnxny

nnynynnxnx

minusminus=minus

minus==

minus=rArrminus=

ωω





[ ] infinlesuminfin

minusinfin=kkh

[ ][ ] nBny

nBnx

y

x

forallinfinltle

forallinfinltle






[ ] [ ] [ ]sum sum= =

minus+minus=K

k

M

mkkk mnxknyny

1000 βα

z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα






[ ]nh( )ωjeH

π2( )ωjeH

ω( ) ( ) ( )

( ) ( )ωω

ωωω

jeHjj

ji

jr

j

eeH

ejHeHeHang=

+=

magnitudephase

( )ωjeH






[ ] infinltsuminfin

minusinfin=nnh

[ ]( )

( )

[ ]mnnmnm

mnmn

emnj

de

mnj

mnj

minus=⎩⎨⎧

ne=

=

minusminus

=

minus= minus

minus

minusminus

int

δ

πππ

ωπ

ππ

ω

ππ

ω

0 1

sin)(2

121

)(

)(

( ) [ ]

[ ] ( ) [ ]


deemhdeeHnh

enheH

m

mnj

m

nj

m

mjnjj

n

njj

=minus==

==

=

sumintsum

int sumint

sum

infin

minusinfin=minus

minusinfin

minusinfin=

minus

infin

minusinfin=

minusminus

infin

minusinfin=

minus

δωπ

ωπ

ωπ

ππ

ω

ππ

ωωππ

ωω

ωω

21

21

21


)(

( ) [ ]

[ ] ( )int

sum

minus

infin

minusinfin=

minus

=

=

ππ

ωω

ωω

ωπ

deeHnh

enheH

njj

n

njj

2

1

absolutely summable

DTFT

Inverse DTFT



( ) [ ]

[ ] [ ] [ ]

( ) [ ] [ ]

[ ] [ ]

( ) ( )

[ ] ( ) ( )ωω

ωω

ωω

ωω

ωω

jj

jj

nj

nk

kj

nj

n k

j

k

n

njj

eHeXnhnx

eHeX

enhekx

eknhkxeY

knhkxnhnxny

enheH

hArrlowastthere4

=

⎟⎠

⎞⎜⎝

⎛=

minus=

minus=lowast=

=

minusinfin

minusinfin=

infin

minusinfin=

minus

minusinfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus

sumsum

sum sum

sum

sum

][

][][

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )ωωω

ωωω

ωωω

jjj

jjj

jjj

eHeXeY

eHeXeY

eHeXeY

ang+ang=angrArr

=rArr

=

knnknnknn






[ ] ( )intsum minus

infin

minusinfin== π

πω ω

πdeXnx j

n

22 21

power spectrum

[ ]nx[ ]

( ) ][][][][

][][

nxnxlnxlx

mxnmxnR

l

mxx


+=

lowastinfin

minusinfin=

infin

minusinfin=

sum

sum

( ) ( ) ( ) ( ) 2 ωωωω XXXS xx ==

hArr


π

π

ωπ

π

ω ωωπ

ωωπ

deXdeSnR njnjxxxx

2

21

21


minusinfin=minus

infin

minusinfin=

===

=

mmxx dXmxmxmxR

nπ

πωω

π22

210

0Set


)(

lnnlmnml



conjugatecomplex





[ ] ( )[ ] [ ] [ ] ( )

[ ] ( ) [ ]

[ ] ( )( )

( ) [ ]( ) ( )( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( )( )[ ] ( ) ( ) ( )( )ωω

ωωωω

φωω

ωφω

ωφω

ωφω

φω

φω

φω

φω

φωφω

φω

ω

ω

jj

jjjj

eHjnjj

eHjjnj

jnj

kj

k

nj

knj

k

nj

nj

eHneHAny

eHneHjAeHneHA

eeHA

eeHAe

eHAe

ekhAe

Aekh

nhAeny

Aenxnxnx

njAnx

j

j

ang++=rArr

ang+++ang++=

=

=

=

sum=

sum=

=

=+=

+=

ang++

ang+

+

minusinfin

minusinfin=

+

+minusinfin

minusinfin=

+

+

cos

sincos

sin

0

10

1

[ ]ny [ ]ny1

( )( ) attenuate 1

amplify 1

lt

gt

ω

ω

j

j

eH

eH





Z-Transform





( ) [ ]suminfin

minusinfin=

minus=n

nznhzH

[ ] ( )zHnh

[ ] ( )ωjeHnh

[ ]nh

ωjrez =

( ) ( ) ωω

jezj zHeH

==

)1( == zez jω

complex plane

unit circle

Im

Re

ω

ωjez =


Z-Transform (cont)


stability



[ ] infinltsuminfin

minusinfin=

minus

n

nznh

complex plan

R1

R2

Re

Im

absolutely summable

1=z


Z-Transform (cont)





[ ] 0for 0 lt= nnh

[ ] 0for 0 gt= nnh


Left-sided sequence

[ ] infinltsuminfin

minusinfin=nnh

[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


Z-Transform (cont)


[ ] [ ] [ ]⎩⎨⎧

ltge

==0for 00for 1

ere wh 1 1 nn

nunuanh n

( ) ( ) 11

1 11

minus

infin

minusinfin=

minusminusinfin


azazzazH

n

nn

n

n

If 11 ltminusaz

azROC gtthere4 is 1

Re

Im

a[ ] 1 if exists

ofansformFourier tr

1 ltanhtimes


az =

the unit cycle

0


Z-Transform (cont)



( ) [ ]

( ) 11

1

10

1

1

12

11

11111

1

minusminus

minus

minus

infin

=

minusinfin

=

minus

minusminus

minusinfin=

minusinfin

minusinfin=

minus=

minusminus

minus=minus

minus=minus=minus=


sumsum

sumsum

azzaza

zazaza

zaznuazH

n

n

n

nn

n

n

nn

n

n If 11 ltminus za

azROC ltthere4 is 2

Re

Im

a[ ] [ ]

minusinfinrarr

lt

nnhnh

a



22times

the unit cycle

0


Z-Transform (cont)


[ ] [ ] [ ]121


⎠⎞

⎜⎝⎛minus⎟

⎠⎞


nn

[ ]

[ ]21

211

1 121

31

311

1 31

1

1

ltminus


⎜⎝⎛minus

gt+


⎜⎝⎛ minus

minus

minus

zz

nu

zz

nu

zn

zn

Re

Im

times

the unit cycle

times21

31



3

3

ROCnh

31 and


( )⎟⎠⎞


⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ minus

=minus

++

=minusminus

21

31

1212

211

1

311

1 11

3

zz

zz

zzzH

0


Z-Transform (cont)



=others 0

10 3 4

Nnanh

n

( ) ( ) ( )azaz

zazazazzazH

NN

N

NN

n

nN

n

nn

minusminus

=minus

minus=== minusminus

minusminus

=

minusminus

=

minus sumsum 11

11

0

11

04

11

1



Im

times

the unit cycle

31

minus


4π

( ) 11 2

minus== N kaez Nkj

k

π

If N=8

0 N-1

Re


Z-Transform (cont)






[ ]nh [ ] 1 0 nnnh le=

01 gen

0rz gtz

infin=z

[ ]nh [ ] 2 0 nnnh ge=

02 ltn 0=z[ ]nh


01 geninfin=zthere4








[ ]nh

[ ] [ ] [ ] [ ]

[ ]

( ) njj

kjnj

knj

eeH

ekhe

ekhnhnxny

ωω

ωω

ω

=

=

=lowast=

minusinfin

infin=

minusinfin

infin=

sum

sum

-k

)(

-k( )



[ ] njenx ω=[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


( )jωeH

( )jωeH

scalar


[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjωnjjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

ang++=

+=

+=

+=

+minusangminus+ang

+minus+

minusminusminus

φω

φωφω

φωφω

ωφωφ




[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0

ωφωφ

φω

minusminus+=

+=

[ ]nh

( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

minus

minus

+=rArr

minus=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω

jωjω

eeHeH

eHeHangminus

minus

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos

sincos

jje

jejyxz

jejyxz

j

j

j

minus=minus+minus=

minus=rArrminus=

+=rArr+=

minus






[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2



Sinusoid Signals




ωφ

A

Tf ππω 22 ==

[ ] ⎟⎠⎞

⎜⎝⎛ minus=

2cos πωnAnx

samples 25=T


lele ff


[ ] [ ]

( )2122

2)(cos

==rArr=rArr

⎟⎠⎞

⎜⎝⎛ minus+=+=

kN

kkN

NnANnxnx

πωπω

πω






[ ]nx[ ] [ ]nxNnx =+


Nπω 2

=

[ ] ( )4cos1 nnx π=[ ] ( )83cos2 nnx π=

[ ] ( )nnx cos3 =



( )

8


44cos)(

4cos

4cos

4cos

1

11

11

1

=there4

sdotrArrsdot=rArr

⎟⎠⎞

⎜⎝⎛ +=⎟

⎠⎞

⎜⎝⎛ +=⎟

⎠⎞

⎜⎝⎛=

=

N

kNkkN

NnNnn

nnx

ππ

πππππ

[ ] ( )

( )

( )

16


1628

3

83

83cos

83cos

83cos

83cos

2

222

22

2

=there4

=rArrsdot=sdotrArr

⎟⎠⎞


⎠⎞

⎜⎝⎛ +=⎟

⎠⎞

⎜⎝⎛ sdot=

=

N

kNkNkN

NnNnn

nnx

ππ

πππππ

[ ] ( )( ) ( )( ) ( )


2 cos1cos1cos

cos

3

3

3

33

3

NkN

kNNnNnn

nnx

there4


=

Q

π




( )φφφ sincos

jAAez

jyxzj +==rArr

+=


Re

Im

φφ

sin cos

AyAx

==

22 yx + 22 yx

x

+22 yx

y

+

Imaginary part

real part




[ ] ( )( )

φω

φω

φω

jnj

nj

eAeAe

nAnx

Re Re

cos

=

=

+=+

real part






( ) ( )

( )

( )φω

φω

φφω

φωφω

+

++

=

=

+=

+

nj

jnj

jjnj

njnj

Ae

Aee

eAeAe

eAeA10

10

10

10






1100

1100coscossinsintan

φφφφ

φAAAA

++

=

( ) ( )( )

( )10102

120

1010102

120

2

21100

21100

2

cos2

sinsincoscos2

sinsincoscos

φφ

φφφφ

φφφφ

minus++=

+++=

+++=

AAAA

AAAAA

AAAAA






[ ]nx

[ ] [ ] [ ]knkxnxk

minussum=infin

minusinfin=δ


[ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ]31211121221

33221101122

3

2



infin

minusinfin=

nnnnnnnxnxnxnxnxnx

knkxknkxnxkk


δδ

321012 minusminus=n


Digital Systems


[ ] [ ] nxTny =

[ ]nx bullT [ ]ny


















DigitalSystem

[ ]nδ [ ]nh




[ ] [ ] [ ]knkxnxk

minus= suminfin

minusinfin=δ

[ ] [ ] [ ] [ ] [ ]

[ ] [ ][ ] [ ]nhnx

knhkx

knTkx

knkxTnxT

k

k

k

lowast=

minus=

minus=

minus=rArr

sum

sum

sum

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

δ

δ

scale


Time invariant

DigitalSystem

[ ]nδ [ ]nh

linear

Time-invariant

[ ] [ ][ ] [ ]knhkn

nhnT

T


⎯rarr⎯

δ

δ

Impulse response

convolution




bull Example

[ ]nδ 0 12

13

-2[ ]nh

LTI[ ]nx

0 1 2

23

1

0

3

0 12

39

-6

1

2

1 23

26

-4

LTI

2

1

2 34

1 3

-2

Length=M=3

Length=L=3

Length=L+M-1

[ ]nδsdot3

[ ]12 minussdot nδ

[ ]21 minussdot nδ

0

3

1

11

2

13

-1

4

-2

Sum up

[ ]ny

[ ]nhsdot3

[ ]12 minussdot nh

[ ]2minusnh






[ ]kh

[ ]kh minus

Reflect Multiply

slide

Sum up

n

[ ] [ ] [ ]

[ ] ( )[ ]nkhkx

knhkxny

k

k

minusminus=

minus=

sum

suminfin

minusinfin=

infin

minusinfin=


0 12

13

-2

0-1-2

13

-2

0 1 2

23

1

0

3

10-1

13

-21

11

210

13

-2

1

2

321

13

-2 -1

3

432

13

-2 -2

4

0

3

1

11

2

13

-1

4

-2

Sum up

[ ]kh

[ ]kx

[ ]kh minus

[ ] 0 =nny

[ ]1+minuskh

[ ]2+minuskh

[ ]3+minuskh

[ ]4+minuskh

[ ] 1 =nny

[ ] 2 =nny

[ ] 3 =nny

[ ] 4 =nny

[ ]ny

Reflect [ ] [ ] [ ][ ] [ ]knhkx

nhnxny

kminus=

lowast=

suminfin

minusinfin=

Convolution




[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=

[ ]nh 1 [ ]nh 2 [ ]nh 2 [ ]nh 1

[ ]nh 2

[ ]nh 1

[ ] [ ]nhnh 21 +



[ ] [ ] [ ][ ] [ ] [ ]nhnhnx

nhnhnx

12

21

=

[ ] [ ] [ ]( )[ ] [ ] [ ] [ ]nhnxnhnx

nhnhnx

21

21

+=+

Commutation

Distribution




[ ] [ ] [ ]

[ ] [ ]

[ ] [ ] ( )

[ ] [ ]nxnh

knmhmnx

knhkx

nhnxny

m

k

mlet

=

minus=sum minus=

minussum=

=

infin

minusinfin=

infin

minusinfin=




[ ] [ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] ( )0000

00011

0

101

cosBut

coscos

suppose

nnωnnxnny

nnnxnnxny

nnynynnxnx

minusminus=minus

minus==

minus=rArrminus=

ωω





[ ] infinlesuminfin

minusinfin=kkh

[ ][ ] nBny

nBnx

y

x

forallinfinltle

forallinfinltle






[ ] [ ] [ ]sum sum= =

minus+minus=K

k

M

mkkk mnxknyny

1000 βα

z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα






[ ]nh( )ωjeH

π2( )ωjeH

ω( ) ( ) ( )

( ) ( )ωω

ωωω

jeHjj

ji

jr

j

eeH

ejHeHeHang=

+=

magnitudephase

( )ωjeH






[ ] infinltsuminfin

minusinfin=nnh

[ ]( )

( )

[ ]mnnmnm

mnmn

emnj

de

mnj

mnj

minus=⎩⎨⎧

ne=

=

minusminus

=

minus= minus

minus

minusminus

int

δ

πππ

ωπ

ππ

ω

ππ

ω

0 1

sin)(2

121

)(

)(

( ) [ ]

[ ] ( ) [ ]


deemhdeeHnh

enheH

m

mnj

m

nj

m

mjnjj

n

njj

=minus==

==

=

sumintsum

int sumint

sum

infin

minusinfin=minus

minusinfin

minusinfin=

minus

infin

minusinfin=

minusminus

infin

minusinfin=

minus

δωπ

ωπ

ωπ

ππ

ω

ππ

ωωππ

ωω

ωω

21

21

21


)(

( ) [ ]

[ ] ( )int

sum

minus

infin

minusinfin=

minus

=

=

ππ

ωω

ωω

ωπ

deeHnh

enheH

njj

n

njj

2

1

absolutely summable

DTFT

Inverse DTFT



( ) [ ]

[ ] [ ] [ ]

( ) [ ] [ ]

[ ] [ ]

( ) ( )

[ ] ( ) ( )ωω

ωω

ωω

ωω

ωω

jj

jj

nj

nk

kj

nj

n k

j

k

n

njj

eHeXnhnx

eHeX

enhekx

eknhkxeY

knhkxnhnxny

enheH

hArrlowastthere4

=

⎟⎠

⎞⎜⎝

⎛=

minus=

minus=lowast=

=

minusinfin

minusinfin=

infin

minusinfin=

minus

minusinfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus

sumsum

sum sum

sum

sum

][

][][

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )ωωω

ωωω

ωωω

jjj

jjj

jjj

eHeXeY

eHeXeY

eHeXeY

ang+ang=angrArr

=rArr

=

knnknnknn






[ ] ( )intsum minus

infin

minusinfin== π

πω ω

πdeXnx j

n

22 21

power spectrum

[ ]nx[ ]

( ) ][][][][

][][

nxnxlnxlx

mxnmxnR

l

mxx


+=

lowastinfin

minusinfin=

infin

minusinfin=

sum

sum

( ) ( ) ( ) ( ) 2 ωωωω XXXS xx ==

hArr


π

π

ωπ

π

ω ωωπ

ωωπ

deXdeSnR njnjxxxx

2

21

21


minusinfin=minus

infin

minusinfin=

===

=

mmxx dXmxmxmxR

nπ

πωω

π22

210

0Set


)(

lnnlmnml



conjugatecomplex





[ ] ( )[ ] [ ] [ ] ( )

[ ] ( ) [ ]

[ ] ( )( )

( ) [ ]( ) ( )( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( )( )[ ] ( ) ( ) ( )( )ωω

ωωωω

φωω

ωφω

ωφω

ωφω

φω

φω

φω

φω

φωφω

φω

ω

ω

jj

jjjj

eHjnjj

eHjjnj

jnj

kj

k

nj

knj

k

nj

nj

eHneHAny

eHneHjAeHneHA

eeHA

eeHAe

eHAe

ekhAe

Aekh

nhAeny

Aenxnxnx

njAnx

j

j

ang++=rArr

ang+++ang++=

=

=

=

sum=

sum=

=

=+=

+=

ang++

ang+

+

minusinfin

minusinfin=

+

+minusinfin

minusinfin=

+

+

cos

sincos

sin

0

10

1

[ ]ny [ ]ny1

( )( ) attenuate 1

amplify 1

lt

gt

ω

ω

j

j

eH

eH





Z-Transform





( ) [ ]suminfin

minusinfin=

minus=n

nznhzH

[ ] ( )zHnh

[ ] ( )ωjeHnh

[ ]nh

ωjrez =

( ) ( ) ωω

jezj zHeH

==

)1( == zez jω

complex plane

unit circle

Im

Re

ω

ωjez =


Z-Transform (cont)


stability



[ ] infinltsuminfin

minusinfin=

minus

n

nznh

complex plan

R1

R2

Re

Im

absolutely summable

1=z


Z-Transform (cont)





[ ] 0for 0 lt= nnh

[ ] 0for 0 gt= nnh


Left-sided sequence

[ ] infinltsuminfin

minusinfin=nnh

[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


Z-Transform (cont)


[ ] [ ] [ ]⎩⎨⎧

ltge

==0for 00for 1

ere wh 1 1 nn

nunuanh n

( ) ( ) 11

1 11

minus

infin

minusinfin=

minusminusinfin


azazzazH

n

nn

n

n

If 11 ltminusaz

azROC gtthere4 is 1

Re

Im

a[ ] 1 if exists

ofansformFourier tr

1 ltanhtimes


az =

the unit cycle

0


Z-Transform (cont)



( ) [ ]

( ) 11

1

10

1

1

12

11

11111

1

minusminus

minus

minus

infin

=

minusinfin

=

minus

minusminus

minusinfin=

minusinfin

minusinfin=

minus=

minusminus

minus=minus

minus=minus=minus=


sumsum

sumsum

azzaza

zazaza

zaznuazH

n

n

n

nn

n

n

nn

n

n If 11 ltminus za

azROC ltthere4 is 2

Re

Im

a[ ] [ ]

minusinfinrarr

lt

nnhnh

a



22times

the unit cycle

0


Z-Transform (cont)


[ ] [ ] [ ]121


⎠⎞

⎜⎝⎛minus⎟

⎠⎞


nn

[ ]

[ ]21

211

1 121

31

311

1 31

1

1

ltminus


⎜⎝⎛minus

gt+


⎜⎝⎛ minus

minus

minus

zz

nu

zz

nu

zn

zn

Re

Im

times

the unit cycle

times21

31



3

3

ROCnh

31 and


( )⎟⎠⎞


⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ minus

=minus

++

=minusminus

21

31

1212

211

1

311

1 11

3

zz

zz

zzzH

0


Z-Transform (cont)



=others 0

10 3 4

Nnanh

n

( ) ( ) ( )azaz

zazazazzazH

NN

N

NN

n

nN

n

nn

minusminus

=minus

minus=== minusminus

minusminus

=

minusminus

=

minus sumsum 11

11

0

11

04

11

1



Im

times

the unit cycle

31

minus


4π

( ) 11 2

minus== N kaez Nkj

k

π

If N=8

0 N-1

Re


Z-Transform (cont)






[ ]nh [ ] 1 0 nnnh le=

01 gen

0rz gtz

infin=z

[ ]nh [ ] 2 0 nnnh ge=

02 ltn 0=z[ ]nh


01 geninfin=zthere4








[ ]nh

[ ] [ ] [ ] [ ]

[ ]

( ) njj

kjnj

knj

eeH

ekhe

ekhnhnxny

ωω

ωω

ω

=

=

=lowast=

minusinfin

infin=

minusinfin

infin=

sum

sum

-k

)(

-k( )



[ ] njenx ω=[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


( )jωeH

( )jωeH

scalar


[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjωnjjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

ang++=

+=

+=

+=

+minusangminus+ang

+minus+

minusminusminus

φω

φωφω

φωφω

ωφωφ




[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0

ωφωφ

φω

minusminus+=

+=

[ ]nh

( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

minus

minus

+=rArr

minus=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω

jωjω

eeHeH

eHeHangminus

minus

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos

sincos

jje

jejyxz

jejyxz

j

j

j

minus=minus+minus=

minus=rArrminus=

+=rArr+=

minus






[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2







[ ]nx[ ] [ ]nxNnx =+


Nπω 2

=

[ ] ( )4cos1 nnx π=[ ] ( )83cos2 nnx π=

[ ] ( )nnx cos3 =



( )

8


44cos)(

4cos

4cos

4cos

1

11

11

1

=there4

sdotrArrsdot=rArr

⎟⎠⎞

⎜⎝⎛ +=⎟

⎠⎞

⎜⎝⎛ +=⎟

⎠⎞

⎜⎝⎛=

=

N

kNkkN

NnNnn

nnx

ππ

πππππ

[ ] ( )

( )

( )

16


1628

3

83

83cos

83cos

83cos

83cos

2

222

22

2

=there4

=rArrsdot=sdotrArr

⎟⎠⎞


⎠⎞

⎜⎝⎛ +=⎟

⎠⎞

⎜⎝⎛ sdot=

=

N

kNkNkN

NnNnn

nnx

ππ

πππππ

[ ] ( )( ) ( )( ) ( )


2 cos1cos1cos

cos

3

3

3

33

3

NkN

kNNnNnn

nnx

there4


=

Q

π




( )φφφ sincos

jAAez

jyxzj +==rArr

+=


Re

Im

φφ

sin cos

AyAx

==

22 yx + 22 yx

x

+22 yx

y

+

Imaginary part

real part




[ ] ( )( )

φω

φω

φω

jnj

nj

eAeAe

nAnx

Re Re

cos

=

=

+=+

real part






( ) ( )

( )

( )φω

φω

φφω

φωφω

+

++

=

=

+=

+

nj

jnj

jjnj

njnj

Ae

Aee

eAeAe

eAeA10

10

10

10






1100

1100coscossinsintan

φφφφ

φAAAA

++

=

( ) ( )( )

( )10102

120

1010102

120

2

21100

21100

2

cos2

sinsincoscos2

sinsincoscos

φφ

φφφφ

φφφφ

minus++=

+++=

+++=

AAAA

AAAAA

AAAAA






[ ]nx

[ ] [ ] [ ]knkxnxk

minussum=infin

minusinfin=δ


[ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ]31211121221

33221101122

3

2



infin

minusinfin=

nnnnnnnxnxnxnxnxnx

knkxknkxnxkk


δδ

321012 minusminus=n


Digital Systems


[ ] [ ] nxTny =

[ ]nx bullT [ ]ny


















DigitalSystem

[ ]nδ [ ]nh




[ ] [ ] [ ]knkxnxk

minus= suminfin

minusinfin=δ

[ ] [ ] [ ] [ ] [ ]

[ ] [ ][ ] [ ]nhnx

knhkx

knTkx

knkxTnxT

k

k

k

lowast=

minus=

minus=

minus=rArr

sum

sum

sum

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

δ

δ

scale


Time invariant

DigitalSystem

[ ]nδ [ ]nh

linear

Time-invariant

[ ] [ ][ ] [ ]knhkn

nhnT

T


⎯rarr⎯

δ

δ

Impulse response

convolution




bull Example

[ ]nδ 0 12

13

-2[ ]nh

LTI[ ]nx

0 1 2

23

1

0

3

0 12

39

-6

1

2

1 23

26

-4

LTI

2

1

2 34

1 3

-2

Length=M=3

Length=L=3

Length=L+M-1

[ ]nδsdot3

[ ]12 minussdot nδ

[ ]21 minussdot nδ

0

3

1

11

2

13

-1

4

-2

Sum up

[ ]ny

[ ]nhsdot3

[ ]12 minussdot nh

[ ]2minusnh






[ ]kh

[ ]kh minus

Reflect Multiply

slide

Sum up

n

[ ] [ ] [ ]

[ ] ( )[ ]nkhkx

knhkxny

k

k

minusminus=

minus=

sum

suminfin

minusinfin=

infin

minusinfin=


0 12

13

-2

0-1-2

13

-2

0 1 2

23

1

0

3

10-1

13

-21

11

210

13

-2

1

2

321

13

-2 -1

3

432

13

-2 -2

4

0

3

1

11

2

13

-1

4

-2

Sum up

[ ]kh

[ ]kx

[ ]kh minus

[ ] 0 =nny

[ ]1+minuskh

[ ]2+minuskh

[ ]3+minuskh

[ ]4+minuskh

[ ] 1 =nny

[ ] 2 =nny

[ ] 3 =nny

[ ] 4 =nny

[ ]ny

Reflect [ ] [ ] [ ][ ] [ ]knhkx

nhnxny

kminus=

lowast=

suminfin

minusinfin=

Convolution




[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=

[ ]nh 1 [ ]nh 2 [ ]nh 2 [ ]nh 1

[ ]nh 2

[ ]nh 1

[ ] [ ]nhnh 21 +



[ ] [ ] [ ][ ] [ ] [ ]nhnhnx

nhnhnx

12

21

=

[ ] [ ] [ ]( )[ ] [ ] [ ] [ ]nhnxnhnx

nhnhnx

21

21

+=+

Commutation

Distribution




[ ] [ ] [ ]

[ ] [ ]

[ ] [ ] ( )

[ ] [ ]nxnh

knmhmnx

knhkx

nhnxny

m

k

mlet

=

minus=sum minus=

minussum=

=

infin

minusinfin=

infin

minusinfin=




[ ] [ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] ( )0000

00011

0

101

cosBut

coscos

suppose

nnωnnxnny

nnnxnnxny

nnynynnxnx

minusminus=minus

minus==

minus=rArrminus=

ωω





[ ] infinlesuminfin

minusinfin=kkh

[ ][ ] nBny

nBnx

y

x

forallinfinltle

forallinfinltle






[ ] [ ] [ ]sum sum= =

minus+minus=K

k

M

mkkk mnxknyny

1000 βα

z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα






[ ]nh( )ωjeH

π2( )ωjeH

ω( ) ( ) ( )

( ) ( )ωω

ωωω

jeHjj

ji

jr

j

eeH

ejHeHeHang=

+=

magnitudephase

( )ωjeH






[ ] infinltsuminfin

minusinfin=nnh

[ ]( )

( )

[ ]mnnmnm

mnmn

emnj

de

mnj

mnj

minus=⎩⎨⎧

ne=

=

minusminus

=

minus= minus

minus

minusminus

int

δ

πππ

ωπ

ππ

ω

ππ

ω

0 1

sin)(2

121

)(

)(

( ) [ ]

[ ] ( ) [ ]


deemhdeeHnh

enheH

m

mnj

m

nj

m

mjnjj

n

njj

=minus==

==

=

sumintsum

int sumint

sum

infin

minusinfin=minus

minusinfin

minusinfin=

minus

infin

minusinfin=

minusminus

infin

minusinfin=

minus

δωπ

ωπ

ωπ

ππ

ω

ππ

ωωππ

ωω

ωω

21

21

21


)(

( ) [ ]

[ ] ( )int

sum

minus

infin

minusinfin=

minus

=

=

ππ

ωω

ωω

ωπ

deeHnh

enheH

njj

n

njj

2

1

absolutely summable

DTFT

Inverse DTFT



( ) [ ]

[ ] [ ] [ ]

( ) [ ] [ ]

[ ] [ ]

( ) ( )

[ ] ( ) ( )ωω

ωω

ωω

ωω

ωω

jj

jj

nj

nk

kj

nj

n k

j

k

n

njj

eHeXnhnx

eHeX

enhekx

eknhkxeY

knhkxnhnxny

enheH

hArrlowastthere4

=

⎟⎠

⎞⎜⎝

⎛=

minus=

minus=lowast=

=

minusinfin

minusinfin=

infin

minusinfin=

minus

minusinfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus

sumsum

sum sum

sum

sum

][

][][

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )ωωω

ωωω

ωωω

jjj

jjj

jjj

eHeXeY

eHeXeY

eHeXeY

ang+ang=angrArr

=rArr

=

knnknnknn






[ ] ( )intsum minus

infin

minusinfin== π

πω ω

πdeXnx j

n

22 21

power spectrum

[ ]nx[ ]

( ) ][][][][

][][

nxnxlnxlx

mxnmxnR

l

mxx


+=

lowastinfin

minusinfin=

infin

minusinfin=

sum

sum

( ) ( ) ( ) ( ) 2 ωωωω XXXS xx ==

hArr


π

π

ωπ

π

ω ωωπ

ωωπ

deXdeSnR njnjxxxx

2

21

21


minusinfin=minus

infin

minusinfin=

===

=

mmxx dXmxmxmxR

nπ

πωω

π22

210

0Set


)(

lnnlmnml



conjugatecomplex





[ ] ( )[ ] [ ] [ ] ( )

[ ] ( ) [ ]

[ ] ( )( )

( ) [ ]( ) ( )( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( )( )[ ] ( ) ( ) ( )( )ωω

ωωωω

φωω

ωφω

ωφω

ωφω

φω

φω

φω

φω

φωφω

φω

ω

ω

jj

jjjj

eHjnjj

eHjjnj

jnj

kj

k

nj

knj

k

nj

nj

eHneHAny

eHneHjAeHneHA

eeHA

eeHAe

eHAe

ekhAe

Aekh

nhAeny

Aenxnxnx

njAnx

j

j

ang++=rArr

ang+++ang++=

=

=

=

sum=

sum=

=

=+=

+=

ang++

ang+

+

minusinfin

minusinfin=

+

+minusinfin

minusinfin=

+

+

cos

sincos

sin

0

10

1

[ ]ny [ ]ny1

( )( ) attenuate 1

amplify 1

lt

gt

ω

ω

j

j

eH

eH





Z-Transform





( ) [ ]suminfin

minusinfin=

minus=n

nznhzH

[ ] ( )zHnh

[ ] ( )ωjeHnh

[ ]nh

ωjrez =

( ) ( ) ωω

jezj zHeH

==

)1( == zez jω

complex plane

unit circle

Im

Re

ω

ωjez =


Z-Transform (cont)


stability



[ ] infinltsuminfin

minusinfin=

minus

n

nznh

complex plan

R1

R2

Re

Im

absolutely summable

1=z


Z-Transform (cont)





[ ] 0for 0 lt= nnh

[ ] 0for 0 gt= nnh


Left-sided sequence

[ ] infinltsuminfin

minusinfin=nnh

[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


Z-Transform (cont)


[ ] [ ] [ ]⎩⎨⎧

ltge

==0for 00for 1

ere wh 1 1 nn

nunuanh n

( ) ( ) 11

1 11

minus

infin

minusinfin=

minusminusinfin


azazzazH

n

nn

n

n

If 11 ltminusaz

azROC gtthere4 is 1

Re

Im

a[ ] 1 if exists

ofansformFourier tr

1 ltanhtimes


az =

the unit cycle

0


Z-Transform (cont)



( ) [ ]

( ) 11

1

10

1

1

12

11

11111

1

minusminus

minus

minus

infin

=

minusinfin

=

minus

minusminus

minusinfin=

minusinfin

minusinfin=

minus=

minusminus

minus=minus

minus=minus=minus=


sumsum

sumsum

azzaza

zazaza

zaznuazH

n

n

n

nn

n

n

nn

n

n If 11 ltminus za

azROC ltthere4 is 2

Re

Im

a[ ] [ ]

minusinfinrarr

lt

nnhnh

a



22times

the unit cycle

0


Z-Transform (cont)


[ ] [ ] [ ]121


⎠⎞

⎜⎝⎛minus⎟

⎠⎞


nn

[ ]

[ ]21

211

1 121

31

311

1 31

1

1

ltminus


⎜⎝⎛minus

gt+


⎜⎝⎛ minus

minus

minus

zz

nu

zz

nu

zn

zn

Re

Im

times

the unit cycle

times21

31



3

3

ROCnh

31 and


( )⎟⎠⎞


⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ minus

=minus

++

=minusminus

21

31

1212

211

1

311

1 11

3

zz

zz

zzzH

0


Z-Transform (cont)



=others 0

10 3 4

Nnanh

n

( ) ( ) ( )azaz

zazazazzazH

NN

N

NN

n

nN

n

nn

minusminus

=minus

minus=== minusminus

minusminus

=

minusminus

=

minus sumsum 11

11

0

11

04

11

1



Im

times

the unit cycle

31

minus


4π

( ) 11 2

minus== N kaez Nkj

k

π

If N=8

0 N-1

Re


Z-Transform (cont)






[ ]nh [ ] 1 0 nnnh le=

01 gen

0rz gtz

infin=z

[ ]nh [ ] 2 0 nnnh ge=

02 ltn 0=z[ ]nh


01 geninfin=zthere4








[ ]nh

[ ] [ ] [ ] [ ]

[ ]

( ) njj

kjnj

knj

eeH

ekhe

ekhnhnxny

ωω

ωω

ω

=

=

=lowast=

minusinfin

infin=

minusinfin

infin=

sum

sum

-k

)(

-k( )



[ ] njenx ω=[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


( )jωeH

( )jωeH

scalar


[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjωnjjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

ang++=

+=

+=

+=

+minusangminus+ang

+minus+

minusminusminus

φω

φωφω

φωφω

ωφωφ




[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0

ωφωφ

φω

minusminus+=

+=

[ ]nh

( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

minus

minus

+=rArr

minus=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω

jωjω

eeHeH

eHeHangminus

minus

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos

sincos

jje

jejyxz

jejyxz

j

j

j

minus=minus+minus=

minus=rArrminus=

+=rArr+=

minus






[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2




( )

8


44cos)(

4cos

4cos

4cos

1

11

11

1

=there4

sdotrArrsdot=rArr

⎟⎠⎞

⎜⎝⎛ +=⎟

⎠⎞

⎜⎝⎛ +=⎟

⎠⎞

⎜⎝⎛=

=

N

kNkkN

NnNnn

nnx

ππ

πππππ

[ ] ( )

( )

( )

16


1628

3

83

83cos

83cos

83cos

83cos

2

222

22

2

=there4

=rArrsdot=sdotrArr

⎟⎠⎞


⎠⎞

⎜⎝⎛ +=⎟

⎠⎞

⎜⎝⎛ sdot=

=

N

kNkNkN

NnNnn

nnx

ππ

πππππ

[ ] ( )( ) ( )( ) ( )


2 cos1cos1cos

cos

3

3

3

33

3

NkN

kNNnNnn

nnx

there4


=

Q

π




( )φφφ sincos

jAAez

jyxzj +==rArr

+=


Re

Im

φφ

sin cos

AyAx

==

22 yx + 22 yx

x

+22 yx

y

+

Imaginary part

real part




[ ] ( )( )

φω

φω

φω

jnj

nj

eAeAe

nAnx

Re Re

cos

=

=

+=+

real part






( ) ( )

( )

( )φω

φω

φφω

φωφω

+

++

=

=

+=

+

nj

jnj

jjnj

njnj

Ae

Aee

eAeAe

eAeA10

10

10

10






1100

1100coscossinsintan

φφφφ

φAAAA

++

=

( ) ( )( )

( )10102

120

1010102

120

2

21100

21100

2

cos2

sinsincoscos2

sinsincoscos

φφ

φφφφ

φφφφ

minus++=

+++=

+++=

AAAA

AAAAA

AAAAA






[ ]nx

[ ] [ ] [ ]knkxnxk

minussum=infin

minusinfin=δ


[ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ]31211121221

33221101122

3

2



infin

minusinfin=

nnnnnnnxnxnxnxnxnx

knkxknkxnxkk


δδ

321012 minusminus=n


Digital Systems


[ ] [ ] nxTny =

[ ]nx bullT [ ]ny


















DigitalSystem

[ ]nδ [ ]nh




[ ] [ ] [ ]knkxnxk

minus= suminfin

minusinfin=δ

[ ] [ ] [ ] [ ] [ ]

[ ] [ ][ ] [ ]nhnx

knhkx

knTkx

knkxTnxT

k

k

k

lowast=

minus=

minus=

minus=rArr

sum

sum

sum

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

δ

δ

scale


Time invariant

DigitalSystem

[ ]nδ [ ]nh

linear

Time-invariant

[ ] [ ][ ] [ ]knhkn

nhnT

T


⎯rarr⎯

δ

δ

Impulse response

convolution




bull Example

[ ]nδ 0 12

13

-2[ ]nh

LTI[ ]nx

0 1 2

23

1

0

3

0 12

39

-6

1

2

1 23

26

-4

LTI

2

1

2 34

1 3

-2

Length=M=3

Length=L=3

Length=L+M-1

[ ]nδsdot3

[ ]12 minussdot nδ

[ ]21 minussdot nδ

0

3

1

11

2

13

-1

4

-2

Sum up

[ ]ny

[ ]nhsdot3

[ ]12 minussdot nh

[ ]2minusnh






[ ]kh

[ ]kh minus

Reflect Multiply

slide

Sum up

n

[ ] [ ] [ ]

[ ] ( )[ ]nkhkx

knhkxny

k

k

minusminus=

minus=

sum

suminfin

minusinfin=

infin

minusinfin=


0 12

13

-2

0-1-2

13

-2

0 1 2

23

1

0

3

10-1

13

-21

11

210

13

-2

1

2

321

13

-2 -1

3

432

13

-2 -2

4

0

3

1

11

2

13

-1

4

-2

Sum up

[ ]kh

[ ]kx

[ ]kh minus

[ ] 0 =nny

[ ]1+minuskh

[ ]2+minuskh

[ ]3+minuskh

[ ]4+minuskh

[ ] 1 =nny

[ ] 2 =nny

[ ] 3 =nny

[ ] 4 =nny

[ ]ny

Reflect [ ] [ ] [ ][ ] [ ]knhkx

nhnxny

kminus=

lowast=

suminfin

minusinfin=

Convolution




[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=

[ ]nh 1 [ ]nh 2 [ ]nh 2 [ ]nh 1

[ ]nh 2

[ ]nh 1

[ ] [ ]nhnh 21 +



[ ] [ ] [ ][ ] [ ] [ ]nhnhnx

nhnhnx

12

21

=

[ ] [ ] [ ]( )[ ] [ ] [ ] [ ]nhnxnhnx

nhnhnx

21

21

+=+

Commutation

Distribution




[ ] [ ] [ ]

[ ] [ ]

[ ] [ ] ( )

[ ] [ ]nxnh

knmhmnx

knhkx

nhnxny

m

k

mlet

=

minus=sum minus=

minussum=

=

infin

minusinfin=

infin

minusinfin=




[ ] [ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] ( )0000

00011

0

101

cosBut

coscos

suppose

nnωnnxnny

nnnxnnxny

nnynynnxnx

minusminus=minus

minus==

minus=rArrminus=

ωω





[ ] infinlesuminfin

minusinfin=kkh

[ ][ ] nBny

nBnx

y

x

forallinfinltle

forallinfinltle






[ ] [ ] [ ]sum sum= =

minus+minus=K

k

M

mkkk mnxknyny

1000 βα

z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα






[ ]nh( )ωjeH

π2( )ωjeH

ω( ) ( ) ( )

( ) ( )ωω

ωωω

jeHjj

ji

jr

j

eeH

ejHeHeHang=

+=

magnitudephase

( )ωjeH






[ ] infinltsuminfin

minusinfin=nnh

[ ]( )

( )

[ ]mnnmnm

mnmn

emnj

de

mnj

mnj

minus=⎩⎨⎧

ne=

=

minusminus

=

minus= minus

minus

minusminus

int

δ

πππ

ωπ

ππ

ω

ππ

ω

0 1

sin)(2

121

)(

)(

( ) [ ]

[ ] ( ) [ ]


deemhdeeHnh

enheH

m

mnj

m

nj

m

mjnjj

n

njj

=minus==

==

=

sumintsum

int sumint

sum

infin

minusinfin=minus

minusinfin

minusinfin=

minus

infin

minusinfin=

minusminus

infin

minusinfin=

minus

δωπ

ωπ

ωπ

ππ

ω

ππ

ωωππ

ωω

ωω

21

21

21


)(

( ) [ ]

[ ] ( )int

sum

minus

infin

minusinfin=

minus

=

=

ππ

ωω

ωω

ωπ

deeHnh

enheH

njj

n

njj

2

1

absolutely summable

DTFT

Inverse DTFT



( ) [ ]

[ ] [ ] [ ]

( ) [ ] [ ]

[ ] [ ]

( ) ( )

[ ] ( ) ( )ωω

ωω

ωω

ωω

ωω

jj

jj

nj

nk

kj

nj

n k

j

k

n

njj

eHeXnhnx

eHeX

enhekx

eknhkxeY

knhkxnhnxny

enheH

hArrlowastthere4

=

⎟⎠

⎞⎜⎝

⎛=

minus=

minus=lowast=

=

minusinfin

minusinfin=

infin

minusinfin=

minus

minusinfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus

sumsum

sum sum

sum

sum

][

][][

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )ωωω

ωωω

ωωω

jjj

jjj

jjj

eHeXeY

eHeXeY

eHeXeY

ang+ang=angrArr

=rArr

=

knnknnknn






[ ] ( )intsum minus

infin

minusinfin== π

πω ω

πdeXnx j

n

22 21

power spectrum

[ ]nx[ ]

( ) ][][][][

][][

nxnxlnxlx

mxnmxnR

l

mxx


+=

lowastinfin

minusinfin=

infin

minusinfin=

sum

sum

( ) ( ) ( ) ( ) 2 ωωωω XXXS xx ==

hArr


π

π

ωπ

π

ω ωωπ

ωωπ

deXdeSnR njnjxxxx

2

21

21


minusinfin=minus

infin

minusinfin=

===

=

mmxx dXmxmxmxR

nπ

πωω

π22

210

0Set


)(

lnnlmnml



conjugatecomplex





[ ] ( )[ ] [ ] [ ] ( )

[ ] ( ) [ ]

[ ] ( )( )

( ) [ ]( ) ( )( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( )( )[ ] ( ) ( ) ( )( )ωω

ωωωω

φωω

ωφω

ωφω

ωφω

φω

φω

φω

φω

φωφω

φω

ω

ω

jj

jjjj

eHjnjj

eHjjnj

jnj

kj

k

nj

knj

k

nj

nj

eHneHAny

eHneHjAeHneHA

eeHA

eeHAe

eHAe

ekhAe

Aekh

nhAeny

Aenxnxnx

njAnx

j

j

ang++=rArr

ang+++ang++=

=

=

=

sum=

sum=

=

=+=

+=

ang++

ang+

+

minusinfin

minusinfin=

+

+minusinfin

minusinfin=

+

+

cos

sincos

sin

0

10

1

[ ]ny [ ]ny1

( )( ) attenuate 1

amplify 1

lt

gt

ω

ω

j

j

eH

eH





Z-Transform





( ) [ ]suminfin

minusinfin=

minus=n

nznhzH

[ ] ( )zHnh

[ ] ( )ωjeHnh

[ ]nh

ωjrez =

( ) ( ) ωω

jezj zHeH

==

)1( == zez jω

complex plane

unit circle

Im

Re

ω

ωjez =


Z-Transform (cont)


stability



[ ] infinltsuminfin

minusinfin=

minus

n

nznh

complex plan

R1

R2

Re

Im

absolutely summable

1=z


Z-Transform (cont)





[ ] 0for 0 lt= nnh

[ ] 0for 0 gt= nnh


Left-sided sequence

[ ] infinltsuminfin

minusinfin=nnh

[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


Z-Transform (cont)


[ ] [ ] [ ]⎩⎨⎧

ltge

==0for 00for 1

ere wh 1 1 nn

nunuanh n

( ) ( ) 11

1 11

minus

infin

minusinfin=

minusminusinfin


azazzazH

n

nn

n

n

If 11 ltminusaz

azROC gtthere4 is 1

Re

Im

a[ ] 1 if exists

ofansformFourier tr

1 ltanhtimes


az =

the unit cycle

0


Z-Transform (cont)



( ) [ ]

( ) 11

1

10

1

1

12

11

11111

1

minusminus

minus

minus

infin

=

minusinfin

=

minus

minusminus

minusinfin=

minusinfin

minusinfin=

minus=

minusminus

minus=minus

minus=minus=minus=


sumsum

sumsum

azzaza

zazaza

zaznuazH

n

n

n

nn

n

n

nn

n

n If 11 ltminus za

azROC ltthere4 is 2

Re

Im

a[ ] [ ]

minusinfinrarr

lt

nnhnh

a



22times

the unit cycle

0


Z-Transform (cont)


[ ] [ ] [ ]121


⎠⎞

⎜⎝⎛minus⎟

⎠⎞


nn

[ ]

[ ]21

211

1 121

31

311

1 31

1

1

ltminus


⎜⎝⎛minus

gt+


⎜⎝⎛ minus

minus

minus

zz

nu

zz

nu

zn

zn

Re

Im

times

the unit cycle

times21

31



3

3

ROCnh

31 and


( )⎟⎠⎞


⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ minus

=minus

++

=minusminus

21

31

1212

211

1

311

1 11

3

zz

zz

zzzH

0


Z-Transform (cont)



=others 0

10 3 4

Nnanh

n

( ) ( ) ( )azaz

zazazazzazH

NN

N

NN

n

nN

n

nn

minusminus

=minus

minus=== minusminus

minusminus

=

minusminus

=

minus sumsum 11

11

0

11

04

11

1



Im

times

the unit cycle

31

minus


4π

( ) 11 2

minus== N kaez Nkj

k

π

If N=8

0 N-1

Re


Z-Transform (cont)






[ ]nh [ ] 1 0 nnnh le=

01 gen

0rz gtz

infin=z

[ ]nh [ ] 2 0 nnnh ge=

02 ltn 0=z[ ]nh


01 geninfin=zthere4








[ ]nh

[ ] [ ] [ ] [ ]

[ ]

( ) njj

kjnj

knj

eeH

ekhe

ekhnhnxny

ωω

ωω

ω

=

=

=lowast=

minusinfin

infin=

minusinfin

infin=

sum

sum

-k

)(

-k( )



[ ] njenx ω=[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


( )jωeH

( )jωeH

scalar


[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjωnjjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

ang++=

+=

+=

+=

+minusangminus+ang

+minus+

minusminusminus

φω

φωφω

φωφω

ωφωφ




[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0

ωφωφ

φω

minusminus+=

+=

[ ]nh

( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

minus

minus

+=rArr

minus=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω

jωjω

eeHeH

eHeHangminus

minus

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos

sincos

jje

jejyxz

jejyxz

j

j

j

minus=minus+minus=

minus=rArrminus=

+=rArr+=

minus






[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2





( )φφφ sincos

jAAez

jyxzj +==rArr

+=


Re

Im

φφ

sin cos

AyAx

==

22 yx + 22 yx

x

+22 yx

y

+

Imaginary part

real part




[ ] ( )( )

φω

φω

φω

jnj

nj

eAeAe

nAnx

Re Re

cos

=

=

+=+

real part






( ) ( )

( )

( )φω

φω

φφω

φωφω

+

++

=

=

+=

+

nj

jnj

jjnj

njnj

Ae

Aee

eAeAe

eAeA10

10

10

10






1100

1100coscossinsintan

φφφφ

φAAAA

++

=

( ) ( )( )

( )10102

120

1010102

120

2

21100

21100

2

cos2

sinsincoscos2

sinsincoscos

φφ

φφφφ

φφφφ

minus++=

+++=

+++=

AAAA

AAAAA

AAAAA






[ ]nx

[ ] [ ] [ ]knkxnxk

minussum=infin

minusinfin=δ


[ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ]31211121221

33221101122

3

2



infin

minusinfin=

nnnnnnnxnxnxnxnxnx

knkxknkxnxkk


δδ

321012 minusminus=n


Digital Systems


[ ] [ ] nxTny =

[ ]nx bullT [ ]ny


















DigitalSystem

[ ]nδ [ ]nh




[ ] [ ] [ ]knkxnxk

minus= suminfin

minusinfin=δ

[ ] [ ] [ ] [ ] [ ]

[ ] [ ][ ] [ ]nhnx

knhkx

knTkx

knkxTnxT

k

k

k

lowast=

minus=

minus=

minus=rArr

sum

sum

sum

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

δ

δ

scale


Time invariant

DigitalSystem

[ ]nδ [ ]nh

linear

Time-invariant

[ ] [ ][ ] [ ]knhkn

nhnT

T


⎯rarr⎯

δ

δ

Impulse response

convolution




bull Example

[ ]nδ 0 12

13

-2[ ]nh

LTI[ ]nx

0 1 2

23

1

0

3

0 12

39

-6

1

2

1 23

26

-4

LTI

2

1

2 34

1 3

-2

Length=M=3

Length=L=3

Length=L+M-1

[ ]nδsdot3

[ ]12 minussdot nδ

[ ]21 minussdot nδ

0

3

1

11

2

13

-1

4

-2

Sum up

[ ]ny

[ ]nhsdot3

[ ]12 minussdot nh

[ ]2minusnh






[ ]kh

[ ]kh minus

Reflect Multiply

slide

Sum up

n

[ ] [ ] [ ]

[ ] ( )[ ]nkhkx

knhkxny

k

k

minusminus=

minus=

sum

suminfin

minusinfin=

infin

minusinfin=


0 12

13

-2

0-1-2

13

-2

0 1 2

23

1

0

3

10-1

13

-21

11

210

13

-2

1

2

321

13

-2 -1

3

432

13

-2 -2

4

0

3

1

11

2

13

-1

4

-2

Sum up

[ ]kh

[ ]kx

[ ]kh minus

[ ] 0 =nny

[ ]1+minuskh

[ ]2+minuskh

[ ]3+minuskh

[ ]4+minuskh

[ ] 1 =nny

[ ] 2 =nny

[ ] 3 =nny

[ ] 4 =nny

[ ]ny

Reflect [ ] [ ] [ ][ ] [ ]knhkx

nhnxny

kminus=

lowast=

suminfin

minusinfin=

Convolution




[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=

[ ]nh 1 [ ]nh 2 [ ]nh 2 [ ]nh 1

[ ]nh 2

[ ]nh 1

[ ] [ ]nhnh 21 +



[ ] [ ] [ ][ ] [ ] [ ]nhnhnx

nhnhnx

12

21

=

[ ] [ ] [ ]( )[ ] [ ] [ ] [ ]nhnxnhnx

nhnhnx

21

21

+=+

Commutation

Distribution




[ ] [ ] [ ]

[ ] [ ]

[ ] [ ] ( )

[ ] [ ]nxnh

knmhmnx

knhkx

nhnxny

m

k

mlet

=

minus=sum minus=

minussum=

=

infin

minusinfin=

infin

minusinfin=




[ ] [ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] ( )0000

00011

0

101

cosBut

coscos

suppose

nnωnnxnny

nnnxnnxny

nnynynnxnx

minusminus=minus

minus==

minus=rArrminus=

ωω





[ ] infinlesuminfin

minusinfin=kkh

[ ][ ] nBny

nBnx

y

x

forallinfinltle

forallinfinltle






[ ] [ ] [ ]sum sum= =

minus+minus=K

k

M

mkkk mnxknyny

1000 βα

z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα






[ ]nh( )ωjeH

π2( )ωjeH

ω( ) ( ) ( )

( ) ( )ωω

ωωω

jeHjj

ji

jr

j

eeH

ejHeHeHang=

+=

magnitudephase

( )ωjeH






[ ] infinltsuminfin

minusinfin=nnh

[ ]( )

( )

[ ]mnnmnm

mnmn

emnj

de

mnj

mnj

minus=⎩⎨⎧

ne=

=

minusminus

=

minus= minus

minus

minusminus

int

δ

πππ

ωπ

ππ

ω

ππ

ω

0 1

sin)(2

121

)(

)(

( ) [ ]

[ ] ( ) [ ]


deemhdeeHnh

enheH

m

mnj

m

nj

m

mjnjj

n

njj

=minus==

==

=

sumintsum

int sumint

sum

infin

minusinfin=minus

minusinfin

minusinfin=

minus

infin

minusinfin=

minusminus

infin

minusinfin=

minus

δωπ

ωπ

ωπ

ππ

ω

ππ

ωωππ

ωω

ωω

21

21

21


)(

( ) [ ]

[ ] ( )int

sum

minus

infin

minusinfin=

minus

=

=

ππ

ωω

ωω

ωπ

deeHnh

enheH

njj

n

njj

2

1

absolutely summable

DTFT

Inverse DTFT



( ) [ ]

[ ] [ ] [ ]

( ) [ ] [ ]

[ ] [ ]

( ) ( )

[ ] ( ) ( )ωω

ωω

ωω

ωω

ωω

jj

jj

nj

nk

kj

nj

n k

j

k

n

njj

eHeXnhnx

eHeX

enhekx

eknhkxeY

knhkxnhnxny

enheH

hArrlowastthere4

=

⎟⎠

⎞⎜⎝

⎛=

minus=

minus=lowast=

=

minusinfin

minusinfin=

infin

minusinfin=

minus

minusinfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus

sumsum

sum sum

sum

sum

][

][][

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )ωωω

ωωω

ωωω

jjj

jjj

jjj

eHeXeY

eHeXeY

eHeXeY

ang+ang=angrArr

=rArr

=

knnknnknn






[ ] ( )intsum minus

infin

minusinfin== π

πω ω

πdeXnx j

n

22 21

power spectrum

[ ]nx[ ]

( ) ][][][][

][][

nxnxlnxlx

mxnmxnR

l

mxx


+=

lowastinfin

minusinfin=

infin

minusinfin=

sum

sum

( ) ( ) ( ) ( ) 2 ωωωω XXXS xx ==

hArr


π

π

ωπ

π

ω ωωπ

ωωπ

deXdeSnR njnjxxxx

2

21

21


minusinfin=minus

infin

minusinfin=

===

=

mmxx dXmxmxmxR

nπ

πωω

π22

210

0Set


)(

lnnlmnml



conjugatecomplex





[ ] ( )[ ] [ ] [ ] ( )

[ ] ( ) [ ]

[ ] ( )( )

( ) [ ]( ) ( )( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( )( )[ ] ( ) ( ) ( )( )ωω

ωωωω

φωω

ωφω

ωφω

ωφω

φω

φω

φω

φω

φωφω

φω

ω

ω

jj

jjjj

eHjnjj

eHjjnj

jnj

kj

k

nj

knj

k

nj

nj

eHneHAny

eHneHjAeHneHA

eeHA

eeHAe

eHAe

ekhAe

Aekh

nhAeny

Aenxnxnx

njAnx

j

j

ang++=rArr

ang+++ang++=

=

=

=

sum=

sum=

=

=+=

+=

ang++

ang+

+

minusinfin

minusinfin=

+

+minusinfin

minusinfin=

+

+

cos

sincos

sin

0

10

1

[ ]ny [ ]ny1

( )( ) attenuate 1

amplify 1

lt

gt

ω

ω

j

j

eH

eH





Z-Transform





( ) [ ]suminfin

minusinfin=

minus=n

nznhzH

[ ] ( )zHnh

[ ] ( )ωjeHnh

[ ]nh

ωjrez =

( ) ( ) ωω

jezj zHeH

==

)1( == zez jω

complex plane

unit circle

Im

Re

ω

ωjez =


Z-Transform (cont)


stability



[ ] infinltsuminfin

minusinfin=

minus

n

nznh

complex plan

R1

R2

Re

Im

absolutely summable

1=z


Z-Transform (cont)





[ ] 0for 0 lt= nnh

[ ] 0for 0 gt= nnh


Left-sided sequence

[ ] infinltsuminfin

minusinfin=nnh

[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


Z-Transform (cont)


[ ] [ ] [ ]⎩⎨⎧

ltge

==0for 00for 1

ere wh 1 1 nn

nunuanh n

( ) ( ) 11

1 11

minus

infin

minusinfin=

minusminusinfin


azazzazH

n

nn

n

n

If 11 ltminusaz

azROC gtthere4 is 1

Re

Im

a[ ] 1 if exists

ofansformFourier tr

1 ltanhtimes


az =

the unit cycle

0


Z-Transform (cont)



( ) [ ]

( ) 11

1

10

1

1

12

11

11111

1

minusminus

minus

minus

infin

=

minusinfin

=

minus

minusminus

minusinfin=

minusinfin

minusinfin=

minus=

minusminus

minus=minus

minus=minus=minus=


sumsum

sumsum

azzaza

zazaza

zaznuazH

n

n

n

nn

n

n

nn

n

n If 11 ltminus za

azROC ltthere4 is 2

Re

Im

a[ ] [ ]

minusinfinrarr

lt

nnhnh

a



22times

the unit cycle

0


Z-Transform (cont)


[ ] [ ] [ ]121


⎠⎞

⎜⎝⎛minus⎟

⎠⎞


nn

[ ]

[ ]21

211

1 121

31

311

1 31

1

1

ltminus


⎜⎝⎛minus

gt+


⎜⎝⎛ minus

minus

minus

zz

nu

zz

nu

zn

zn

Re

Im

times

the unit cycle

times21

31



3

3

ROCnh

31 and


( )⎟⎠⎞


⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ minus

=minus

++

=minusminus

21

31

1212

211

1

311

1 11

3

zz

zz

zzzH

0


Z-Transform (cont)



=others 0

10 3 4

Nnanh

n

( ) ( ) ( )azaz

zazazazzazH

NN

N

NN

n

nN

n

nn

minusminus

=minus

minus=== minusminus

minusminus

=

minusminus

=

minus sumsum 11

11

0

11

04

11

1



Im

times

the unit cycle

31

minus


4π

( ) 11 2

minus== N kaez Nkj

k

π

If N=8

0 N-1

Re


Z-Transform (cont)






[ ]nh [ ] 1 0 nnnh le=

01 gen

0rz gtz

infin=z

[ ]nh [ ] 2 0 nnnh ge=

02 ltn 0=z[ ]nh


01 geninfin=zthere4








[ ]nh

[ ] [ ] [ ] [ ]

[ ]

( ) njj

kjnj

knj

eeH

ekhe

ekhnhnxny

ωω

ωω

ω

=

=

=lowast=

minusinfin

infin=

minusinfin

infin=

sum

sum

-k

)(

-k( )



[ ] njenx ω=[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


( )jωeH

( )jωeH

scalar


[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjωnjjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

ang++=

+=

+=

+=

+minusangminus+ang

+minus+

minusminusminus

φω

φωφω

φωφω

ωφωφ




[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0

ωφωφ

φω

minusminus+=

+=

[ ]nh

( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

minus

minus

+=rArr

minus=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω

jωjω

eeHeH

eHeHangminus

minus

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos

sincos

jje

jejyxz

jejyxz

j

j

j

minus=minus+minus=

minus=rArrminus=

+=rArr+=

minus






[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2





[ ] ( )( )

φω

φω

φω

jnj

nj

eAeAe

nAnx

Re Re

cos

=

=

+=+

real part






( ) ( )

( )

( )φω

φω

φφω

φωφω

+

++

=

=

+=

+

nj

jnj

jjnj

njnj

Ae

Aee

eAeAe

eAeA10

10

10

10






1100

1100coscossinsintan

φφφφ

φAAAA

++

=

( ) ( )( )

( )10102

120

1010102

120

2

21100

21100

2

cos2

sinsincoscos2

sinsincoscos

φφ

φφφφ

φφφφ

minus++=

+++=

+++=

AAAA

AAAAA

AAAAA






[ ]nx

[ ] [ ] [ ]knkxnxk

minussum=infin

minusinfin=δ


[ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ]31211121221

33221101122

3

2



infin

minusinfin=

nnnnnnnxnxnxnxnxnx

knkxknkxnxkk


δδ

321012 minusminus=n


Digital Systems


[ ] [ ] nxTny =

[ ]nx bullT [ ]ny


















DigitalSystem

[ ]nδ [ ]nh




[ ] [ ] [ ]knkxnxk

minus= suminfin

minusinfin=δ

[ ] [ ] [ ] [ ] [ ]

[ ] [ ][ ] [ ]nhnx

knhkx

knTkx

knkxTnxT

k

k

k

lowast=

minus=

minus=

minus=rArr

sum

sum

sum

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

δ

δ

scale


Time invariant

DigitalSystem

[ ]nδ [ ]nh

linear

Time-invariant

[ ] [ ][ ] [ ]knhkn

nhnT

T


⎯rarr⎯

δ

δ

Impulse response

convolution




bull Example

[ ]nδ 0 12

13

-2[ ]nh

LTI[ ]nx

0 1 2

23

1

0

3

0 12

39

-6

1

2

1 23

26

-4

LTI

2

1

2 34

1 3

-2

Length=M=3

Length=L=3

Length=L+M-1

[ ]nδsdot3

[ ]12 minussdot nδ

[ ]21 minussdot nδ

0

3

1

11

2

13

-1

4

-2

Sum up

[ ]ny

[ ]nhsdot3

[ ]12 minussdot nh

[ ]2minusnh






[ ]kh

[ ]kh minus

Reflect Multiply

slide

Sum up

n

[ ] [ ] [ ]

[ ] ( )[ ]nkhkx

knhkxny

k

k

minusminus=

minus=

sum

suminfin

minusinfin=

infin

minusinfin=


0 12

13

-2

0-1-2

13

-2

0 1 2

23

1

0

3

10-1

13

-21

11

210

13

-2

1

2

321

13

-2 -1

3

432

13

-2 -2

4

0

3

1

11

2

13

-1

4

-2

Sum up

[ ]kh

[ ]kx

[ ]kh minus

[ ] 0 =nny

[ ]1+minuskh

[ ]2+minuskh

[ ]3+minuskh

[ ]4+minuskh

[ ] 1 =nny

[ ] 2 =nny

[ ] 3 =nny

[ ] 4 =nny

[ ]ny

Reflect [ ] [ ] [ ][ ] [ ]knhkx

nhnxny

kminus=

lowast=

suminfin

minusinfin=

Convolution




[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=

[ ]nh 1 [ ]nh 2 [ ]nh 2 [ ]nh 1

[ ]nh 2

[ ]nh 1

[ ] [ ]nhnh 21 +



[ ] [ ] [ ][ ] [ ] [ ]nhnhnx

nhnhnx

12

21

=

[ ] [ ] [ ]( )[ ] [ ] [ ] [ ]nhnxnhnx

nhnhnx

21

21

+=+

Commutation

Distribution




[ ] [ ] [ ]

[ ] [ ]

[ ] [ ] ( )

[ ] [ ]nxnh

knmhmnx

knhkx

nhnxny

m

k

mlet

=

minus=sum minus=

minussum=

=

infin

minusinfin=

infin

minusinfin=




[ ] [ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] ( )0000

00011

0

101

cosBut

coscos

suppose

nnωnnxnny

nnnxnnxny

nnynynnxnx

minusminus=minus

minus==

minus=rArrminus=

ωω





[ ] infinlesuminfin

minusinfin=kkh

[ ][ ] nBny

nBnx

y

x

forallinfinltle

forallinfinltle






[ ] [ ] [ ]sum sum= =

minus+minus=K

k

M

mkkk mnxknyny

1000 βα

z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα






[ ]nh( )ωjeH

π2( )ωjeH

ω( ) ( ) ( )

( ) ( )ωω

ωωω

jeHjj

ji

jr

j

eeH

ejHeHeHang=

+=

magnitudephase

( )ωjeH






[ ] infinltsuminfin

minusinfin=nnh

[ ]( )

( )

[ ]mnnmnm

mnmn

emnj

de

mnj

mnj

minus=⎩⎨⎧

ne=

=

minusminus

=

minus= minus

minus

minusminus

int

δ

πππ

ωπ

ππ

ω

ππ

ω

0 1

sin)(2

121

)(

)(

( ) [ ]

[ ] ( ) [ ]


deemhdeeHnh

enheH

m

mnj

m

nj

m

mjnjj

n

njj

=minus==

==

=

sumintsum

int sumint

sum

infin

minusinfin=minus

minusinfin

minusinfin=

minus

infin

minusinfin=

minusminus

infin

minusinfin=

minus

δωπ

ωπ

ωπ

ππ

ω

ππ

ωωππ

ωω

ωω

21

21

21


)(

( ) [ ]

[ ] ( )int

sum

minus

infin

minusinfin=

minus

=

=

ππ

ωω

ωω

ωπ

deeHnh

enheH

njj

n

njj

2

1

absolutely summable

DTFT

Inverse DTFT



( ) [ ]

[ ] [ ] [ ]

( ) [ ] [ ]

[ ] [ ]

( ) ( )

[ ] ( ) ( )ωω

ωω

ωω

ωω

ωω

jj

jj

nj

nk

kj

nj

n k

j

k

n

njj

eHeXnhnx

eHeX

enhekx

eknhkxeY

knhkxnhnxny

enheH

hArrlowastthere4

=

⎟⎠

⎞⎜⎝

⎛=

minus=

minus=lowast=

=

minusinfin

minusinfin=

infin

minusinfin=

minus

minusinfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus

sumsum

sum sum

sum

sum

][

][][

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )ωωω

ωωω

ωωω

jjj

jjj

jjj

eHeXeY

eHeXeY

eHeXeY

ang+ang=angrArr

=rArr

=

knnknnknn






[ ] ( )intsum minus

infin

minusinfin== π

πω ω

πdeXnx j

n

22 21

power spectrum

[ ]nx[ ]

( ) ][][][][

][][

nxnxlnxlx

mxnmxnR

l

mxx


+=

lowastinfin

minusinfin=

infin

minusinfin=

sum

sum

( ) ( ) ( ) ( ) 2 ωωωω XXXS xx ==

hArr


π

π

ωπ

π

ω ωωπ

ωωπ

deXdeSnR njnjxxxx

2

21

21


minusinfin=minus

infin

minusinfin=

===

=

mmxx dXmxmxmxR

nπ

πωω

π22

210

0Set


)(

lnnlmnml



conjugatecomplex





[ ] ( )[ ] [ ] [ ] ( )

[ ] ( ) [ ]

[ ] ( )( )

( ) [ ]( ) ( )( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( )( )[ ] ( ) ( ) ( )( )ωω

ωωωω

φωω

ωφω

ωφω

ωφω

φω

φω

φω

φω

φωφω

φω

ω

ω

jj

jjjj

eHjnjj

eHjjnj

jnj

kj

k

nj

knj

k

nj

nj

eHneHAny

eHneHjAeHneHA

eeHA

eeHAe

eHAe

ekhAe

Aekh

nhAeny

Aenxnxnx

njAnx

j

j

ang++=rArr

ang+++ang++=

=

=

=

sum=

sum=

=

=+=

+=

ang++

ang+

+

minusinfin

minusinfin=

+

+minusinfin

minusinfin=

+

+

cos

sincos

sin

0

10

1

[ ]ny [ ]ny1

( )( ) attenuate 1

amplify 1

lt

gt

ω

ω

j

j

eH

eH





Z-Transform





( ) [ ]suminfin

minusinfin=

minus=n

nznhzH

[ ] ( )zHnh

[ ] ( )ωjeHnh

[ ]nh

ωjrez =

( ) ( ) ωω

jezj zHeH

==

)1( == zez jω

complex plane

unit circle

Im

Re

ω

ωjez =


Z-Transform (cont)


stability



[ ] infinltsuminfin

minusinfin=

minus

n

nznh

complex plan

R1

R2

Re

Im

absolutely summable

1=z


Z-Transform (cont)





[ ] 0for 0 lt= nnh

[ ] 0for 0 gt= nnh


Left-sided sequence

[ ] infinltsuminfin

minusinfin=nnh

[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


Z-Transform (cont)


[ ] [ ] [ ]⎩⎨⎧

ltge

==0for 00for 1

ere wh 1 1 nn

nunuanh n

( ) ( ) 11

1 11

minus

infin

minusinfin=

minusminusinfin


azazzazH

n

nn

n

n

If 11 ltminusaz

azROC gtthere4 is 1

Re

Im

a[ ] 1 if exists

ofansformFourier tr

1 ltanhtimes


az =

the unit cycle

0


Z-Transform (cont)



( ) [ ]

( ) 11

1

10

1

1

12

11

11111

1

minusminus

minus

minus

infin

=

minusinfin

=

minus

minusminus

minusinfin=

minusinfin

minusinfin=

minus=

minusminus

minus=minus

minus=minus=minus=


sumsum

sumsum

azzaza

zazaza

zaznuazH

n

n

n

nn

n

n

nn

n

n If 11 ltminus za

azROC ltthere4 is 2

Re

Im

a[ ] [ ]

minusinfinrarr

lt

nnhnh

a



22times

the unit cycle

0


Z-Transform (cont)


[ ] [ ] [ ]121


⎠⎞

⎜⎝⎛minus⎟

⎠⎞


nn

[ ]

[ ]21

211

1 121

31

311

1 31

1

1

ltminus


⎜⎝⎛minus

gt+


⎜⎝⎛ minus

minus

minus

zz

nu

zz

nu

zn

zn

Re

Im

times

the unit cycle

times21

31



3

3

ROCnh

31 and


( )⎟⎠⎞


⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ minus

=minus

++

=minusminus

21

31

1212

211

1

311

1 11

3

zz

zz

zzzH

0


Z-Transform (cont)



=others 0

10 3 4

Nnanh

n

( ) ( ) ( )azaz

zazazazzazH

NN

N

NN

n

nN

n

nn

minusminus

=minus

minus=== minusminus

minusminus

=

minusminus

=

minus sumsum 11

11

0

11

04

11

1



Im

times

the unit cycle

31

minus


4π

( ) 11 2

minus== N kaez Nkj

k

π

If N=8

0 N-1

Re


Z-Transform (cont)






[ ]nh [ ] 1 0 nnnh le=

01 gen

0rz gtz

infin=z

[ ]nh [ ] 2 0 nnnh ge=

02 ltn 0=z[ ]nh


01 geninfin=zthere4








[ ]nh

[ ] [ ] [ ] [ ]

[ ]

( ) njj

kjnj

knj

eeH

ekhe

ekhnhnxny

ωω

ωω

ω

=

=

=lowast=

minusinfin

infin=

minusinfin

infin=

sum

sum

-k

)(

-k( )



[ ] njenx ω=[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


( )jωeH

( )jωeH

scalar


[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjωnjjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

ang++=

+=

+=

+=

+minusangminus+ang

+minus+

minusminusminus

φω

φωφω

φωφω

ωφωφ




[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0

ωφωφ

φω

minusminus+=

+=

[ ]nh

( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

minus

minus

+=rArr

minus=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω

jωjω

eeHeH

eHeHangminus

minus

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos

sincos

jje

jejyxz

jejyxz

j

j

j

minus=minus+minus=

minus=rArrminus=

+=rArr+=

minus






[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2







( ) ( )

( )

( )φω

φω

φφω

φωφω

+

++

=

=

+=

+

nj

jnj

jjnj

njnj

Ae

Aee

eAeAe

eAeA10

10

10

10






1100

1100coscossinsintan

φφφφ

φAAAA

++

=

( ) ( )( )

( )10102

120

1010102

120

2

21100

21100

2

cos2

sinsincoscos2

sinsincoscos

φφ

φφφφ

φφφφ

minus++=

+++=

+++=

AAAA

AAAAA

AAAAA






[ ]nx

[ ] [ ] [ ]knkxnxk

minussum=infin

minusinfin=δ


[ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ]31211121221

33221101122

3

2



infin

minusinfin=

nnnnnnnxnxnxnxnxnx

knkxknkxnxkk


δδ

321012 minusminus=n


Digital Systems


[ ] [ ] nxTny =

[ ]nx bullT [ ]ny


















DigitalSystem

[ ]nδ [ ]nh




[ ] [ ] [ ]knkxnxk

minus= suminfin

minusinfin=δ

[ ] [ ] [ ] [ ] [ ]

[ ] [ ][ ] [ ]nhnx

knhkx

knTkx

knkxTnxT

k

k

k

lowast=

minus=

minus=

minus=rArr

sum

sum

sum

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

δ

δ

scale


Time invariant

DigitalSystem

[ ]nδ [ ]nh

linear

Time-invariant

[ ] [ ][ ] [ ]knhkn

nhnT

T


⎯rarr⎯

δ

δ

Impulse response

convolution




bull Example

[ ]nδ 0 12

13

-2[ ]nh

LTI[ ]nx

0 1 2

23

1

0

3

0 12

39

-6

1

2

1 23

26

-4

LTI

2

1

2 34

1 3

-2

Length=M=3

Length=L=3

Length=L+M-1

[ ]nδsdot3

[ ]12 minussdot nδ

[ ]21 minussdot nδ

0

3

1

11

2

13

-1

4

-2

Sum up

[ ]ny

[ ]nhsdot3

[ ]12 minussdot nh

[ ]2minusnh






[ ]kh

[ ]kh minus

Reflect Multiply

slide

Sum up

n

[ ] [ ] [ ]

[ ] ( )[ ]nkhkx

knhkxny

k

k

minusminus=

minus=

sum

suminfin

minusinfin=

infin

minusinfin=


0 12

13

-2

0-1-2

13

-2

0 1 2

23

1

0

3

10-1

13

-21

11

210

13

-2

1

2

321

13

-2 -1

3

432

13

-2 -2

4

0

3

1

11

2

13

-1

4

-2

Sum up

[ ]kh

[ ]kx

[ ]kh minus

[ ] 0 =nny

[ ]1+minuskh

[ ]2+minuskh

[ ]3+minuskh

[ ]4+minuskh

[ ] 1 =nny

[ ] 2 =nny

[ ] 3 =nny

[ ] 4 =nny

[ ]ny

Reflect [ ] [ ] [ ][ ] [ ]knhkx

nhnxny

kminus=

lowast=

suminfin

minusinfin=

Convolution




[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=

[ ]nh 1 [ ]nh 2 [ ]nh 2 [ ]nh 1

[ ]nh 2

[ ]nh 1

[ ] [ ]nhnh 21 +



[ ] [ ] [ ][ ] [ ] [ ]nhnhnx

nhnhnx

12

21

=

[ ] [ ] [ ]( )[ ] [ ] [ ] [ ]nhnxnhnx

nhnhnx

21

21

+=+

Commutation

Distribution




[ ] [ ] [ ]

[ ] [ ]

[ ] [ ] ( )

[ ] [ ]nxnh

knmhmnx

knhkx

nhnxny

m

k

mlet

=

minus=sum minus=

minussum=

=

infin

minusinfin=

infin

minusinfin=




[ ] [ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] ( )0000

00011

0

101

cosBut

coscos

suppose

nnωnnxnny

nnnxnnxny

nnynynnxnx

minusminus=minus

minus==

minus=rArrminus=

ωω





[ ] infinlesuminfin

minusinfin=kkh

[ ][ ] nBny

nBnx

y

x

forallinfinltle

forallinfinltle






[ ] [ ] [ ]sum sum= =

minus+minus=K

k

M

mkkk mnxknyny

1000 βα

z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα






[ ]nh( )ωjeH

π2( )ωjeH

ω( ) ( ) ( )

( ) ( )ωω

ωωω

jeHjj

ji

jr

j

eeH

ejHeHeHang=

+=

magnitudephase

( )ωjeH






[ ] infinltsuminfin

minusinfin=nnh

[ ]( )

( )

[ ]mnnmnm

mnmn

emnj

de

mnj

mnj

minus=⎩⎨⎧

ne=

=

minusminus

=

minus= minus

minus

minusminus

int

δ

πππ

ωπ

ππ

ω

ππ

ω

0 1

sin)(2

121

)(

)(

( ) [ ]

[ ] ( ) [ ]


deemhdeeHnh

enheH

m

mnj

m

nj

m

mjnjj

n

njj

=minus==

==

=

sumintsum

int sumint

sum

infin

minusinfin=minus

minusinfin

minusinfin=

minus

infin

minusinfin=

minusminus

infin

minusinfin=

minus

δωπ

ωπ

ωπ

ππ

ω

ππ

ωωππ

ωω

ωω

21

21

21


)(

( ) [ ]

[ ] ( )int

sum

minus

infin

minusinfin=

minus

=

=

ππ

ωω

ωω

ωπ

deeHnh

enheH

njj

n

njj

2

1

absolutely summable

DTFT

Inverse DTFT



( ) [ ]

[ ] [ ] [ ]

( ) [ ] [ ]

[ ] [ ]

( ) ( )

[ ] ( ) ( )ωω

ωω

ωω

ωω

ωω

jj

jj

nj

nk

kj

nj

n k

j

k

n

njj

eHeXnhnx

eHeX

enhekx

eknhkxeY

knhkxnhnxny

enheH

hArrlowastthere4

=

⎟⎠

⎞⎜⎝

⎛=

minus=

minus=lowast=

=

minusinfin

minusinfin=

infin

minusinfin=

minus

minusinfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus

sumsum

sum sum

sum

sum

][

][][

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )ωωω

ωωω

ωωω

jjj

jjj

jjj

eHeXeY

eHeXeY

eHeXeY

ang+ang=angrArr

=rArr

=

knnknnknn






[ ] ( )intsum minus

infin

minusinfin== π

πω ω

πdeXnx j

n

22 21

power spectrum

[ ]nx[ ]

( ) ][][][][

][][

nxnxlnxlx

mxnmxnR

l

mxx


+=

lowastinfin

minusinfin=

infin

minusinfin=

sum

sum

( ) ( ) ( ) ( ) 2 ωωωω XXXS xx ==

hArr


π

π

ωπ

π

ω ωωπ

ωωπ

deXdeSnR njnjxxxx

2

21

21


minusinfin=minus

infin

minusinfin=

===

=

mmxx dXmxmxmxR

nπ

πωω

π22

210

0Set


)(

lnnlmnml



conjugatecomplex





[ ] ( )[ ] [ ] [ ] ( )

[ ] ( ) [ ]

[ ] ( )( )

( ) [ ]( ) ( )( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( )( )[ ] ( ) ( ) ( )( )ωω

ωωωω

φωω

ωφω

ωφω

ωφω

φω

φω

φω

φω

φωφω

φω

ω

ω

jj

jjjj

eHjnjj

eHjjnj

jnj

kj

k

nj

knj

k

nj

nj

eHneHAny

eHneHjAeHneHA

eeHA

eeHAe

eHAe

ekhAe

Aekh

nhAeny

Aenxnxnx

njAnx

j

j

ang++=rArr

ang+++ang++=

=

=

=

sum=

sum=

=

=+=

+=

ang++

ang+

+

minusinfin

minusinfin=

+

+minusinfin

minusinfin=

+

+

cos

sincos

sin

0

10

1

[ ]ny [ ]ny1

( )( ) attenuate 1

amplify 1

lt

gt

ω

ω

j

j

eH

eH





Z-Transform





( ) [ ]suminfin

minusinfin=

minus=n

nznhzH

[ ] ( )zHnh

[ ] ( )ωjeHnh

[ ]nh

ωjrez =

( ) ( ) ωω

jezj zHeH

==

)1( == zez jω

complex plane

unit circle

Im

Re

ω

ωjez =


Z-Transform (cont)


stability



[ ] infinltsuminfin

minusinfin=

minus

n

nznh

complex plan

R1

R2

Re

Im

absolutely summable

1=z


Z-Transform (cont)





[ ] 0for 0 lt= nnh

[ ] 0for 0 gt= nnh


Left-sided sequence

[ ] infinltsuminfin

minusinfin=nnh

[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


Z-Transform (cont)


[ ] [ ] [ ]⎩⎨⎧

ltge

==0for 00for 1

ere wh 1 1 nn

nunuanh n

( ) ( ) 11

1 11

minus

infin

minusinfin=

minusminusinfin


azazzazH

n

nn

n

n

If 11 ltminusaz

azROC gtthere4 is 1

Re

Im

a[ ] 1 if exists

ofansformFourier tr

1 ltanhtimes


az =

the unit cycle

0


Z-Transform (cont)



( ) [ ]

( ) 11

1

10

1

1

12

11

11111

1

minusminus

minus

minus

infin

=

minusinfin

=

minus

minusminus

minusinfin=

minusinfin

minusinfin=

minus=

minusminus

minus=minus

minus=minus=minus=


sumsum

sumsum

azzaza

zazaza

zaznuazH

n

n

n

nn

n

n

nn

n

n If 11 ltminus za

azROC ltthere4 is 2

Re

Im

a[ ] [ ]

minusinfinrarr

lt

nnhnh

a



22times

the unit cycle

0


Z-Transform (cont)


[ ] [ ] [ ]121


⎠⎞

⎜⎝⎛minus⎟

⎠⎞


nn

[ ]

[ ]21

211

1 121

31

311

1 31

1

1

ltminus


⎜⎝⎛minus

gt+


⎜⎝⎛ minus

minus

minus

zz

nu

zz

nu

zn

zn

Re

Im

times

the unit cycle

times21

31



3

3

ROCnh

31 and


( )⎟⎠⎞


⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ minus

=minus

++

=minusminus

21

31

1212

211

1

311

1 11

3

zz

zz

zzzH

0


Z-Transform (cont)



=others 0

10 3 4

Nnanh

n

( ) ( ) ( )azaz

zazazazzazH

NN

N

NN

n

nN

n

nn

minusminus

=minus

minus=== minusminus

minusminus

=

minusminus

=

minus sumsum 11

11

0

11

04

11

1



Im

times

the unit cycle

31

minus


4π

( ) 11 2

minus== N kaez Nkj

k

π

If N=8

0 N-1

Re


Z-Transform (cont)






[ ]nh [ ] 1 0 nnnh le=

01 gen

0rz gtz

infin=z

[ ]nh [ ] 2 0 nnnh ge=

02 ltn 0=z[ ]nh


01 geninfin=zthere4








[ ]nh

[ ] [ ] [ ] [ ]

[ ]

( ) njj

kjnj

knj

eeH

ekhe

ekhnhnxny

ωω

ωω

ω

=

=

=lowast=

minusinfin

infin=

minusinfin

infin=

sum

sum

-k

)(

-k( )



[ ] njenx ω=[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


( )jωeH

( )jωeH

scalar


[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjωnjjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

ang++=

+=

+=

+=

+minusangminus+ang

+minus+

minusminusminus

φω

φωφω

φωφω

ωφωφ




[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0

ωφωφ

φω

minusminus+=

+=

[ ]nh

( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

minus

minus

+=rArr

minus=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω

jωjω

eeHeH

eHeHangminus

minus

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos

sincos

jje

jejyxz

jejyxz

j

j

j

minus=minus+minus=

minus=rArrminus=

+=rArr+=

minus






[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2





1100

1100coscossinsintan

φφφφ

φAAAA

++

=

( ) ( )( )

( )10102

120

1010102

120

2

21100

21100

2

cos2

sinsincoscos2

sinsincoscos

φφ

φφφφ

φφφφ

minus++=

+++=

+++=

AAAA

AAAAA

AAAAA






[ ]nx

[ ] [ ] [ ]knkxnxk

minussum=infin

minusinfin=δ


[ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ]31211121221

33221101122

3

2



infin

minusinfin=

nnnnnnnxnxnxnxnxnx

knkxknkxnxkk


δδ

321012 minusminus=n


Digital Systems


[ ] [ ] nxTny =

[ ]nx bullT [ ]ny


















DigitalSystem

[ ]nδ [ ]nh




[ ] [ ] [ ]knkxnxk

minus= suminfin

minusinfin=δ

[ ] [ ] [ ] [ ] [ ]

[ ] [ ][ ] [ ]nhnx

knhkx

knTkx

knkxTnxT

k

k

k

lowast=

minus=

minus=

minus=rArr

sum

sum

sum

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

δ

δ

scale


Time invariant

DigitalSystem

[ ]nδ [ ]nh

linear

Time-invariant

[ ] [ ][ ] [ ]knhkn

nhnT

T


⎯rarr⎯

δ

δ

Impulse response

convolution




bull Example

[ ]nδ 0 12

13

-2[ ]nh

LTI[ ]nx

0 1 2

23

1

0

3

0 12

39

-6

1

2

1 23

26

-4

LTI

2

1

2 34

1 3

-2

Length=M=3

Length=L=3

Length=L+M-1

[ ]nδsdot3

[ ]12 minussdot nδ

[ ]21 minussdot nδ

0

3

1

11

2

13

-1

4

-2

Sum up

[ ]ny

[ ]nhsdot3

[ ]12 minussdot nh

[ ]2minusnh






[ ]kh

[ ]kh minus

Reflect Multiply

slide

Sum up

n

[ ] [ ] [ ]

[ ] ( )[ ]nkhkx

knhkxny

k

k

minusminus=

minus=

sum

suminfin

minusinfin=

infin

minusinfin=


0 12

13

-2

0-1-2

13

-2

0 1 2

23

1

0

3

10-1

13

-21

11

210

13

-2

1

2

321

13

-2 -1

3

432

13

-2 -2

4

0

3

1

11

2

13

-1

4

-2

Sum up

[ ]kh

[ ]kx

[ ]kh minus

[ ] 0 =nny

[ ]1+minuskh

[ ]2+minuskh

[ ]3+minuskh

[ ]4+minuskh

[ ] 1 =nny

[ ] 2 =nny

[ ] 3 =nny

[ ] 4 =nny

[ ]ny

Reflect [ ] [ ] [ ][ ] [ ]knhkx

nhnxny

kminus=

lowast=

suminfin

minusinfin=

Convolution




[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=

[ ]nh 1 [ ]nh 2 [ ]nh 2 [ ]nh 1

[ ]nh 2

[ ]nh 1

[ ] [ ]nhnh 21 +



[ ] [ ] [ ][ ] [ ] [ ]nhnhnx

nhnhnx

12

21

=

[ ] [ ] [ ]( )[ ] [ ] [ ] [ ]nhnxnhnx

nhnhnx

21

21

+=+

Commutation

Distribution




[ ] [ ] [ ]

[ ] [ ]

[ ] [ ] ( )

[ ] [ ]nxnh

knmhmnx

knhkx

nhnxny

m

k

mlet

=

minus=sum minus=

minussum=

=

infin

minusinfin=

infin

minusinfin=




[ ] [ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] ( )0000

00011

0

101

cosBut

coscos

suppose

nnωnnxnny

nnnxnnxny

nnynynnxnx

minusminus=minus

minus==

minus=rArrminus=

ωω





[ ] infinlesuminfin

minusinfin=kkh

[ ][ ] nBny

nBnx

y

x

forallinfinltle

forallinfinltle






[ ] [ ] [ ]sum sum= =

minus+minus=K

k

M

mkkk mnxknyny

1000 βα

z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα






[ ]nh( )ωjeH

π2( )ωjeH

ω( ) ( ) ( )

( ) ( )ωω

ωωω

jeHjj

ji

jr

j

eeH

ejHeHeHang=

+=

magnitudephase

( )ωjeH






[ ] infinltsuminfin

minusinfin=nnh

[ ]( )

( )

[ ]mnnmnm

mnmn

emnj

de

mnj

mnj

minus=⎩⎨⎧

ne=

=

minusminus

=

minus= minus

minus

minusminus

int

δ

πππ

ωπ

ππ

ω

ππ

ω

0 1

sin)(2

121

)(

)(

( ) [ ]

[ ] ( ) [ ]


deemhdeeHnh

enheH

m

mnj

m

nj

m

mjnjj

n

njj

=minus==

==

=

sumintsum

int sumint

sum

infin

minusinfin=minus

minusinfin

minusinfin=

minus

infin

minusinfin=

minusminus

infin

minusinfin=

minus

δωπ

ωπ

ωπ

ππ

ω

ππ

ωωππ

ωω

ωω

21

21

21


)(

( ) [ ]

[ ] ( )int

sum

minus

infin

minusinfin=

minus

=

=

ππ

ωω

ωω

ωπ

deeHnh

enheH

njj

n

njj

2

1

absolutely summable

DTFT

Inverse DTFT



( ) [ ]

[ ] [ ] [ ]

( ) [ ] [ ]

[ ] [ ]

( ) ( )

[ ] ( ) ( )ωω

ωω

ωω

ωω

ωω

jj

jj

nj

nk

kj

nj

n k

j

k

n

njj

eHeXnhnx

eHeX

enhekx

eknhkxeY

knhkxnhnxny

enheH

hArrlowastthere4

=

⎟⎠

⎞⎜⎝

⎛=

minus=

minus=lowast=

=

minusinfin

minusinfin=

infin

minusinfin=

minus

minusinfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus

sumsum

sum sum

sum

sum

][

][][

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )ωωω

ωωω

ωωω

jjj

jjj

jjj

eHeXeY

eHeXeY

eHeXeY

ang+ang=angrArr

=rArr

=

knnknnknn






[ ] ( )intsum minus

infin

minusinfin== π

πω ω

πdeXnx j

n

22 21

power spectrum

[ ]nx[ ]

( ) ][][][][

][][

nxnxlnxlx

mxnmxnR

l

mxx


+=

lowastinfin

minusinfin=

infin

minusinfin=

sum

sum

( ) ( ) ( ) ( ) 2 ωωωω XXXS xx ==

hArr


π

π

ωπ

π

ω ωωπ

ωωπ

deXdeSnR njnjxxxx

2

21

21


minusinfin=minus

infin

minusinfin=

===

=

mmxx dXmxmxmxR

nπ

πωω

π22

210

0Set


)(

lnnlmnml



conjugatecomplex





[ ] ( )[ ] [ ] [ ] ( )

[ ] ( ) [ ]

[ ] ( )( )

( ) [ ]( ) ( )( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( )( )[ ] ( ) ( ) ( )( )ωω

ωωωω

φωω

ωφω

ωφω

ωφω

φω

φω

φω

φω

φωφω

φω

ω

ω

jj

jjjj

eHjnjj

eHjjnj

jnj

kj

k

nj

knj

k

nj

nj

eHneHAny

eHneHjAeHneHA

eeHA

eeHAe

eHAe

ekhAe

Aekh

nhAeny

Aenxnxnx

njAnx

j

j

ang++=rArr

ang+++ang++=

=

=

=

sum=

sum=

=

=+=

+=

ang++

ang+

+

minusinfin

minusinfin=

+

+minusinfin

minusinfin=

+

+

cos

sincos

sin

0

10

1

[ ]ny [ ]ny1

( )( ) attenuate 1

amplify 1

lt

gt

ω

ω

j

j

eH

eH





Z-Transform





( ) [ ]suminfin

minusinfin=

minus=n

nznhzH

[ ] ( )zHnh

[ ] ( )ωjeHnh

[ ]nh

ωjrez =

( ) ( ) ωω

jezj zHeH

==

)1( == zez jω

complex plane

unit circle

Im

Re

ω

ωjez =


Z-Transform (cont)


stability



[ ] infinltsuminfin

minusinfin=

minus

n

nznh

complex plan

R1

R2

Re

Im

absolutely summable

1=z


Z-Transform (cont)





[ ] 0for 0 lt= nnh

[ ] 0for 0 gt= nnh


Left-sided sequence

[ ] infinltsuminfin

minusinfin=nnh

[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


Z-Transform (cont)


[ ] [ ] [ ]⎩⎨⎧

ltge

==0for 00for 1

ere wh 1 1 nn

nunuanh n

( ) ( ) 11

1 11

minus

infin

minusinfin=

minusminusinfin


azazzazH

n

nn

n

n

If 11 ltminusaz

azROC gtthere4 is 1

Re

Im

a[ ] 1 if exists

ofansformFourier tr

1 ltanhtimes


az =

the unit cycle

0


Z-Transform (cont)



( ) [ ]

( ) 11

1

10

1

1

12

11

11111

1

minusminus

minus

minus

infin

=

minusinfin

=

minus

minusminus

minusinfin=

minusinfin

minusinfin=

minus=

minusminus

minus=minus

minus=minus=minus=


sumsum

sumsum

azzaza

zazaza

zaznuazH

n

n

n

nn

n

n

nn

n

n If 11 ltminus za

azROC ltthere4 is 2

Re

Im

a[ ] [ ]

minusinfinrarr

lt

nnhnh

a



22times

the unit cycle

0


Z-Transform (cont)


[ ] [ ] [ ]121


⎠⎞

⎜⎝⎛minus⎟

⎠⎞


nn

[ ]

[ ]21

211

1 121

31

311

1 31

1

1

ltminus


⎜⎝⎛minus

gt+


⎜⎝⎛ minus

minus

minus

zz

nu

zz

nu

zn

zn

Re

Im

times

the unit cycle

times21

31



3

3

ROCnh

31 and


( )⎟⎠⎞


⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ minus

=minus

++

=minusminus

21

31

1212

211

1

311

1 11

3

zz

zz

zzzH

0


Z-Transform (cont)



=others 0

10 3 4

Nnanh

n

( ) ( ) ( )azaz

zazazazzazH

NN

N

NN

n

nN

n

nn

minusminus

=minus

minus=== minusminus

minusminus

=

minusminus

=

minus sumsum 11

11

0

11

04

11

1



Im

times

the unit cycle

31

minus


4π

( ) 11 2

minus== N kaez Nkj

k

π

If N=8

0 N-1

Re


Z-Transform (cont)






[ ]nh [ ] 1 0 nnnh le=

01 gen

0rz gtz

infin=z

[ ]nh [ ] 2 0 nnnh ge=

02 ltn 0=z[ ]nh


01 geninfin=zthere4








[ ]nh

[ ] [ ] [ ] [ ]

[ ]

( ) njj

kjnj

knj

eeH

ekhe

ekhnhnxny

ωω

ωω

ω

=

=

=lowast=

minusinfin

infin=

minusinfin

infin=

sum

sum

-k

)(

-k( )



[ ] njenx ω=[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


( )jωeH

( )jωeH

scalar


[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjωnjjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

ang++=

+=

+=

+=

+minusangminus+ang

+minus+

minusminusminus

φω

φωφω

φωφω

ωφωφ




[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0

ωφωφ

φω

minusminus+=

+=

[ ]nh

( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

minus

minus

+=rArr

minus=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω

jωjω

eeHeH

eHeHangminus

minus

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos

sincos

jje

jejyxz

jejyxz

j

j

j

minus=minus+minus=

minus=rArrminus=

+=rArr+=

minus






[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2







[ ]nx

[ ] [ ] [ ]knkxnxk

minussum=infin

minusinfin=δ


[ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ]31211121221

33221101122

3

2



infin

minusinfin=

nnnnnnnxnxnxnxnxnx

knkxknkxnxkk


δδ

321012 minusminus=n


Digital Systems


[ ] [ ] nxTny =

[ ]nx bullT [ ]ny


















DigitalSystem

[ ]nδ [ ]nh




[ ] [ ] [ ]knkxnxk

minus= suminfin

minusinfin=δ

[ ] [ ] [ ] [ ] [ ]

[ ] [ ][ ] [ ]nhnx

knhkx

knTkx

knkxTnxT

k

k

k

lowast=

minus=

minus=

minus=rArr

sum

sum

sum

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

δ

δ

scale


Time invariant

DigitalSystem

[ ]nδ [ ]nh

linear

Time-invariant

[ ] [ ][ ] [ ]knhkn

nhnT

T


⎯rarr⎯

δ

δ

Impulse response

convolution




bull Example

[ ]nδ 0 12

13

-2[ ]nh

LTI[ ]nx

0 1 2

23

1

0

3

0 12

39

-6

1

2

1 23

26

-4

LTI

2

1

2 34

1 3

-2

Length=M=3

Length=L=3

Length=L+M-1

[ ]nδsdot3

[ ]12 minussdot nδ

[ ]21 minussdot nδ

0

3

1

11

2

13

-1

4

-2

Sum up

[ ]ny

[ ]nhsdot3

[ ]12 minussdot nh

[ ]2minusnh






[ ]kh

[ ]kh minus

Reflect Multiply

slide

Sum up

n

[ ] [ ] [ ]

[ ] ( )[ ]nkhkx

knhkxny

k

k

minusminus=

minus=

sum

suminfin

minusinfin=

infin

minusinfin=


0 12

13

-2

0-1-2

13

-2

0 1 2

23

1

0

3

10-1

13

-21

11

210

13

-2

1

2

321

13

-2 -1

3

432

13

-2 -2

4

0

3

1

11

2

13

-1

4

-2

Sum up

[ ]kh

[ ]kx

[ ]kh minus

[ ] 0 =nny

[ ]1+minuskh

[ ]2+minuskh

[ ]3+minuskh

[ ]4+minuskh

[ ] 1 =nny

[ ] 2 =nny

[ ] 3 =nny

[ ] 4 =nny

[ ]ny

Reflect [ ] [ ] [ ][ ] [ ]knhkx

nhnxny

kminus=

lowast=

suminfin

minusinfin=

Convolution




[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=

[ ]nh 1 [ ]nh 2 [ ]nh 2 [ ]nh 1

[ ]nh 2

[ ]nh 1

[ ] [ ]nhnh 21 +



[ ] [ ] [ ][ ] [ ] [ ]nhnhnx

nhnhnx

12

21

=

[ ] [ ] [ ]( )[ ] [ ] [ ] [ ]nhnxnhnx

nhnhnx

21

21

+=+

Commutation

Distribution




[ ] [ ] [ ]

[ ] [ ]

[ ] [ ] ( )

[ ] [ ]nxnh

knmhmnx

knhkx

nhnxny

m

k

mlet

=

minus=sum minus=

minussum=

=

infin

minusinfin=

infin

minusinfin=




[ ] [ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] ( )0000

00011

0

101

cosBut

coscos

suppose

nnωnnxnny

nnnxnnxny

nnynynnxnx

minusminus=minus

minus==

minus=rArrminus=

ωω





[ ] infinlesuminfin

minusinfin=kkh

[ ][ ] nBny

nBnx

y

x

forallinfinltle

forallinfinltle






[ ] [ ] [ ]sum sum= =

minus+minus=K

k

M

mkkk mnxknyny

1000 βα

z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα






[ ]nh( )ωjeH

π2( )ωjeH

ω( ) ( ) ( )

( ) ( )ωω

ωωω

jeHjj

ji

jr

j

eeH

ejHeHeHang=

+=

magnitudephase

( )ωjeH






[ ] infinltsuminfin

minusinfin=nnh

[ ]( )

( )

[ ]mnnmnm

mnmn

emnj

de

mnj

mnj

minus=⎩⎨⎧

ne=

=

minusminus

=

minus= minus

minus

minusminus

int

δ

πππ

ωπ

ππ

ω

ππ

ω

0 1

sin)(2

121

)(

)(

( ) [ ]

[ ] ( ) [ ]


deemhdeeHnh

enheH

m

mnj

m

nj

m

mjnjj

n

njj

=minus==

==

=

sumintsum

int sumint

sum

infin

minusinfin=minus

minusinfin

minusinfin=

minus

infin

minusinfin=

minusminus

infin

minusinfin=

minus

δωπ

ωπ

ωπ

ππ

ω

ππ

ωωππ

ωω

ωω

21

21

21


)(

( ) [ ]

[ ] ( )int

sum

minus

infin

minusinfin=

minus

=

=

ππ

ωω

ωω

ωπ

deeHnh

enheH

njj

n

njj

2

1

absolutely summable

DTFT

Inverse DTFT



( ) [ ]

[ ] [ ] [ ]

( ) [ ] [ ]

[ ] [ ]

( ) ( )

[ ] ( ) ( )ωω

ωω

ωω

ωω

ωω

jj

jj

nj

nk

kj

nj

n k

j

k

n

njj

eHeXnhnx

eHeX

enhekx

eknhkxeY

knhkxnhnxny

enheH

hArrlowastthere4

=

⎟⎠

⎞⎜⎝

⎛=

minus=

minus=lowast=

=

minusinfin

minusinfin=

infin

minusinfin=

minus

minusinfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus

sumsum

sum sum

sum

sum

][

][][

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )ωωω

ωωω

ωωω

jjj

jjj

jjj

eHeXeY

eHeXeY

eHeXeY

ang+ang=angrArr

=rArr

=

knnknnknn






[ ] ( )intsum minus

infin

minusinfin== π

πω ω

πdeXnx j

n

22 21

power spectrum

[ ]nx[ ]

( ) ][][][][

][][

nxnxlnxlx

mxnmxnR

l

mxx


+=

lowastinfin

minusinfin=

infin

minusinfin=

sum

sum

( ) ( ) ( ) ( ) 2 ωωωω XXXS xx ==

hArr


π

π

ωπ

π

ω ωωπ

ωωπ

deXdeSnR njnjxxxx

2

21

21


minusinfin=minus

infin

minusinfin=

===

=

mmxx dXmxmxmxR

nπ

πωω

π22

210

0Set


)(

lnnlmnml



conjugatecomplex





[ ] ( )[ ] [ ] [ ] ( )

[ ] ( ) [ ]

[ ] ( )( )

( ) [ ]( ) ( )( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( )( )[ ] ( ) ( ) ( )( )ωω

ωωωω

φωω

ωφω

ωφω

ωφω

φω

φω

φω

φω

φωφω

φω

ω

ω

jj

jjjj

eHjnjj

eHjjnj

jnj

kj

k

nj

knj

k

nj

nj

eHneHAny

eHneHjAeHneHA

eeHA

eeHAe

eHAe

ekhAe

Aekh

nhAeny

Aenxnxnx

njAnx

j

j

ang++=rArr

ang+++ang++=

=

=

=

sum=

sum=

=

=+=

+=

ang++

ang+

+

minusinfin

minusinfin=

+

+minusinfin

minusinfin=

+

+

cos

sincos

sin

0

10

1

[ ]ny [ ]ny1

( )( ) attenuate 1

amplify 1

lt

gt

ω

ω

j

j

eH

eH





Z-Transform





( ) [ ]suminfin

minusinfin=

minus=n

nznhzH

[ ] ( )zHnh

[ ] ( )ωjeHnh

[ ]nh

ωjrez =

( ) ( ) ωω

jezj zHeH

==

)1( == zez jω

complex plane

unit circle

Im

Re

ω

ωjez =


Z-Transform (cont)


stability



[ ] infinltsuminfin

minusinfin=

minus

n

nznh

complex plan

R1

R2

Re

Im

absolutely summable

1=z


Z-Transform (cont)





[ ] 0for 0 lt= nnh

[ ] 0for 0 gt= nnh


Left-sided sequence

[ ] infinltsuminfin

minusinfin=nnh

[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


Z-Transform (cont)


[ ] [ ] [ ]⎩⎨⎧

ltge

==0for 00for 1

ere wh 1 1 nn

nunuanh n

( ) ( ) 11

1 11

minus

infin

minusinfin=

minusminusinfin


azazzazH

n

nn

n

n

If 11 ltminusaz

azROC gtthere4 is 1

Re

Im

a[ ] 1 if exists

ofansformFourier tr

1 ltanhtimes


az =

the unit cycle

0


Z-Transform (cont)



( ) [ ]

( ) 11

1

10

1

1

12

11

11111

1

minusminus

minus

minus

infin

=

minusinfin

=

minus

minusminus

minusinfin=

minusinfin

minusinfin=

minus=

minusminus

minus=minus

minus=minus=minus=


sumsum

sumsum

azzaza

zazaza

zaznuazH

n

n

n

nn

n

n

nn

n

n If 11 ltminus za

azROC ltthere4 is 2

Re

Im

a[ ] [ ]

minusinfinrarr

lt

nnhnh

a



22times

the unit cycle

0


Z-Transform (cont)


[ ] [ ] [ ]121


⎠⎞

⎜⎝⎛minus⎟

⎠⎞


nn

[ ]

[ ]21

211

1 121

31

311

1 31

1

1

ltminus


⎜⎝⎛minus

gt+


⎜⎝⎛ minus

minus

minus

zz

nu

zz

nu

zn

zn

Re

Im

times

the unit cycle

times21

31



3

3

ROCnh

31 and


( )⎟⎠⎞


⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ minus

=minus

++

=minusminus

21

31

1212

211

1

311

1 11

3

zz

zz

zzzH

0


Z-Transform (cont)



=others 0

10 3 4

Nnanh

n

( ) ( ) ( )azaz

zazazazzazH

NN

N

NN

n

nN

n

nn

minusminus

=minus

minus=== minusminus

minusminus

=

minusminus

=

minus sumsum 11

11

0

11

04

11

1



Im

times

the unit cycle

31

minus


4π

( ) 11 2

minus== N kaez Nkj

k

π

If N=8

0 N-1

Re


Z-Transform (cont)






[ ]nh [ ] 1 0 nnnh le=

01 gen

0rz gtz

infin=z

[ ]nh [ ] 2 0 nnnh ge=

02 ltn 0=z[ ]nh


01 geninfin=zthere4








[ ]nh

[ ] [ ] [ ] [ ]

[ ]

( ) njj

kjnj

knj

eeH

ekhe

ekhnhnxny

ωω

ωω

ω

=

=

=lowast=

minusinfin

infin=

minusinfin

infin=

sum

sum

-k

)(

-k( )



[ ] njenx ω=[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


( )jωeH

( )jωeH

scalar


[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjωnjjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

ang++=

+=

+=

+=

+minusangminus+ang

+minus+

minusminusminus

φω

φωφω

φωφω

ωφωφ




[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0

ωφωφ

φω

minusminus+=

+=

[ ]nh

( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

minus

minus

+=rArr

minus=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω

jωjω

eeHeH

eHeHangminus

minus

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos

sincos

jje

jejyxz

jejyxz

j

j

j

minus=minus+minus=

minus=rArrminus=

+=rArr+=

minus






[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2





[ ]nx

[ ] [ ] [ ]knkxnxk

minussum=infin

minusinfin=δ


[ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ]31211121221

33221101122

3

2



infin

minusinfin=

nnnnnnnxnxnxnxnxnx

knkxknkxnxkk


δδ

321012 minusminus=n


Digital Systems


[ ] [ ] nxTny =

[ ]nx bullT [ ]ny


















DigitalSystem

[ ]nδ [ ]nh




[ ] [ ] [ ]knkxnxk

minus= suminfin

minusinfin=δ

[ ] [ ] [ ] [ ] [ ]

[ ] [ ][ ] [ ]nhnx

knhkx

knTkx

knkxTnxT

k

k

k

lowast=

minus=

minus=

minus=rArr

sum

sum

sum

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

δ

δ

scale


Time invariant

DigitalSystem

[ ]nδ [ ]nh

linear

Time-invariant

[ ] [ ][ ] [ ]knhkn

nhnT

T


⎯rarr⎯

δ

δ

Impulse response

convolution




bull Example

[ ]nδ 0 12

13

-2[ ]nh

LTI[ ]nx

0 1 2

23

1

0

3

0 12

39

-6

1

2

1 23

26

-4

LTI

2

1

2 34

1 3

-2

Length=M=3

Length=L=3

Length=L+M-1

[ ]nδsdot3

[ ]12 minussdot nδ

[ ]21 minussdot nδ

0

3

1

11

2

13

-1

4

-2

Sum up

[ ]ny

[ ]nhsdot3

[ ]12 minussdot nh

[ ]2minusnh






[ ]kh

[ ]kh minus

Reflect Multiply

slide

Sum up

n

[ ] [ ] [ ]

[ ] ( )[ ]nkhkx

knhkxny

k

k

minusminus=

minus=

sum

suminfin

minusinfin=

infin

minusinfin=


0 12

13

-2

0-1-2

13

-2

0 1 2

23

1

0

3

10-1

13

-21

11

210

13

-2

1

2

321

13

-2 -1

3

432

13

-2 -2

4

0

3

1

11

2

13

-1

4

-2

Sum up

[ ]kh

[ ]kx

[ ]kh minus

[ ] 0 =nny

[ ]1+minuskh

[ ]2+minuskh

[ ]3+minuskh

[ ]4+minuskh

[ ] 1 =nny

[ ] 2 =nny

[ ] 3 =nny

[ ] 4 =nny

[ ]ny

Reflect [ ] [ ] [ ][ ] [ ]knhkx

nhnxny

kminus=

lowast=

suminfin

minusinfin=

Convolution




[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=

[ ]nh 1 [ ]nh 2 [ ]nh 2 [ ]nh 1

[ ]nh 2

[ ]nh 1

[ ] [ ]nhnh 21 +



[ ] [ ] [ ][ ] [ ] [ ]nhnhnx

nhnhnx

12

21

=

[ ] [ ] [ ]( )[ ] [ ] [ ] [ ]nhnxnhnx

nhnhnx

21

21

+=+

Commutation

Distribution




[ ] [ ] [ ]

[ ] [ ]

[ ] [ ] ( )

[ ] [ ]nxnh

knmhmnx

knhkx

nhnxny

m

k

mlet

=

minus=sum minus=

minussum=

=

infin

minusinfin=

infin

minusinfin=




[ ] [ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] ( )0000

00011

0

101

cosBut

coscos

suppose

nnωnnxnny

nnnxnnxny

nnynynnxnx

minusminus=minus

minus==

minus=rArrminus=

ωω





[ ] infinlesuminfin

minusinfin=kkh

[ ][ ] nBny

nBnx

y

x

forallinfinltle

forallinfinltle






[ ] [ ] [ ]sum sum= =

minus+minus=K

k

M

mkkk mnxknyny

1000 βα

z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα






[ ]nh( )ωjeH

π2( )ωjeH

ω( ) ( ) ( )

( ) ( )ωω

ωωω

jeHjj

ji

jr

j

eeH

ejHeHeHang=

+=

magnitudephase

( )ωjeH






[ ] infinltsuminfin

minusinfin=nnh

[ ]( )

( )

[ ]mnnmnm

mnmn

emnj

de

mnj

mnj

minus=⎩⎨⎧

ne=

=

minusminus

=

minus= minus

minus

minusminus

int

δ

πππ

ωπ

ππ

ω

ππ

ω

0 1

sin)(2

121

)(

)(

( ) [ ]

[ ] ( ) [ ]


deemhdeeHnh

enheH

m

mnj

m

nj

m

mjnjj

n

njj

=minus==

==

=

sumintsum

int sumint

sum

infin

minusinfin=minus

minusinfin

minusinfin=

minus

infin

minusinfin=

minusminus

infin

minusinfin=

minus

δωπ

ωπ

ωπ

ππ

ω

ππ

ωωππ

ωω

ωω

21

21

21


)(

( ) [ ]

[ ] ( )int

sum

minus

infin

minusinfin=

minus

=

=

ππ

ωω

ωω

ωπ

deeHnh

enheH

njj

n

njj

2

1

absolutely summable

DTFT

Inverse DTFT



( ) [ ]

[ ] [ ] [ ]

( ) [ ] [ ]

[ ] [ ]

( ) ( )

[ ] ( ) ( )ωω

ωω

ωω

ωω

ωω

jj

jj

nj

nk

kj

nj

n k

j

k

n

njj

eHeXnhnx

eHeX

enhekx

eknhkxeY

knhkxnhnxny

enheH

hArrlowastthere4

=

⎟⎠

⎞⎜⎝

⎛=

minus=

minus=lowast=

=

minusinfin

minusinfin=

infin

minusinfin=

minus

minusinfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus

sumsum

sum sum

sum

sum

][

][][

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )ωωω

ωωω

ωωω

jjj

jjj

jjj

eHeXeY

eHeXeY

eHeXeY

ang+ang=angrArr

=rArr

=

knnknnknn






[ ] ( )intsum minus

infin

minusinfin== π

πω ω

πdeXnx j

n

22 21

power spectrum

[ ]nx[ ]

( ) ][][][][

][][

nxnxlnxlx

mxnmxnR

l

mxx


+=

lowastinfin

minusinfin=

infin

minusinfin=

sum

sum

( ) ( ) ( ) ( ) 2 ωωωω XXXS xx ==

hArr


π

π

ωπ

π

ω ωωπ

ωωπ

deXdeSnR njnjxxxx

2

21

21


minusinfin=minus

infin

minusinfin=

===

=

mmxx dXmxmxmxR

nπ

πωω

π22

210

0Set


)(

lnnlmnml



conjugatecomplex





[ ] ( )[ ] [ ] [ ] ( )

[ ] ( ) [ ]

[ ] ( )( )

( ) [ ]( ) ( )( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( )( )[ ] ( ) ( ) ( )( )ωω

ωωωω

φωω

ωφω

ωφω

ωφω

φω

φω

φω

φω

φωφω

φω

ω

ω

jj

jjjj

eHjnjj

eHjjnj

jnj

kj

k

nj

knj

k

nj

nj

eHneHAny

eHneHjAeHneHA

eeHA

eeHAe

eHAe

ekhAe

Aekh

nhAeny

Aenxnxnx

njAnx

j

j

ang++=rArr

ang+++ang++=

=

=

=

sum=

sum=

=

=+=

+=

ang++

ang+

+

minusinfin

minusinfin=

+

+minusinfin

minusinfin=

+

+

cos

sincos

sin

0

10

1

[ ]ny [ ]ny1

( )( ) attenuate 1

amplify 1

lt

gt

ω

ω

j

j

eH

eH





Z-Transform





( ) [ ]suminfin

minusinfin=

minus=n

nznhzH

[ ] ( )zHnh

[ ] ( )ωjeHnh

[ ]nh

ωjrez =

( ) ( ) ωω

jezj zHeH

==

)1( == zez jω

complex plane

unit circle

Im

Re

ω

ωjez =


Z-Transform (cont)


stability



[ ] infinltsuminfin

minusinfin=

minus

n

nznh

complex plan

R1

R2

Re

Im

absolutely summable

1=z


Z-Transform (cont)





[ ] 0for 0 lt= nnh

[ ] 0for 0 gt= nnh


Left-sided sequence

[ ] infinltsuminfin

minusinfin=nnh

[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


Z-Transform (cont)


[ ] [ ] [ ]⎩⎨⎧

ltge

==0for 00for 1

ere wh 1 1 nn

nunuanh n

( ) ( ) 11

1 11

minus

infin

minusinfin=

minusminusinfin


azazzazH

n

nn

n

n

If 11 ltminusaz

azROC gtthere4 is 1

Re

Im

a[ ] 1 if exists

ofansformFourier tr

1 ltanhtimes


az =

the unit cycle

0


Z-Transform (cont)



( ) [ ]

( ) 11

1

10

1

1

12

11

11111

1

minusminus

minus

minus

infin

=

minusinfin

=

minus

minusminus

minusinfin=

minusinfin

minusinfin=

minus=

minusminus

minus=minus

minus=minus=minus=


sumsum

sumsum

azzaza

zazaza

zaznuazH

n

n

n

nn

n

n

nn

n

n If 11 ltminus za

azROC ltthere4 is 2

Re

Im

a[ ] [ ]

minusinfinrarr

lt

nnhnh

a



22times

the unit cycle

0


Z-Transform (cont)


[ ] [ ] [ ]121


⎠⎞

⎜⎝⎛minus⎟

⎠⎞


nn

[ ]

[ ]21

211

1 121

31

311

1 31

1

1

ltminus


⎜⎝⎛minus

gt+


⎜⎝⎛ minus

minus

minus

zz

nu

zz

nu

zn

zn

Re

Im

times

the unit cycle

times21

31



3

3

ROCnh

31 and


( )⎟⎠⎞


⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ minus

=minus

++

=minusminus

21

31

1212

211

1

311

1 11

3

zz

zz

zzzH

0


Z-Transform (cont)



=others 0

10 3 4

Nnanh

n

( ) ( ) ( )azaz

zazazazzazH

NN

N

NN

n

nN

n

nn

minusminus

=minus

minus=== minusminus

minusminus

=

minusminus

=

minus sumsum 11

11

0

11

04

11

1



Im

times

the unit cycle

31

minus


4π

( ) 11 2

minus== N kaez Nkj

k

π

If N=8

0 N-1

Re


Z-Transform (cont)






[ ]nh [ ] 1 0 nnnh le=

01 gen

0rz gtz

infin=z

[ ]nh [ ] 2 0 nnnh ge=

02 ltn 0=z[ ]nh


01 geninfin=zthere4








[ ]nh

[ ] [ ] [ ] [ ]

[ ]

( ) njj

kjnj

knj

eeH

ekhe

ekhnhnxny

ωω

ωω

ω

=

=

=lowast=

minusinfin

infin=

minusinfin

infin=

sum

sum

-k

)(

-k( )



[ ] njenx ω=[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


( )jωeH

( )jωeH

scalar


[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjωnjjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

ang++=

+=

+=

+=

+minusangminus+ang

+minus+

minusminusminus

φω

φωφω

φωφω

ωφωφ




[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0

ωφωφ

φω

minusminus+=

+=

[ ]nh

( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

minus

minus

+=rArr

minus=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω

jωjω

eeHeH

eHeHangminus

minus

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos

sincos

jje

jejyxz

jejyxz

j

j

j

minus=minus+minus=

minus=rArrminus=

+=rArr+=

minus






[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2



Digital Systems


[ ] [ ] nxTny =

[ ]nx bullT [ ]ny


















DigitalSystem

[ ]nδ [ ]nh




[ ] [ ] [ ]knkxnxk

minus= suminfin

minusinfin=δ

[ ] [ ] [ ] [ ] [ ]

[ ] [ ][ ] [ ]nhnx

knhkx

knTkx

knkxTnxT

k

k

k

lowast=

minus=

minus=

minus=rArr

sum

sum

sum

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

δ

δ

scale


Time invariant

DigitalSystem

[ ]nδ [ ]nh

linear

Time-invariant

[ ] [ ][ ] [ ]knhkn

nhnT

T


⎯rarr⎯

δ

δ

Impulse response

convolution




bull Example

[ ]nδ 0 12

13

-2[ ]nh

LTI[ ]nx

0 1 2

23

1

0

3

0 12

39

-6

1

2

1 23

26

-4

LTI

2

1

2 34

1 3

-2

Length=M=3

Length=L=3

Length=L+M-1

[ ]nδsdot3

[ ]12 minussdot nδ

[ ]21 minussdot nδ

0

3

1

11

2

13

-1

4

-2

Sum up

[ ]ny

[ ]nhsdot3

[ ]12 minussdot nh

[ ]2minusnh






[ ]kh

[ ]kh minus

Reflect Multiply

slide

Sum up

n

[ ] [ ] [ ]

[ ] ( )[ ]nkhkx

knhkxny

k

k

minusminus=

minus=

sum

suminfin

minusinfin=

infin

minusinfin=


0 12

13

-2

0-1-2

13

-2

0 1 2

23

1

0

3

10-1

13

-21

11

210

13

-2

1

2

321

13

-2 -1

3

432

13

-2 -2

4

0

3

1

11

2

13

-1

4

-2

Sum up

[ ]kh

[ ]kx

[ ]kh minus

[ ] 0 =nny

[ ]1+minuskh

[ ]2+minuskh

[ ]3+minuskh

[ ]4+minuskh

[ ] 1 =nny

[ ] 2 =nny

[ ] 3 =nny

[ ] 4 =nny

[ ]ny

Reflect [ ] [ ] [ ][ ] [ ]knhkx

nhnxny

kminus=

lowast=

suminfin

minusinfin=

Convolution




[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=

[ ]nh 1 [ ]nh 2 [ ]nh 2 [ ]nh 1

[ ]nh 2

[ ]nh 1

[ ] [ ]nhnh 21 +



[ ] [ ] [ ][ ] [ ] [ ]nhnhnx

nhnhnx

12

21

=

[ ] [ ] [ ]( )[ ] [ ] [ ] [ ]nhnxnhnx

nhnhnx

21

21

+=+

Commutation

Distribution




[ ] [ ] [ ]

[ ] [ ]

[ ] [ ] ( )

[ ] [ ]nxnh

knmhmnx

knhkx

nhnxny

m

k

mlet

=

minus=sum minus=

minussum=

=

infin

minusinfin=

infin

minusinfin=




[ ] [ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] ( )0000

00011

0

101

cosBut

coscos

suppose

nnωnnxnny

nnnxnnxny

nnynynnxnx

minusminus=minus

minus==

minus=rArrminus=

ωω





[ ] infinlesuminfin

minusinfin=kkh

[ ][ ] nBny

nBnx

y

x

forallinfinltle

forallinfinltle






[ ] [ ] [ ]sum sum= =

minus+minus=K

k

M

mkkk mnxknyny

1000 βα

z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα






[ ]nh( )ωjeH

π2( )ωjeH

ω( ) ( ) ( )

( ) ( )ωω

ωωω

jeHjj

ji

jr

j

eeH

ejHeHeHang=

+=

magnitudephase

( )ωjeH






[ ] infinltsuminfin

minusinfin=nnh

[ ]( )

( )

[ ]mnnmnm

mnmn

emnj

de

mnj

mnj

minus=⎩⎨⎧

ne=

=

minusminus

=

minus= minus

minus

minusminus

int

δ

πππ

ωπ

ππ

ω

ππ

ω

0 1

sin)(2

121

)(

)(

( ) [ ]

[ ] ( ) [ ]


deemhdeeHnh

enheH

m

mnj

m

nj

m

mjnjj

n

njj

=minus==

==

=

sumintsum

int sumint

sum

infin

minusinfin=minus

minusinfin

minusinfin=

minus

infin

minusinfin=

minusminus

infin

minusinfin=

minus

δωπ

ωπ

ωπ

ππ

ω

ππ

ωωππ

ωω

ωω

21

21

21


)(

( ) [ ]

[ ] ( )int

sum

minus

infin

minusinfin=

minus

=

=

ππ

ωω

ωω

ωπ

deeHnh

enheH

njj

n

njj

2

1

absolutely summable

DTFT

Inverse DTFT



( ) [ ]

[ ] [ ] [ ]

( ) [ ] [ ]

[ ] [ ]

( ) ( )

[ ] ( ) ( )ωω

ωω

ωω

ωω

ωω

jj

jj

nj

nk

kj

nj

n k

j

k

n

njj

eHeXnhnx

eHeX

enhekx

eknhkxeY

knhkxnhnxny

enheH

hArrlowastthere4

=

⎟⎠

⎞⎜⎝

⎛=

minus=

minus=lowast=

=

minusinfin

minusinfin=

infin

minusinfin=

minus

minusinfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus

sumsum

sum sum

sum

sum

][

][][

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )ωωω

ωωω

ωωω

jjj

jjj

jjj

eHeXeY

eHeXeY

eHeXeY

ang+ang=angrArr

=rArr

=

knnknnknn






[ ] ( )intsum minus

infin

minusinfin== π

πω ω

πdeXnx j

n

22 21

power spectrum

[ ]nx[ ]

( ) ][][][][

][][

nxnxlnxlx

mxnmxnR

l

mxx


+=

lowastinfin

minusinfin=

infin

minusinfin=

sum

sum

( ) ( ) ( ) ( ) 2 ωωωω XXXS xx ==

hArr


π

π

ωπ

π

ω ωωπ

ωωπ

deXdeSnR njnjxxxx

2

21

21


minusinfin=minus

infin

minusinfin=

===

=

mmxx dXmxmxmxR

nπ

πωω

π22

210

0Set


)(

lnnlmnml



conjugatecomplex





[ ] ( )[ ] [ ] [ ] ( )

[ ] ( ) [ ]

[ ] ( )( )

( ) [ ]( ) ( )( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( )( )[ ] ( ) ( ) ( )( )ωω

ωωωω

φωω

ωφω

ωφω

ωφω

φω

φω

φω

φω

φωφω

φω

ω

ω

jj

jjjj

eHjnjj

eHjjnj

jnj

kj

k

nj

knj

k

nj

nj

eHneHAny

eHneHjAeHneHA

eeHA

eeHAe

eHAe

ekhAe

Aekh

nhAeny

Aenxnxnx

njAnx

j

j

ang++=rArr

ang+++ang++=

=

=

=

sum=

sum=

=

=+=

+=

ang++

ang+

+

minusinfin

minusinfin=

+

+minusinfin

minusinfin=

+

+

cos

sincos

sin

0

10

1

[ ]ny [ ]ny1

( )( ) attenuate 1

amplify 1

lt

gt

ω

ω

j

j

eH

eH





Z-Transform





( ) [ ]suminfin

minusinfin=

minus=n

nznhzH

[ ] ( )zHnh

[ ] ( )ωjeHnh

[ ]nh

ωjrez =

( ) ( ) ωω

jezj zHeH

==

)1( == zez jω

complex plane

unit circle

Im

Re

ω

ωjez =


Z-Transform (cont)


stability



[ ] infinltsuminfin

minusinfin=

minus

n

nznh

complex plan

R1

R2

Re

Im

absolutely summable

1=z


Z-Transform (cont)





[ ] 0for 0 lt= nnh

[ ] 0for 0 gt= nnh


Left-sided sequence

[ ] infinltsuminfin

minusinfin=nnh

[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


Z-Transform (cont)


[ ] [ ] [ ]⎩⎨⎧

ltge

==0for 00for 1

ere wh 1 1 nn

nunuanh n

( ) ( ) 11

1 11

minus

infin

minusinfin=

minusminusinfin


azazzazH

n

nn

n

n

If 11 ltminusaz

azROC gtthere4 is 1

Re

Im

a[ ] 1 if exists

ofansformFourier tr

1 ltanhtimes


az =

the unit cycle

0


Z-Transform (cont)



( ) [ ]

( ) 11

1

10

1

1

12

11

11111

1

minusminus

minus

minus

infin

=

minusinfin

=

minus

minusminus

minusinfin=

minusinfin

minusinfin=

minus=

minusminus

minus=minus

minus=minus=minus=


sumsum

sumsum

azzaza

zazaza

zaznuazH

n

n

n

nn

n

n

nn

n

n If 11 ltminus za

azROC ltthere4 is 2

Re

Im

a[ ] [ ]

minusinfinrarr

lt

nnhnh

a



22times

the unit cycle

0


Z-Transform (cont)


[ ] [ ] [ ]121


⎠⎞

⎜⎝⎛minus⎟

⎠⎞


nn

[ ]

[ ]21

211

1 121

31

311

1 31

1

1

ltminus


⎜⎝⎛minus

gt+


⎜⎝⎛ minus

minus

minus

zz

nu

zz

nu

zn

zn

Re

Im

times

the unit cycle

times21

31



3

3

ROCnh

31 and


( )⎟⎠⎞


⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ minus

=minus

++

=minusminus

21

31

1212

211

1

311

1 11

3

zz

zz

zzzH

0


Z-Transform (cont)



=others 0

10 3 4

Nnanh

n

( ) ( ) ( )azaz

zazazazzazH

NN

N

NN

n

nN

n

nn

minusminus

=minus

minus=== minusminus

minusminus

=

minusminus

=

minus sumsum 11

11

0

11

04

11

1



Im

times

the unit cycle

31

minus


4π

( ) 11 2

minus== N kaez Nkj

k

π

If N=8

0 N-1

Re


Z-Transform (cont)






[ ]nh [ ] 1 0 nnnh le=

01 gen

0rz gtz

infin=z

[ ]nh [ ] 2 0 nnnh ge=

02 ltn 0=z[ ]nh


01 geninfin=zthere4








[ ]nh

[ ] [ ] [ ] [ ]

[ ]

( ) njj

kjnj

knj

eeH

ekhe

ekhnhnxny

ωω

ωω

ω

=

=

=lowast=

minusinfin

infin=

minusinfin

infin=

sum

sum

-k

)(

-k( )



[ ] njenx ω=[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


( )jωeH

( )jωeH

scalar


[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjωnjjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

ang++=

+=

+=

+=

+minusangminus+ang

+minus+

minusminusminus

φω

φωφω

φωφω

ωφωφ




[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0

ωφωφ

φω

minusminus+=

+=

[ ]nh

( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

minus

minus

+=rArr

minus=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω

jωjω

eeHeH

eHeHangminus

minus

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos

sincos

jje

jejyxz

jejyxz

j

j

j

minus=minus+minus=

minus=rArrminus=

+=rArr+=

minus






[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2



















DigitalSystem

[ ]nδ [ ]nh




[ ] [ ] [ ]knkxnxk

minus= suminfin

minusinfin=δ

[ ] [ ] [ ] [ ] [ ]

[ ] [ ][ ] [ ]nhnx

knhkx

knTkx

knkxTnxT

k

k

k

lowast=

minus=

minus=

minus=rArr

sum

sum

sum

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

δ

δ

scale


Time invariant

DigitalSystem

[ ]nδ [ ]nh

linear

Time-invariant

[ ] [ ][ ] [ ]knhkn

nhnT

T


⎯rarr⎯

δ

δ

Impulse response

convolution




bull Example

[ ]nδ 0 12

13

-2[ ]nh

LTI[ ]nx

0 1 2

23

1

0

3

0 12

39

-6

1

2

1 23

26

-4

LTI

2

1

2 34

1 3

-2

Length=M=3

Length=L=3

Length=L+M-1

[ ]nδsdot3

[ ]12 minussdot nδ

[ ]21 minussdot nδ

0

3

1

11

2

13

-1

4

-2

Sum up

[ ]ny

[ ]nhsdot3

[ ]12 minussdot nh

[ ]2minusnh






[ ]kh

[ ]kh minus

Reflect Multiply

slide

Sum up

n

[ ] [ ] [ ]

[ ] ( )[ ]nkhkx

knhkxny

k

k

minusminus=

minus=

sum

suminfin

minusinfin=

infin

minusinfin=


0 12

13

-2

0-1-2

13

-2

0 1 2

23

1

0

3

10-1

13

-21

11

210

13

-2

1

2

321

13

-2 -1

3

432

13

-2 -2

4

0

3

1

11

2

13

-1

4

-2

Sum up

[ ]kh

[ ]kx

[ ]kh minus

[ ] 0 =nny

[ ]1+minuskh

[ ]2+minuskh

[ ]3+minuskh

[ ]4+minuskh

[ ] 1 =nny

[ ] 2 =nny

[ ] 3 =nny

[ ] 4 =nny

[ ]ny

Reflect [ ] [ ] [ ][ ] [ ]knhkx

nhnxny

kminus=

lowast=

suminfin

minusinfin=

Convolution




[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=

[ ]nh 1 [ ]nh 2 [ ]nh 2 [ ]nh 1

[ ]nh 2

[ ]nh 1

[ ] [ ]nhnh 21 +



[ ] [ ] [ ][ ] [ ] [ ]nhnhnx

nhnhnx

12

21

=

[ ] [ ] [ ]( )[ ] [ ] [ ] [ ]nhnxnhnx

nhnhnx

21

21

+=+

Commutation

Distribution




[ ] [ ] [ ]

[ ] [ ]

[ ] [ ] ( )

[ ] [ ]nxnh

knmhmnx

knhkx

nhnxny

m

k

mlet

=

minus=sum minus=

minussum=

=

infin

minusinfin=

infin

minusinfin=




[ ] [ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] ( )0000

00011

0

101

cosBut

coscos

suppose

nnωnnxnny

nnnxnnxny

nnynynnxnx

minusminus=minus

minus==

minus=rArrminus=

ωω





[ ] infinlesuminfin

minusinfin=kkh

[ ][ ] nBny

nBnx

y

x

forallinfinltle

forallinfinltle






[ ] [ ] [ ]sum sum= =

minus+minus=K

k

M

mkkk mnxknyny

1000 βα

z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα






[ ]nh( )ωjeH

π2( )ωjeH

ω( ) ( ) ( )

( ) ( )ωω

ωωω

jeHjj

ji

jr

j

eeH

ejHeHeHang=

+=

magnitudephase

( )ωjeH






[ ] infinltsuminfin

minusinfin=nnh

[ ]( )

( )

[ ]mnnmnm

mnmn

emnj

de

mnj

mnj

minus=⎩⎨⎧

ne=

=

minusminus

=

minus= minus

minus

minusminus

int

δ

πππ

ωπ

ππ

ω

ππ

ω

0 1

sin)(2

121

)(

)(

( ) [ ]

[ ] ( ) [ ]


deemhdeeHnh

enheH

m

mnj

m

nj

m

mjnjj

n

njj

=minus==

==

=

sumintsum

int sumint

sum

infin

minusinfin=minus

minusinfin

minusinfin=

minus

infin

minusinfin=

minusminus

infin

minusinfin=

minus

δωπ

ωπ

ωπ

ππ

ω

ππ

ωωππ

ωω

ωω

21

21

21


)(

( ) [ ]

[ ] ( )int

sum

minus

infin

minusinfin=

minus

=

=

ππ

ωω

ωω

ωπ

deeHnh

enheH

njj

n

njj

2

1

absolutely summable

DTFT

Inverse DTFT



( ) [ ]

[ ] [ ] [ ]

( ) [ ] [ ]

[ ] [ ]

( ) ( )

[ ] ( ) ( )ωω

ωω

ωω

ωω

ωω

jj

jj

nj

nk

kj

nj

n k

j

k

n

njj

eHeXnhnx

eHeX

enhekx

eknhkxeY

knhkxnhnxny

enheH

hArrlowastthere4

=

⎟⎠

⎞⎜⎝

⎛=

minus=

minus=lowast=

=

minusinfin

minusinfin=

infin

minusinfin=

minus

minusinfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus

sumsum

sum sum

sum

sum

][

][][

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )ωωω

ωωω

ωωω

jjj

jjj

jjj

eHeXeY

eHeXeY

eHeXeY

ang+ang=angrArr

=rArr

=

knnknnknn






[ ] ( )intsum minus

infin

minusinfin== π

πω ω

πdeXnx j

n

22 21

power spectrum

[ ]nx[ ]

( ) ][][][][

][][

nxnxlnxlx

mxnmxnR

l

mxx


+=

lowastinfin

minusinfin=

infin

minusinfin=

sum

sum

( ) ( ) ( ) ( ) 2 ωωωω XXXS xx ==

hArr


π

π

ωπ

π

ω ωωπ

ωωπ

deXdeSnR njnjxxxx

2

21

21


minusinfin=minus

infin

minusinfin=

===

=

mmxx dXmxmxmxR

nπ

πωω

π22

210

0Set


)(

lnnlmnml



conjugatecomplex





[ ] ( )[ ] [ ] [ ] ( )

[ ] ( ) [ ]

[ ] ( )( )

( ) [ ]( ) ( )( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( )( )[ ] ( ) ( ) ( )( )ωω

ωωωω

φωω

ωφω

ωφω

ωφω

φω

φω

φω

φω

φωφω

φω

ω

ω

jj

jjjj

eHjnjj

eHjjnj

jnj

kj

k

nj

knj

k

nj

nj

eHneHAny

eHneHjAeHneHA

eeHA

eeHAe

eHAe

ekhAe

Aekh

nhAeny

Aenxnxnx

njAnx

j

j

ang++=rArr

ang+++ang++=

=

=

=

sum=

sum=

=

=+=

+=

ang++

ang+

+

minusinfin

minusinfin=

+

+minusinfin

minusinfin=

+

+

cos

sincos

sin

0

10

1

[ ]ny [ ]ny1

( )( ) attenuate 1

amplify 1

lt

gt

ω

ω

j

j

eH

eH





Z-Transform





( ) [ ]suminfin

minusinfin=

minus=n

nznhzH

[ ] ( )zHnh

[ ] ( )ωjeHnh

[ ]nh

ωjrez =

( ) ( ) ωω

jezj zHeH

==

)1( == zez jω

complex plane

unit circle

Im

Re

ω

ωjez =


Z-Transform (cont)


stability



[ ] infinltsuminfin

minusinfin=

minus

n

nznh

complex plan

R1

R2

Re

Im

absolutely summable

1=z


Z-Transform (cont)





[ ] 0for 0 lt= nnh

[ ] 0for 0 gt= nnh


Left-sided sequence

[ ] infinltsuminfin

minusinfin=nnh

[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


Z-Transform (cont)


[ ] [ ] [ ]⎩⎨⎧

ltge

==0for 00for 1

ere wh 1 1 nn

nunuanh n

( ) ( ) 11

1 11

minus

infin

minusinfin=

minusminusinfin


azazzazH

n

nn

n

n

If 11 ltminusaz

azROC gtthere4 is 1

Re

Im

a[ ] 1 if exists

ofansformFourier tr

1 ltanhtimes


az =

the unit cycle

0


Z-Transform (cont)



( ) [ ]

( ) 11

1

10

1

1

12

11

11111

1

minusminus

minus

minus

infin

=

minusinfin

=

minus

minusminus

minusinfin=

minusinfin

minusinfin=

minus=

minusminus

minus=minus

minus=minus=minus=


sumsum

sumsum

azzaza

zazaza

zaznuazH

n

n

n

nn

n

n

nn

n

n If 11 ltminus za

azROC ltthere4 is 2

Re

Im

a[ ] [ ]

minusinfinrarr

lt

nnhnh

a



22times

the unit cycle

0


Z-Transform (cont)


[ ] [ ] [ ]121


⎠⎞

⎜⎝⎛minus⎟

⎠⎞


nn

[ ]

[ ]21

211

1 121

31

311

1 31

1

1

ltminus


⎜⎝⎛minus

gt+


⎜⎝⎛ minus

minus

minus

zz

nu

zz

nu

zn

zn

Re

Im

times

the unit cycle

times21

31



3

3

ROCnh

31 and


( )⎟⎠⎞


⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ minus

=minus

++

=minusminus

21

31

1212

211

1

311

1 11

3

zz

zz

zzzH

0


Z-Transform (cont)



=others 0

10 3 4

Nnanh

n

( ) ( ) ( )azaz

zazazazzazH

NN

N

NN

n

nN

n

nn

minusminus

=minus

minus=== minusminus

minusminus

=

minusminus

=

minus sumsum 11

11

0

11

04

11

1



Im

times

the unit cycle

31

minus


4π

( ) 11 2

minus== N kaez Nkj

k

π

If N=8

0 N-1

Re


Z-Transform (cont)






[ ]nh [ ] 1 0 nnnh le=

01 gen

0rz gtz

infin=z

[ ]nh [ ] 2 0 nnnh ge=

02 ltn 0=z[ ]nh


01 geninfin=zthere4








[ ]nh

[ ] [ ] [ ] [ ]

[ ]

( ) njj

kjnj

knj

eeH

ekhe

ekhnhnxny

ωω

ωω

ω

=

=

=lowast=

minusinfin

infin=

minusinfin

infin=

sum

sum

-k

)(

-k( )



[ ] njenx ω=[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


( )jωeH

( )jωeH

scalar


[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjωnjjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

ang++=

+=

+=

+=

+minusangminus+ang

+minus+

minusminusminus

φω

φωφω

φωφω

ωφωφ




[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0

ωφωφ

φω

minusminus+=

+=

[ ]nh

( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

minus

minus

+=rArr

minus=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω

jωjω

eeHeH

eHeHangminus

minus

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos

sincos

jje

jejyxz

jejyxz

j

j

j

minus=minus+minus=

minus=rArrminus=

+=rArr+=

minus






[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2








DigitalSystem

[ ]nδ [ ]nh




[ ] [ ] [ ]knkxnxk

minus= suminfin

minusinfin=δ

[ ] [ ] [ ] [ ] [ ]

[ ] [ ][ ] [ ]nhnx

knhkx

knTkx

knkxTnxT

k

k

k

lowast=

minus=

minus=

minus=rArr

sum

sum

sum

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

δ

δ

scale


Time invariant

DigitalSystem

[ ]nδ [ ]nh

linear

Time-invariant

[ ] [ ][ ] [ ]knhkn

nhnT

T


⎯rarr⎯

δ

δ

Impulse response

convolution




bull Example

[ ]nδ 0 12

13

-2[ ]nh

LTI[ ]nx

0 1 2

23

1

0

3

0 12

39

-6

1

2

1 23

26

-4

LTI

2

1

2 34

1 3

-2

Length=M=3

Length=L=3

Length=L+M-1

[ ]nδsdot3

[ ]12 minussdot nδ

[ ]21 minussdot nδ

0

3

1

11

2

13

-1

4

-2

Sum up

[ ]ny

[ ]nhsdot3

[ ]12 minussdot nh

[ ]2minusnh






[ ]kh

[ ]kh minus

Reflect Multiply

slide

Sum up

n

[ ] [ ] [ ]

[ ] ( )[ ]nkhkx

knhkxny

k

k

minusminus=

minus=

sum

suminfin

minusinfin=

infin

minusinfin=


0 12

13

-2

0-1-2

13

-2

0 1 2

23

1

0

3

10-1

13

-21

11

210

13

-2

1

2

321

13

-2 -1

3

432

13

-2 -2

4

0

3

1

11

2

13

-1

4

-2

Sum up

[ ]kh

[ ]kx

[ ]kh minus

[ ] 0 =nny

[ ]1+minuskh

[ ]2+minuskh

[ ]3+minuskh

[ ]4+minuskh

[ ] 1 =nny

[ ] 2 =nny

[ ] 3 =nny

[ ] 4 =nny

[ ]ny

Reflect [ ] [ ] [ ][ ] [ ]knhkx

nhnxny

kminus=

lowast=

suminfin

minusinfin=

Convolution




[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=

[ ]nh 1 [ ]nh 2 [ ]nh 2 [ ]nh 1

[ ]nh 2

[ ]nh 1

[ ] [ ]nhnh 21 +



[ ] [ ] [ ][ ] [ ] [ ]nhnhnx

nhnhnx

12

21

=

[ ] [ ] [ ]( )[ ] [ ] [ ] [ ]nhnxnhnx

nhnhnx

21

21

+=+

Commutation

Distribution




[ ] [ ] [ ]

[ ] [ ]

[ ] [ ] ( )

[ ] [ ]nxnh

knmhmnx

knhkx

nhnxny

m

k

mlet

=

minus=sum minus=

minussum=

=

infin

minusinfin=

infin

minusinfin=




[ ] [ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] ( )0000

00011

0

101

cosBut

coscos

suppose

nnωnnxnny

nnnxnnxny

nnynynnxnx

minusminus=minus

minus==

minus=rArrminus=

ωω





[ ] infinlesuminfin

minusinfin=kkh

[ ][ ] nBny

nBnx

y

x

forallinfinltle

forallinfinltle






[ ] [ ] [ ]sum sum= =

minus+minus=K

k

M

mkkk mnxknyny

1000 βα

z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα






[ ]nh( )ωjeH

π2( )ωjeH

ω( ) ( ) ( )

( ) ( )ωω

ωωω

jeHjj

ji

jr

j

eeH

ejHeHeHang=

+=

magnitudephase

( )ωjeH






[ ] infinltsuminfin

minusinfin=nnh

[ ]( )

( )

[ ]mnnmnm

mnmn

emnj

de

mnj

mnj

minus=⎩⎨⎧

ne=

=

minusminus

=

minus= minus

minus

minusminus

int

δ

πππ

ωπ

ππ

ω

ππ

ω

0 1

sin)(2

121

)(

)(

( ) [ ]

[ ] ( ) [ ]


deemhdeeHnh

enheH

m

mnj

m

nj

m

mjnjj

n

njj

=minus==

==

=

sumintsum

int sumint

sum

infin

minusinfin=minus

minusinfin

minusinfin=

minus

infin

minusinfin=

minusminus

infin

minusinfin=

minus

δωπ

ωπ

ωπ

ππ

ω

ππ

ωωππ

ωω

ωω

21

21

21


)(

( ) [ ]

[ ] ( )int

sum

minus

infin

minusinfin=

minus

=

=

ππ

ωω

ωω

ωπ

deeHnh

enheH

njj

n

njj

2

1

absolutely summable

DTFT

Inverse DTFT



( ) [ ]

[ ] [ ] [ ]

( ) [ ] [ ]

[ ] [ ]

( ) ( )

[ ] ( ) ( )ωω

ωω

ωω

ωω

ωω

jj

jj

nj

nk

kj

nj

n k

j

k

n

njj

eHeXnhnx

eHeX

enhekx

eknhkxeY

knhkxnhnxny

enheH

hArrlowastthere4

=

⎟⎠

⎞⎜⎝

⎛=

minus=

minus=lowast=

=

minusinfin

minusinfin=

infin

minusinfin=

minus

minusinfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus

sumsum

sum sum

sum

sum

][

][][

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )ωωω

ωωω

ωωω

jjj

jjj

jjj

eHeXeY

eHeXeY

eHeXeY

ang+ang=angrArr

=rArr

=

knnknnknn






[ ] ( )intsum minus

infin

minusinfin== π

πω ω

πdeXnx j

n

22 21

power spectrum

[ ]nx[ ]

( ) ][][][][

][][

nxnxlnxlx

mxnmxnR

l

mxx


+=

lowastinfin

minusinfin=

infin

minusinfin=

sum

sum

( ) ( ) ( ) ( ) 2 ωωωω XXXS xx ==

hArr


π

π

ωπ

π

ω ωωπ

ωωπ

deXdeSnR njnjxxxx

2

21

21


minusinfin=minus

infin

minusinfin=

===

=

mmxx dXmxmxmxR

nπ

πωω

π22

210

0Set


)(

lnnlmnml



conjugatecomplex





[ ] ( )[ ] [ ] [ ] ( )

[ ] ( ) [ ]

[ ] ( )( )

( ) [ ]( ) ( )( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( )( )[ ] ( ) ( ) ( )( )ωω

ωωωω

φωω

ωφω

ωφω

ωφω

φω

φω

φω

φω

φωφω

φω

ω

ω

jj

jjjj

eHjnjj

eHjjnj

jnj

kj

k

nj

knj

k

nj

nj

eHneHAny

eHneHjAeHneHA

eeHA

eeHAe

eHAe

ekhAe

Aekh

nhAeny

Aenxnxnx

njAnx

j

j

ang++=rArr

ang+++ang++=

=

=

=

sum=

sum=

=

=+=

+=

ang++

ang+

+

minusinfin

minusinfin=

+

+minusinfin

minusinfin=

+

+

cos

sincos

sin

0

10

1

[ ]ny [ ]ny1

( )( ) attenuate 1

amplify 1

lt

gt

ω

ω

j

j

eH

eH





Z-Transform





( ) [ ]suminfin

minusinfin=

minus=n

nznhzH

[ ] ( )zHnh

[ ] ( )ωjeHnh

[ ]nh

ωjrez =

( ) ( ) ωω

jezj zHeH

==

)1( == zez jω

complex plane

unit circle

Im

Re

ω

ωjez =


Z-Transform (cont)


stability



[ ] infinltsuminfin

minusinfin=

minus

n

nznh

complex plan

R1

R2

Re

Im

absolutely summable

1=z


Z-Transform (cont)





[ ] 0for 0 lt= nnh

[ ] 0for 0 gt= nnh


Left-sided sequence

[ ] infinltsuminfin

minusinfin=nnh

[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


Z-Transform (cont)


[ ] [ ] [ ]⎩⎨⎧

ltge

==0for 00for 1

ere wh 1 1 nn

nunuanh n

( ) ( ) 11

1 11

minus

infin

minusinfin=

minusminusinfin


azazzazH

n

nn

n

n

If 11 ltminusaz

azROC gtthere4 is 1

Re

Im

a[ ] 1 if exists

ofansformFourier tr

1 ltanhtimes


az =

the unit cycle

0


Z-Transform (cont)



( ) [ ]

( ) 11

1

10

1

1

12

11

11111

1

minusminus

minus

minus

infin

=

minusinfin

=

minus

minusminus

minusinfin=

minusinfin

minusinfin=

minus=

minusminus

minus=minus

minus=minus=minus=


sumsum

sumsum

azzaza

zazaza

zaznuazH

n

n

n

nn

n

n

nn

n

n If 11 ltminus za

azROC ltthere4 is 2

Re

Im

a[ ] [ ]

minusinfinrarr

lt

nnhnh

a



22times

the unit cycle

0


Z-Transform (cont)


[ ] [ ] [ ]121


⎠⎞

⎜⎝⎛minus⎟

⎠⎞


nn

[ ]

[ ]21

211

1 121

31

311

1 31

1

1

ltminus


⎜⎝⎛minus

gt+


⎜⎝⎛ minus

minus

minus

zz

nu

zz

nu

zn

zn

Re

Im

times

the unit cycle

times21

31



3

3

ROCnh

31 and


( )⎟⎠⎞


⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ minus

=minus

++

=minusminus

21

31

1212

211

1

311

1 11

3

zz

zz

zzzH

0


Z-Transform (cont)



=others 0

10 3 4

Nnanh

n

( ) ( ) ( )azaz

zazazazzazH

NN

N

NN

n

nN

n

nn

minusminus

=minus

minus=== minusminus

minusminus

=

minusminus

=

minus sumsum 11

11

0

11

04

11

1



Im

times

the unit cycle

31

minus


4π

( ) 11 2

minus== N kaez Nkj

k

π

If N=8

0 N-1

Re


Z-Transform (cont)






[ ]nh [ ] 1 0 nnnh le=

01 gen

0rz gtz

infin=z

[ ]nh [ ] 2 0 nnnh ge=

02 ltn 0=z[ ]nh


01 geninfin=zthere4








[ ]nh

[ ] [ ] [ ] [ ]

[ ]

( ) njj

kjnj

knj

eeH

ekhe

ekhnhnxny

ωω

ωω

ω

=

=

=lowast=

minusinfin

infin=

minusinfin

infin=

sum

sum

-k

)(

-k( )



[ ] njenx ω=[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


( )jωeH

( )jωeH

scalar


[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjωnjjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

ang++=

+=

+=

+=

+minusangminus+ang

+minus+

minusminusminus

φω

φωφω

φωφω

ωφωφ




[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0

ωφωφ

φω

minusminus+=

+=

[ ]nh

( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

minus

minus

+=rArr

minus=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω

jωjω

eeHeH

eHeHangminus

minus

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos

sincos

jje

jejyxz

jejyxz

j

j

j

minus=minus+minus=

minus=rArrminus=

+=rArr+=

minus






[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2





[ ] [ ] [ ]knkxnxk

minus= suminfin

minusinfin=δ

[ ] [ ] [ ] [ ] [ ]

[ ] [ ][ ] [ ]nhnx

knhkx

knTkx

knkxTnxT

k

k

k

lowast=

minus=

minus=

minus=rArr

sum

sum

sum

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

δ

δ

scale


Time invariant

DigitalSystem

[ ]nδ [ ]nh

linear

Time-invariant

[ ] [ ][ ] [ ]knhkn

nhnT

T


⎯rarr⎯

δ

δ

Impulse response

convolution




bull Example

[ ]nδ 0 12

13

-2[ ]nh

LTI[ ]nx

0 1 2

23

1

0

3

0 12

39

-6

1

2

1 23

26

-4

LTI

2

1

2 34

1 3

-2

Length=M=3

Length=L=3

Length=L+M-1

[ ]nδsdot3

[ ]12 minussdot nδ

[ ]21 minussdot nδ

0

3

1

11

2

13

-1

4

-2

Sum up

[ ]ny

[ ]nhsdot3

[ ]12 minussdot nh

[ ]2minusnh






[ ]kh

[ ]kh minus

Reflect Multiply

slide

Sum up

n

[ ] [ ] [ ]

[ ] ( )[ ]nkhkx

knhkxny

k

k

minusminus=

minus=

sum

suminfin

minusinfin=

infin

minusinfin=


0 12

13

-2

0-1-2

13

-2

0 1 2

23

1

0

3

10-1

13

-21

11

210

13

-2

1

2

321

13

-2 -1

3

432

13

-2 -2

4

0

3

1

11

2

13

-1

4

-2

Sum up

[ ]kh

[ ]kx

[ ]kh minus

[ ] 0 =nny

[ ]1+minuskh

[ ]2+minuskh

[ ]3+minuskh

[ ]4+minuskh

[ ] 1 =nny

[ ] 2 =nny

[ ] 3 =nny

[ ] 4 =nny

[ ]ny

Reflect [ ] [ ] [ ][ ] [ ]knhkx

nhnxny

kminus=

lowast=

suminfin

minusinfin=

Convolution




[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=

[ ]nh 1 [ ]nh 2 [ ]nh 2 [ ]nh 1

[ ]nh 2

[ ]nh 1

[ ] [ ]nhnh 21 +



[ ] [ ] [ ][ ] [ ] [ ]nhnhnx

nhnhnx

12

21

=

[ ] [ ] [ ]( )[ ] [ ] [ ] [ ]nhnxnhnx

nhnhnx

21

21

+=+

Commutation

Distribution




[ ] [ ] [ ]

[ ] [ ]

[ ] [ ] ( )

[ ] [ ]nxnh

knmhmnx

knhkx

nhnxny

m

k

mlet

=

minus=sum minus=

minussum=

=

infin

minusinfin=

infin

minusinfin=




[ ] [ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] ( )0000

00011

0

101

cosBut

coscos

suppose

nnωnnxnny

nnnxnnxny

nnynynnxnx

minusminus=minus

minus==

minus=rArrminus=

ωω





[ ] infinlesuminfin

minusinfin=kkh

[ ][ ] nBny

nBnx

y

x

forallinfinltle

forallinfinltle






[ ] [ ] [ ]sum sum= =

minus+minus=K

k

M

mkkk mnxknyny

1000 βα

z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα






[ ]nh( )ωjeH

π2( )ωjeH

ω( ) ( ) ( )

( ) ( )ωω

ωωω

jeHjj

ji

jr

j

eeH

ejHeHeHang=

+=

magnitudephase

( )ωjeH






[ ] infinltsuminfin

minusinfin=nnh

[ ]( )

( )

[ ]mnnmnm

mnmn

emnj

de

mnj

mnj

minus=⎩⎨⎧

ne=

=

minusminus

=

minus= minus

minus

minusminus

int

δ

πππ

ωπ

ππ

ω

ππ

ω

0 1

sin)(2

121

)(

)(

( ) [ ]

[ ] ( ) [ ]


deemhdeeHnh

enheH

m

mnj

m

nj

m

mjnjj

n

njj

=minus==

==

=

sumintsum

int sumint

sum

infin

minusinfin=minus

minusinfin

minusinfin=

minus

infin

minusinfin=

minusminus

infin

minusinfin=

minus

δωπ

ωπ

ωπ

ππ

ω

ππ

ωωππ

ωω

ωω

21

21

21


)(

( ) [ ]

[ ] ( )int

sum

minus

infin

minusinfin=

minus

=

=

ππ

ωω

ωω

ωπ

deeHnh

enheH

njj

n

njj

2

1

absolutely summable

DTFT

Inverse DTFT



( ) [ ]

[ ] [ ] [ ]

( ) [ ] [ ]

[ ] [ ]

( ) ( )

[ ] ( ) ( )ωω

ωω

ωω

ωω

ωω

jj

jj

nj

nk

kj

nj

n k

j

k

n

njj

eHeXnhnx

eHeX

enhekx

eknhkxeY

knhkxnhnxny

enheH

hArrlowastthere4

=

⎟⎠

⎞⎜⎝

⎛=

minus=

minus=lowast=

=

minusinfin

minusinfin=

infin

minusinfin=

minus

minusinfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus

sumsum

sum sum

sum

sum

][

][][

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )ωωω

ωωω

ωωω

jjj

jjj

jjj

eHeXeY

eHeXeY

eHeXeY

ang+ang=angrArr

=rArr

=

knnknnknn






[ ] ( )intsum minus

infin

minusinfin== π

πω ω

πdeXnx j

n

22 21

power spectrum

[ ]nx[ ]

( ) ][][][][

][][

nxnxlnxlx

mxnmxnR

l

mxx


+=

lowastinfin

minusinfin=

infin

minusinfin=

sum

sum

( ) ( ) ( ) ( ) 2 ωωωω XXXS xx ==

hArr


π

π

ωπ

π

ω ωωπ

ωωπ

deXdeSnR njnjxxxx

2

21

21


minusinfin=minus

infin

minusinfin=

===

=

mmxx dXmxmxmxR

nπ

πωω

π22

210

0Set


)(

lnnlmnml



conjugatecomplex





[ ] ( )[ ] [ ] [ ] ( )

[ ] ( ) [ ]

[ ] ( )( )

( ) [ ]( ) ( )( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( )( )[ ] ( ) ( ) ( )( )ωω

ωωωω

φωω

ωφω

ωφω

ωφω

φω

φω

φω

φω

φωφω

φω

ω

ω

jj

jjjj

eHjnjj

eHjjnj

jnj

kj

k

nj

knj

k

nj

nj

eHneHAny

eHneHjAeHneHA

eeHA

eeHAe

eHAe

ekhAe

Aekh

nhAeny

Aenxnxnx

njAnx

j

j

ang++=rArr

ang+++ang++=

=

=

=

sum=

sum=

=

=+=

+=

ang++

ang+

+

minusinfin

minusinfin=

+

+minusinfin

minusinfin=

+

+

cos

sincos

sin

0

10

1

[ ]ny [ ]ny1

( )( ) attenuate 1

amplify 1

lt

gt

ω

ω

j

j

eH

eH





Z-Transform





( ) [ ]suminfin

minusinfin=

minus=n

nznhzH

[ ] ( )zHnh

[ ] ( )ωjeHnh

[ ]nh

ωjrez =

( ) ( ) ωω

jezj zHeH

==

)1( == zez jω

complex plane

unit circle

Im

Re

ω

ωjez =


Z-Transform (cont)


stability



[ ] infinltsuminfin

minusinfin=

minus

n

nznh

complex plan

R1

R2

Re

Im

absolutely summable

1=z


Z-Transform (cont)





[ ] 0for 0 lt= nnh

[ ] 0for 0 gt= nnh


Left-sided sequence

[ ] infinltsuminfin

minusinfin=nnh

[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


Z-Transform (cont)


[ ] [ ] [ ]⎩⎨⎧

ltge

==0for 00for 1

ere wh 1 1 nn

nunuanh n

( ) ( ) 11

1 11

minus

infin

minusinfin=

minusminusinfin


azazzazH

n

nn

n

n

If 11 ltminusaz

azROC gtthere4 is 1

Re

Im

a[ ] 1 if exists

ofansformFourier tr

1 ltanhtimes


az =

the unit cycle

0


Z-Transform (cont)



( ) [ ]

( ) 11

1

10

1

1

12

11

11111

1

minusminus

minus

minus

infin

=

minusinfin

=

minus

minusminus

minusinfin=

minusinfin

minusinfin=

minus=

minusminus

minus=minus

minus=minus=minus=


sumsum

sumsum

azzaza

zazaza

zaznuazH

n

n

n

nn

n

n

nn

n

n If 11 ltminus za

azROC ltthere4 is 2

Re

Im

a[ ] [ ]

minusinfinrarr

lt

nnhnh

a



22times

the unit cycle

0


Z-Transform (cont)


[ ] [ ] [ ]121


⎠⎞

⎜⎝⎛minus⎟

⎠⎞


nn

[ ]

[ ]21

211

1 121

31

311

1 31

1

1

ltminus


⎜⎝⎛minus

gt+


⎜⎝⎛ minus

minus

minus

zz

nu

zz

nu

zn

zn

Re

Im

times

the unit cycle

times21

31



3

3

ROCnh

31 and


( )⎟⎠⎞


⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ minus

=minus

++

=minusminus

21

31

1212

211

1

311

1 11

3

zz

zz

zzzH

0


Z-Transform (cont)



=others 0

10 3 4

Nnanh

n

( ) ( ) ( )azaz

zazazazzazH

NN

N

NN

n

nN

n

nn

minusminus

=minus

minus=== minusminus

minusminus

=

minusminus

=

minus sumsum 11

11

0

11

04

11

1



Im

times

the unit cycle

31

minus


4π

( ) 11 2

minus== N kaez Nkj

k

π

If N=8

0 N-1

Re


Z-Transform (cont)






[ ]nh [ ] 1 0 nnnh le=

01 gen

0rz gtz

infin=z

[ ]nh [ ] 2 0 nnnh ge=

02 ltn 0=z[ ]nh


01 geninfin=zthere4








[ ]nh

[ ] [ ] [ ] [ ]

[ ]

( ) njj

kjnj

knj

eeH

ekhe

ekhnhnxny

ωω

ωω

ω

=

=

=lowast=

minusinfin

infin=

minusinfin

infin=

sum

sum

-k

)(

-k( )



[ ] njenx ω=[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


( )jωeH

( )jωeH

scalar


[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjωnjjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

ang++=

+=

+=

+=

+minusangminus+ang

+minus+

minusminusminus

φω

φωφω

φωφω

ωφωφ




[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0

ωφωφ

φω

minusminus+=

+=

[ ]nh

( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

minus

minus

+=rArr

minus=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω

jωjω

eeHeH

eHeHangminus

minus

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos

sincos

jje

jejyxz

jejyxz

j

j

j

minus=minus+minus=

minus=rArrminus=

+=rArr+=

minus






[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2





bull Example

[ ]nδ 0 12

13

-2[ ]nh

LTI[ ]nx

0 1 2

23

1

0

3

0 12

39

-6

1

2

1 23

26

-4

LTI

2

1

2 34

1 3

-2

Length=M=3

Length=L=3

Length=L+M-1

[ ]nδsdot3

[ ]12 minussdot nδ

[ ]21 minussdot nδ

0

3

1

11

2

13

-1

4

-2

Sum up

[ ]ny

[ ]nhsdot3

[ ]12 minussdot nh

[ ]2minusnh






[ ]kh

[ ]kh minus

Reflect Multiply

slide

Sum up

n

[ ] [ ] [ ]

[ ] ( )[ ]nkhkx

knhkxny

k

k

minusminus=

minus=

sum

suminfin

minusinfin=

infin

minusinfin=


0 12

13

-2

0-1-2

13

-2

0 1 2

23

1

0

3

10-1

13

-21

11

210

13

-2

1

2

321

13

-2 -1

3

432

13

-2 -2

4

0

3

1

11

2

13

-1

4

-2

Sum up

[ ]kh

[ ]kx

[ ]kh minus

[ ] 0 =nny

[ ]1+minuskh

[ ]2+minuskh

[ ]3+minuskh

[ ]4+minuskh

[ ] 1 =nny

[ ] 2 =nny

[ ] 3 =nny

[ ] 4 =nny

[ ]ny

Reflect [ ] [ ] [ ][ ] [ ]knhkx

nhnxny

kminus=

lowast=

suminfin

minusinfin=

Convolution




[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=

[ ]nh 1 [ ]nh 2 [ ]nh 2 [ ]nh 1

[ ]nh 2

[ ]nh 1

[ ] [ ]nhnh 21 +



[ ] [ ] [ ][ ] [ ] [ ]nhnhnx

nhnhnx

12

21

=

[ ] [ ] [ ]( )[ ] [ ] [ ] [ ]nhnxnhnx

nhnhnx

21

21

+=+

Commutation

Distribution




[ ] [ ] [ ]

[ ] [ ]

[ ] [ ] ( )

[ ] [ ]nxnh

knmhmnx

knhkx

nhnxny

m

k

mlet

=

minus=sum minus=

minussum=

=

infin

minusinfin=

infin

minusinfin=




[ ] [ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] ( )0000

00011

0

101

cosBut

coscos

suppose

nnωnnxnny

nnnxnnxny

nnynynnxnx

minusminus=minus

minus==

minus=rArrminus=

ωω





[ ] infinlesuminfin

minusinfin=kkh

[ ][ ] nBny

nBnx

y

x

forallinfinltle

forallinfinltle






[ ] [ ] [ ]sum sum= =

minus+minus=K

k

M

mkkk mnxknyny

1000 βα

z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα






[ ]nh( )ωjeH

π2( )ωjeH

ω( ) ( ) ( )

( ) ( )ωω

ωωω

jeHjj

ji

jr

j

eeH

ejHeHeHang=

+=

magnitudephase

( )ωjeH






[ ] infinltsuminfin

minusinfin=nnh

[ ]( )

( )

[ ]mnnmnm

mnmn

emnj

de

mnj

mnj

minus=⎩⎨⎧

ne=

=

minusminus

=

minus= minus

minus

minusminus

int

δ

πππ

ωπ

ππ

ω

ππ

ω

0 1

sin)(2

121

)(

)(

( ) [ ]

[ ] ( ) [ ]


deemhdeeHnh

enheH

m

mnj

m

nj

m

mjnjj

n

njj

=minus==

==

=

sumintsum

int sumint

sum

infin

minusinfin=minus

minusinfin

minusinfin=

minus

infin

minusinfin=

minusminus

infin

minusinfin=

minus

δωπ

ωπ

ωπ

ππ

ω

ππ

ωωππ

ωω

ωω

21

21

21


)(

( ) [ ]

[ ] ( )int

sum

minus

infin

minusinfin=

minus

=

=

ππ

ωω

ωω

ωπ

deeHnh

enheH

njj

n

njj

2

1

absolutely summable

DTFT

Inverse DTFT



( ) [ ]

[ ] [ ] [ ]

( ) [ ] [ ]

[ ] [ ]

( ) ( )

[ ] ( ) ( )ωω

ωω

ωω

ωω

ωω

jj

jj

nj

nk

kj

nj

n k

j

k

n

njj

eHeXnhnx

eHeX

enhekx

eknhkxeY

knhkxnhnxny

enheH

hArrlowastthere4

=

⎟⎠

⎞⎜⎝

⎛=

minus=

minus=lowast=

=

minusinfin

minusinfin=

infin

minusinfin=

minus

minusinfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus

sumsum

sum sum

sum

sum

][

][][

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )ωωω

ωωω

ωωω

jjj

jjj

jjj

eHeXeY

eHeXeY

eHeXeY

ang+ang=angrArr

=rArr

=

knnknnknn






[ ] ( )intsum minus

infin

minusinfin== π

πω ω

πdeXnx j

n

22 21

power spectrum

[ ]nx[ ]

( ) ][][][][

][][

nxnxlnxlx

mxnmxnR

l

mxx


+=

lowastinfin

minusinfin=

infin

minusinfin=

sum

sum

( ) ( ) ( ) ( ) 2 ωωωω XXXS xx ==

hArr


π

π

ωπ

π

ω ωωπ

ωωπ

deXdeSnR njnjxxxx

2

21

21


minusinfin=minus

infin

minusinfin=

===

=

mmxx dXmxmxmxR

nπ

πωω

π22

210

0Set


)(

lnnlmnml



conjugatecomplex





[ ] ( )[ ] [ ] [ ] ( )

[ ] ( ) [ ]

[ ] ( )( )

( ) [ ]( ) ( )( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( )( )[ ] ( ) ( ) ( )( )ωω

ωωωω

φωω

ωφω

ωφω

ωφω

φω

φω

φω

φω

φωφω

φω

ω

ω

jj

jjjj

eHjnjj

eHjjnj

jnj

kj

k

nj

knj

k

nj

nj

eHneHAny

eHneHjAeHneHA

eeHA

eeHAe

eHAe

ekhAe

Aekh

nhAeny

Aenxnxnx

njAnx

j

j

ang++=rArr

ang+++ang++=

=

=

=

sum=

sum=

=

=+=

+=

ang++

ang+

+

minusinfin

minusinfin=

+

+minusinfin

minusinfin=

+

+

cos

sincos

sin

0

10

1

[ ]ny [ ]ny1

( )( ) attenuate 1

amplify 1

lt

gt

ω

ω

j

j

eH

eH





Z-Transform





( ) [ ]suminfin

minusinfin=

minus=n

nznhzH

[ ] ( )zHnh

[ ] ( )ωjeHnh

[ ]nh

ωjrez =

( ) ( ) ωω

jezj zHeH

==

)1( == zez jω

complex plane

unit circle

Im

Re

ω

ωjez =


Z-Transform (cont)


stability



[ ] infinltsuminfin

minusinfin=

minus

n

nznh

complex plan

R1

R2

Re

Im

absolutely summable

1=z


Z-Transform (cont)





[ ] 0for 0 lt= nnh

[ ] 0for 0 gt= nnh


Left-sided sequence

[ ] infinltsuminfin

minusinfin=nnh

[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


Z-Transform (cont)


[ ] [ ] [ ]⎩⎨⎧

ltge

==0for 00for 1

ere wh 1 1 nn

nunuanh n

( ) ( ) 11

1 11

minus

infin

minusinfin=

minusminusinfin


azazzazH

n

nn

n

n

If 11 ltminusaz

azROC gtthere4 is 1

Re

Im

a[ ] 1 if exists

ofansformFourier tr

1 ltanhtimes


az =

the unit cycle

0


Z-Transform (cont)



( ) [ ]

( ) 11

1

10

1

1

12

11

11111

1

minusminus

minus

minus

infin

=

minusinfin

=

minus

minusminus

minusinfin=

minusinfin

minusinfin=

minus=

minusminus

minus=minus

minus=minus=minus=


sumsum

sumsum

azzaza

zazaza

zaznuazH

n

n

n

nn

n

n

nn

n

n If 11 ltminus za

azROC ltthere4 is 2

Re

Im

a[ ] [ ]

minusinfinrarr

lt

nnhnh

a



22times

the unit cycle

0


Z-Transform (cont)


[ ] [ ] [ ]121


⎠⎞

⎜⎝⎛minus⎟

⎠⎞


nn

[ ]

[ ]21

211

1 121

31

311

1 31

1

1

ltminus


⎜⎝⎛minus

gt+


⎜⎝⎛ minus

minus

minus

zz

nu

zz

nu

zn

zn

Re

Im

times

the unit cycle

times21

31



3

3

ROCnh

31 and


( )⎟⎠⎞


⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ minus

=minus

++

=minusminus

21

31

1212

211

1

311

1 11

3

zz

zz

zzzH

0


Z-Transform (cont)



=others 0

10 3 4

Nnanh

n

( ) ( ) ( )azaz

zazazazzazH

NN

N

NN

n

nN

n

nn

minusminus

=minus

minus=== minusminus

minusminus

=

minusminus

=

minus sumsum 11

11

0

11

04

11

1



Im

times

the unit cycle

31

minus


4π

( ) 11 2

minus== N kaez Nkj

k

π

If N=8

0 N-1

Re


Z-Transform (cont)






[ ]nh [ ] 1 0 nnnh le=

01 gen

0rz gtz

infin=z

[ ]nh [ ] 2 0 nnnh ge=

02 ltn 0=z[ ]nh


01 geninfin=zthere4








[ ]nh

[ ] [ ] [ ] [ ]

[ ]

( ) njj

kjnj

knj

eeH

ekhe

ekhnhnxny

ωω

ωω

ω

=

=

=lowast=

minusinfin

infin=

minusinfin

infin=

sum

sum

-k

)(

-k( )



[ ] njenx ω=[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


( )jωeH

( )jωeH

scalar


[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjωnjjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

ang++=

+=

+=

+=

+minusangminus+ang

+minus+

minusminusminus

φω

φωφω

φωφω

ωφωφ




[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0

ωφωφ

φω

minusminus+=

+=

[ ]nh

( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

minus

minus

+=rArr

minus=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω

jωjω

eeHeH

eHeHangminus

minus

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos

sincos

jje

jejyxz

jejyxz

j

j

j

minus=minus+minus=

minus=rArrminus=

+=rArr+=

minus






[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2







[ ]kh

[ ]kh minus

Reflect Multiply

slide

Sum up

n

[ ] [ ] [ ]

[ ] ( )[ ]nkhkx

knhkxny

k

k

minusminus=

minus=

sum

suminfin

minusinfin=

infin

minusinfin=


0 12

13

-2

0-1-2

13

-2

0 1 2

23

1

0

3

10-1

13

-21

11

210

13

-2

1

2

321

13

-2 -1

3

432

13

-2 -2

4

0

3

1

11

2

13

-1

4

-2

Sum up

[ ]kh

[ ]kx

[ ]kh minus

[ ] 0 =nny

[ ]1+minuskh

[ ]2+minuskh

[ ]3+minuskh

[ ]4+minuskh

[ ] 1 =nny

[ ] 2 =nny

[ ] 3 =nny

[ ] 4 =nny

[ ]ny

Reflect [ ] [ ] [ ][ ] [ ]knhkx

nhnxny

kminus=

lowast=

suminfin

minusinfin=

Convolution




[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=

[ ]nh 1 [ ]nh 2 [ ]nh 2 [ ]nh 1

[ ]nh 2

[ ]nh 1

[ ] [ ]nhnh 21 +



[ ] [ ] [ ][ ] [ ] [ ]nhnhnx

nhnhnx

12

21

=

[ ] [ ] [ ]( )[ ] [ ] [ ] [ ]nhnxnhnx

nhnhnx

21

21

+=+

Commutation

Distribution




[ ] [ ] [ ]

[ ] [ ]

[ ] [ ] ( )

[ ] [ ]nxnh

knmhmnx

knhkx

nhnxny

m

k

mlet

=

minus=sum minus=

minussum=

=

infin

minusinfin=

infin

minusinfin=




[ ] [ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] ( )0000

00011

0

101

cosBut

coscos

suppose

nnωnnxnny

nnnxnnxny

nnynynnxnx

minusminus=minus

minus==

minus=rArrminus=

ωω





[ ] infinlesuminfin

minusinfin=kkh

[ ][ ] nBny

nBnx

y

x

forallinfinltle

forallinfinltle






[ ] [ ] [ ]sum sum= =

minus+minus=K

k

M

mkkk mnxknyny

1000 βα

z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα






[ ]nh( )ωjeH

π2( )ωjeH

ω( ) ( ) ( )

( ) ( )ωω

ωωω

jeHjj

ji

jr

j

eeH

ejHeHeHang=

+=

magnitudephase

( )ωjeH






[ ] infinltsuminfin

minusinfin=nnh

[ ]( )

( )

[ ]mnnmnm

mnmn

emnj

de

mnj

mnj

minus=⎩⎨⎧

ne=

=

minusminus

=

minus= minus

minus

minusminus

int

δ

πππ

ωπ

ππ

ω

ππ

ω

0 1

sin)(2

121

)(

)(

( ) [ ]

[ ] ( ) [ ]


deemhdeeHnh

enheH

m

mnj

m

nj

m

mjnjj

n

njj

=minus==

==

=

sumintsum

int sumint

sum

infin

minusinfin=minus

minusinfin

minusinfin=

minus

infin

minusinfin=

minusminus

infin

minusinfin=

minus

δωπ

ωπ

ωπ

ππ

ω

ππ

ωωππ

ωω

ωω

21

21

21


)(

( ) [ ]

[ ] ( )int

sum

minus

infin

minusinfin=

minus

=

=

ππ

ωω

ωω

ωπ

deeHnh

enheH

njj

n

njj

2

1

absolutely summable

DTFT

Inverse DTFT



( ) [ ]

[ ] [ ] [ ]

( ) [ ] [ ]

[ ] [ ]

( ) ( )

[ ] ( ) ( )ωω

ωω

ωω

ωω

ωω

jj

jj

nj

nk

kj

nj

n k

j

k

n

njj

eHeXnhnx

eHeX

enhekx

eknhkxeY

knhkxnhnxny

enheH

hArrlowastthere4

=

⎟⎠

⎞⎜⎝

⎛=

minus=

minus=lowast=

=

minusinfin

minusinfin=

infin

minusinfin=

minus

minusinfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus

sumsum

sum sum

sum

sum

][

][][

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )ωωω

ωωω

ωωω

jjj

jjj

jjj

eHeXeY

eHeXeY

eHeXeY

ang+ang=angrArr

=rArr

=

knnknnknn






[ ] ( )intsum minus

infin

minusinfin== π

πω ω

πdeXnx j

n

22 21

power spectrum

[ ]nx[ ]

( ) ][][][][

][][

nxnxlnxlx

mxnmxnR

l

mxx


+=

lowastinfin

minusinfin=

infin

minusinfin=

sum

sum

( ) ( ) ( ) ( ) 2 ωωωω XXXS xx ==

hArr


π

π

ωπ

π

ω ωωπ

ωωπ

deXdeSnR njnjxxxx

2

21

21


minusinfin=minus

infin

minusinfin=

===

=

mmxx dXmxmxmxR

nπ

πωω

π22

210

0Set


)(

lnnlmnml



conjugatecomplex





[ ] ( )[ ] [ ] [ ] ( )

[ ] ( ) [ ]

[ ] ( )( )

( ) [ ]( ) ( )( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( )( )[ ] ( ) ( ) ( )( )ωω

ωωωω

φωω

ωφω

ωφω

ωφω

φω

φω

φω

φω

φωφω

φω

ω

ω

jj

jjjj

eHjnjj

eHjjnj

jnj

kj

k

nj

knj

k

nj

nj

eHneHAny

eHneHjAeHneHA

eeHA

eeHAe

eHAe

ekhAe

Aekh

nhAeny

Aenxnxnx

njAnx

j

j

ang++=rArr

ang+++ang++=

=

=

=

sum=

sum=

=

=+=

+=

ang++

ang+

+

minusinfin

minusinfin=

+

+minusinfin

minusinfin=

+

+

cos

sincos

sin

0

10

1

[ ]ny [ ]ny1

( )( ) attenuate 1

amplify 1

lt

gt

ω

ω

j

j

eH

eH





Z-Transform





( ) [ ]suminfin

minusinfin=

minus=n

nznhzH

[ ] ( )zHnh

[ ] ( )ωjeHnh

[ ]nh

ωjrez =

( ) ( ) ωω

jezj zHeH

==

)1( == zez jω

complex plane

unit circle

Im

Re

ω

ωjez =


Z-Transform (cont)


stability



[ ] infinltsuminfin

minusinfin=

minus

n

nznh

complex plan

R1

R2

Re

Im

absolutely summable

1=z


Z-Transform (cont)





[ ] 0for 0 lt= nnh

[ ] 0for 0 gt= nnh


Left-sided sequence

[ ] infinltsuminfin

minusinfin=nnh

[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


Z-Transform (cont)


[ ] [ ] [ ]⎩⎨⎧

ltge

==0for 00for 1

ere wh 1 1 nn

nunuanh n

( ) ( ) 11

1 11

minus

infin

minusinfin=

minusminusinfin


azazzazH

n

nn

n

n

If 11 ltminusaz

azROC gtthere4 is 1

Re

Im

a[ ] 1 if exists

ofansformFourier tr

1 ltanhtimes


az =

the unit cycle

0


Z-Transform (cont)



( ) [ ]

( ) 11

1

10

1

1

12

11

11111

1

minusminus

minus

minus

infin

=

minusinfin

=

minus

minusminus

minusinfin=

minusinfin

minusinfin=

minus=

minusminus

minus=minus

minus=minus=minus=


sumsum

sumsum

azzaza

zazaza

zaznuazH

n

n

n

nn

n

n

nn

n

n If 11 ltminus za

azROC ltthere4 is 2

Re

Im

a[ ] [ ]

minusinfinrarr

lt

nnhnh

a



22times

the unit cycle

0


Z-Transform (cont)


[ ] [ ] [ ]121


⎠⎞

⎜⎝⎛minus⎟

⎠⎞


nn

[ ]

[ ]21

211

1 121

31

311

1 31

1

1

ltminus


⎜⎝⎛minus

gt+


⎜⎝⎛ minus

minus

minus

zz

nu

zz

nu

zn

zn

Re

Im

times

the unit cycle

times21

31



3

3

ROCnh

31 and


( )⎟⎠⎞


⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ minus

=minus

++

=minusminus

21

31

1212

211

1

311

1 11

3

zz

zz

zzzH

0


Z-Transform (cont)



=others 0

10 3 4

Nnanh

n

( ) ( ) ( )azaz

zazazazzazH

NN

N

NN

n

nN

n

nn

minusminus

=minus

minus=== minusminus

minusminus

=

minusminus

=

minus sumsum 11

11

0

11

04

11

1



Im

times

the unit cycle

31

minus


4π

( ) 11 2

minus== N kaez Nkj

k

π

If N=8

0 N-1

Re


Z-Transform (cont)






[ ]nh [ ] 1 0 nnnh le=

01 gen

0rz gtz

infin=z

[ ]nh [ ] 2 0 nnnh ge=

02 ltn 0=z[ ]nh


01 geninfin=zthere4








[ ]nh

[ ] [ ] [ ] [ ]

[ ]

( ) njj

kjnj

knj

eeH

ekhe

ekhnhnxny

ωω

ωω

ω

=

=

=lowast=

minusinfin

infin=

minusinfin

infin=

sum

sum

-k

)(

-k( )



[ ] njenx ω=[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


( )jωeH

( )jωeH

scalar


[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjωnjjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

ang++=

+=

+=

+=

+minusangminus+ang

+minus+

minusminusminus

φω

φωφω

φωφω

ωφωφ




[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0

ωφωφ

φω

minusminus+=

+=

[ ]nh

( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

minus

minus

+=rArr

minus=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω

jωjω

eeHeH

eHeHangminus

minus

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos

sincos

jje

jejyxz

jejyxz

j

j

j

minus=minus+minus=

minus=rArrminus=

+=rArr+=

minus






[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2



0 12

13

-2

0-1-2

13

-2

0 1 2

23

1

0

3

10-1

13

-21

11

210

13

-2

1

2

321

13

-2 -1

3

432

13

-2 -2

4

0

3

1

11

2

13

-1

4

-2

Sum up

[ ]kh

[ ]kx

[ ]kh minus

[ ] 0 =nny

[ ]1+minuskh

[ ]2+minuskh

[ ]3+minuskh

[ ]4+minuskh

[ ] 1 =nny

[ ] 2 =nny

[ ] 3 =nny

[ ] 4 =nny

[ ]ny

Reflect [ ] [ ] [ ][ ] [ ]knhkx

nhnxny

kminus=

lowast=

suminfin

minusinfin=

Convolution




[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=

[ ]nh 1 [ ]nh 2 [ ]nh 2 [ ]nh 1

[ ]nh 2

[ ]nh 1

[ ] [ ]nhnh 21 +



[ ] [ ] [ ][ ] [ ] [ ]nhnhnx

nhnhnx

12

21

=

[ ] [ ] [ ]( )[ ] [ ] [ ] [ ]nhnxnhnx

nhnhnx

21

21

+=+

Commutation

Distribution




[ ] [ ] [ ]

[ ] [ ]

[ ] [ ] ( )

[ ] [ ]nxnh

knmhmnx

knhkx

nhnxny

m

k

mlet

=

minus=sum minus=

minussum=

=

infin

minusinfin=

infin

minusinfin=




[ ] [ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] ( )0000

00011

0

101

cosBut

coscos

suppose

nnωnnxnny

nnnxnnxny

nnynynnxnx

minusminus=minus

minus==

minus=rArrminus=

ωω





[ ] infinlesuminfin

minusinfin=kkh

[ ][ ] nBny

nBnx

y

x

forallinfinltle

forallinfinltle






[ ] [ ] [ ]sum sum= =

minus+minus=K

k

M

mkkk mnxknyny

1000 βα

z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα






[ ]nh( )ωjeH

π2( )ωjeH

ω( ) ( ) ( )

( ) ( )ωω

ωωω

jeHjj

ji

jr

j

eeH

ejHeHeHang=

+=

magnitudephase

( )ωjeH






[ ] infinltsuminfin

minusinfin=nnh

[ ]( )

( )

[ ]mnnmnm

mnmn

emnj

de

mnj

mnj

minus=⎩⎨⎧

ne=

=

minusminus

=

minus= minus

minus

minusminus

int

δ

πππ

ωπ

ππ

ω

ππ

ω

0 1

sin)(2

121

)(

)(

( ) [ ]

[ ] ( ) [ ]


deemhdeeHnh

enheH

m

mnj

m

nj

m

mjnjj

n

njj

=minus==

==

=

sumintsum

int sumint

sum

infin

minusinfin=minus

minusinfin

minusinfin=

minus

infin

minusinfin=

minusminus

infin

minusinfin=

minus

δωπ

ωπ

ωπ

ππ

ω

ππ

ωωππ

ωω

ωω

21

21

21


)(

( ) [ ]

[ ] ( )int

sum

minus

infin

minusinfin=

minus

=

=

ππ

ωω

ωω

ωπ

deeHnh

enheH

njj

n

njj

2

1

absolutely summable

DTFT

Inverse DTFT



( ) [ ]

[ ] [ ] [ ]

( ) [ ] [ ]

[ ] [ ]

( ) ( )

[ ] ( ) ( )ωω

ωω

ωω

ωω

ωω

jj

jj

nj

nk

kj

nj

n k

j

k

n

njj

eHeXnhnx

eHeX

enhekx

eknhkxeY

knhkxnhnxny

enheH

hArrlowastthere4

=

⎟⎠

⎞⎜⎝

⎛=

minus=

minus=lowast=

=

minusinfin

minusinfin=

infin

minusinfin=

minus

minusinfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus

sumsum

sum sum

sum

sum

][

][][

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )ωωω

ωωω

ωωω

jjj

jjj

jjj

eHeXeY

eHeXeY

eHeXeY

ang+ang=angrArr

=rArr

=

knnknnknn






[ ] ( )intsum minus

infin

minusinfin== π

πω ω

πdeXnx j

n

22 21

power spectrum

[ ]nx[ ]

( ) ][][][][

][][

nxnxlnxlx

mxnmxnR

l

mxx


+=

lowastinfin

minusinfin=

infin

minusinfin=

sum

sum

( ) ( ) ( ) ( ) 2 ωωωω XXXS xx ==

hArr


π

π

ωπ

π

ω ωωπ

ωωπ

deXdeSnR njnjxxxx

2

21

21


minusinfin=minus

infin

minusinfin=

===

=

mmxx dXmxmxmxR

nπ

πωω

π22

210

0Set


)(

lnnlmnml



conjugatecomplex





[ ] ( )[ ] [ ] [ ] ( )

[ ] ( ) [ ]

[ ] ( )( )

( ) [ ]( ) ( )( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( )( )[ ] ( ) ( ) ( )( )ωω

ωωωω

φωω

ωφω

ωφω

ωφω

φω

φω

φω

φω

φωφω

φω

ω

ω

jj

jjjj

eHjnjj

eHjjnj

jnj

kj

k

nj

knj

k

nj

nj

eHneHAny

eHneHjAeHneHA

eeHA

eeHAe

eHAe

ekhAe

Aekh

nhAeny

Aenxnxnx

njAnx

j

j

ang++=rArr

ang+++ang++=

=

=

=

sum=

sum=

=

=+=

+=

ang++

ang+

+

minusinfin

minusinfin=

+

+minusinfin

minusinfin=

+

+

cos

sincos

sin

0

10

1

[ ]ny [ ]ny1

( )( ) attenuate 1

amplify 1

lt

gt

ω

ω

j

j

eH

eH





Z-Transform





( ) [ ]suminfin

minusinfin=

minus=n

nznhzH

[ ] ( )zHnh

[ ] ( )ωjeHnh

[ ]nh

ωjrez =

( ) ( ) ωω

jezj zHeH

==

)1( == zez jω

complex plane

unit circle

Im

Re

ω

ωjez =


Z-Transform (cont)


stability



[ ] infinltsuminfin

minusinfin=

minus

n

nznh

complex plan

R1

R2

Re

Im

absolutely summable

1=z


Z-Transform (cont)





[ ] 0for 0 lt= nnh

[ ] 0for 0 gt= nnh


Left-sided sequence

[ ] infinltsuminfin

minusinfin=nnh

[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


Z-Transform (cont)


[ ] [ ] [ ]⎩⎨⎧

ltge

==0for 00for 1

ere wh 1 1 nn

nunuanh n

( ) ( ) 11

1 11

minus

infin

minusinfin=

minusminusinfin


azazzazH

n

nn

n

n

If 11 ltminusaz

azROC gtthere4 is 1

Re

Im

a[ ] 1 if exists

ofansformFourier tr

1 ltanhtimes


az =

the unit cycle

0


Z-Transform (cont)



( ) [ ]

( ) 11

1

10

1

1

12

11

11111

1

minusminus

minus

minus

infin

=

minusinfin

=

minus

minusminus

minusinfin=

minusinfin

minusinfin=

minus=

minusminus

minus=minus

minus=minus=minus=


sumsum

sumsum

azzaza

zazaza

zaznuazH

n

n

n

nn

n

n

nn

n

n If 11 ltminus za

azROC ltthere4 is 2

Re

Im

a[ ] [ ]

minusinfinrarr

lt

nnhnh

a



22times

the unit cycle

0


Z-Transform (cont)


[ ] [ ] [ ]121


⎠⎞

⎜⎝⎛minus⎟

⎠⎞


nn

[ ]

[ ]21

211

1 121

31

311

1 31

1

1

ltminus


⎜⎝⎛minus

gt+


⎜⎝⎛ minus

minus

minus

zz

nu

zz

nu

zn

zn

Re

Im

times

the unit cycle

times21

31



3

3

ROCnh

31 and


( )⎟⎠⎞


⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ minus

=minus

++

=minusminus

21

31

1212

211

1

311

1 11

3

zz

zz

zzzH

0


Z-Transform (cont)



=others 0

10 3 4

Nnanh

n

( ) ( ) ( )azaz

zazazazzazH

NN

N

NN

n

nN

n

nn

minusminus

=minus

minus=== minusminus

minusminus

=

minusminus

=

minus sumsum 11

11

0

11

04

11

1



Im

times

the unit cycle

31

minus


4π

( ) 11 2

minus== N kaez Nkj

k

π

If N=8

0 N-1

Re


Z-Transform (cont)






[ ]nh [ ] 1 0 nnnh le=

01 gen

0rz gtz

infin=z

[ ]nh [ ] 2 0 nnnh ge=

02 ltn 0=z[ ]nh


01 geninfin=zthere4








[ ]nh

[ ] [ ] [ ] [ ]

[ ]

( ) njj

kjnj

knj

eeH

ekhe

ekhnhnxny

ωω

ωω

ω

=

=

=lowast=

minusinfin

infin=

minusinfin

infin=

sum

sum

-k

)(

-k( )



[ ] njenx ω=[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


( )jωeH

( )jωeH

scalar


[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjωnjjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

ang++=

+=

+=

+=

+minusangminus+ang

+minus+

minusminusminus

φω

φωφω

φωφω

ωφωφ




[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0

ωφωφ

φω

minusminus+=

+=

[ ]nh

( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

minus

minus

+=rArr

minus=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω

jωjω

eeHeH

eHeHangminus

minus

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos

sincos

jje

jejyxz

jejyxz

j

j

j

minus=minus+minus=

minus=rArrminus=

+=rArr+=

minus






[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2





[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=

[ ]nh 1 [ ]nh 2 [ ]nh 2 [ ]nh 1

[ ]nh 2

[ ]nh 1

[ ] [ ]nhnh 21 +



[ ] [ ] [ ][ ] [ ] [ ]nhnhnx

nhnhnx

12

21

=

[ ] [ ] [ ]( )[ ] [ ] [ ] [ ]nhnxnhnx

nhnhnx

21

21

+=+

Commutation

Distribution




[ ] [ ] [ ]

[ ] [ ]

[ ] [ ] ( )

[ ] [ ]nxnh

knmhmnx

knhkx

nhnxny

m

k

mlet

=

minus=sum minus=

minussum=

=

infin

minusinfin=

infin

minusinfin=




[ ] [ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] ( )0000

00011

0

101

cosBut

coscos

suppose

nnωnnxnny

nnnxnnxny

nnynynnxnx

minusminus=minus

minus==

minus=rArrminus=

ωω





[ ] infinlesuminfin

minusinfin=kkh

[ ][ ] nBny

nBnx

y

x

forallinfinltle

forallinfinltle






[ ] [ ] [ ]sum sum= =

minus+minus=K

k

M

mkkk mnxknyny

1000 βα

z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα






[ ]nh( )ωjeH

π2( )ωjeH

ω( ) ( ) ( )

( ) ( )ωω

ωωω

jeHjj

ji

jr

j

eeH

ejHeHeHang=

+=

magnitudephase

( )ωjeH






[ ] infinltsuminfin

minusinfin=nnh

[ ]( )

( )

[ ]mnnmnm

mnmn

emnj

de

mnj

mnj

minus=⎩⎨⎧

ne=

=

minusminus

=

minus= minus

minus

minusminus

int

δ

πππ

ωπ

ππ

ω

ππ

ω

0 1

sin)(2

121

)(

)(

( ) [ ]

[ ] ( ) [ ]


deemhdeeHnh

enheH

m

mnj

m

nj

m

mjnjj

n

njj

=minus==

==

=

sumintsum

int sumint

sum

infin

minusinfin=minus

minusinfin

minusinfin=

minus

infin

minusinfin=

minusminus

infin

minusinfin=

minus

δωπ

ωπ

ωπ

ππ

ω

ππ

ωωππ

ωω

ωω

21

21

21


)(

( ) [ ]

[ ] ( )int

sum

minus

infin

minusinfin=

minus

=

=

ππ

ωω

ωω

ωπ

deeHnh

enheH

njj

n

njj

2

1

absolutely summable

DTFT

Inverse DTFT



( ) [ ]

[ ] [ ] [ ]

( ) [ ] [ ]

[ ] [ ]

( ) ( )

[ ] ( ) ( )ωω

ωω

ωω

ωω

ωω

jj

jj

nj

nk

kj

nj

n k

j

k

n

njj

eHeXnhnx

eHeX

enhekx

eknhkxeY

knhkxnhnxny

enheH

hArrlowastthere4

=

⎟⎠

⎞⎜⎝

⎛=

minus=

minus=lowast=

=

minusinfin

minusinfin=

infin

minusinfin=

minus

minusinfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus

sumsum

sum sum

sum

sum

][

][][

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )ωωω

ωωω

ωωω

jjj

jjj

jjj

eHeXeY

eHeXeY

eHeXeY

ang+ang=angrArr

=rArr

=

knnknnknn






[ ] ( )intsum minus

infin

minusinfin== π

πω ω

πdeXnx j

n

22 21

power spectrum

[ ]nx[ ]

( ) ][][][][

][][

nxnxlnxlx

mxnmxnR

l

mxx


+=

lowastinfin

minusinfin=

infin

minusinfin=

sum

sum

( ) ( ) ( ) ( ) 2 ωωωω XXXS xx ==

hArr


π

π

ωπ

π

ω ωωπ

ωωπ

deXdeSnR njnjxxxx

2

21

21


minusinfin=minus

infin

minusinfin=

===

=

mmxx dXmxmxmxR

nπ

πωω

π22

210

0Set


)(

lnnlmnml



conjugatecomplex





[ ] ( )[ ] [ ] [ ] ( )

[ ] ( ) [ ]

[ ] ( )( )

( ) [ ]( ) ( )( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( )( )[ ] ( ) ( ) ( )( )ωω

ωωωω

φωω

ωφω

ωφω

ωφω

φω

φω

φω

φω

φωφω

φω

ω

ω

jj

jjjj

eHjnjj

eHjjnj

jnj

kj

k

nj

knj

k

nj

nj

eHneHAny

eHneHjAeHneHA

eeHA

eeHAe

eHAe

ekhAe

Aekh

nhAeny

Aenxnxnx

njAnx

j

j

ang++=rArr

ang+++ang++=

=

=

=

sum=

sum=

=

=+=

+=

ang++

ang+

+

minusinfin

minusinfin=

+

+minusinfin

minusinfin=

+

+

cos

sincos

sin

0

10

1

[ ]ny [ ]ny1

( )( ) attenuate 1

amplify 1

lt

gt

ω

ω

j

j

eH

eH





Z-Transform





( ) [ ]suminfin

minusinfin=

minus=n

nznhzH

[ ] ( )zHnh

[ ] ( )ωjeHnh

[ ]nh

ωjrez =

( ) ( ) ωω

jezj zHeH

==

)1( == zez jω

complex plane

unit circle

Im

Re

ω

ωjez =


Z-Transform (cont)


stability



[ ] infinltsuminfin

minusinfin=

minus

n

nznh

complex plan

R1

R2

Re

Im

absolutely summable

1=z


Z-Transform (cont)





[ ] 0for 0 lt= nnh

[ ] 0for 0 gt= nnh


Left-sided sequence

[ ] infinltsuminfin

minusinfin=nnh

[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


Z-Transform (cont)


[ ] [ ] [ ]⎩⎨⎧

ltge

==0for 00for 1

ere wh 1 1 nn

nunuanh n

( ) ( ) 11

1 11

minus

infin

minusinfin=

minusminusinfin


azazzazH

n

nn

n

n

If 11 ltminusaz

azROC gtthere4 is 1

Re

Im

a[ ] 1 if exists

ofansformFourier tr

1 ltanhtimes


az =

the unit cycle

0


Z-Transform (cont)



( ) [ ]

( ) 11

1

10

1

1

12

11

11111

1

minusminus

minus

minus

infin

=

minusinfin

=

minus

minusminus

minusinfin=

minusinfin

minusinfin=

minus=

minusminus

minus=minus

minus=minus=minus=


sumsum

sumsum

azzaza

zazaza

zaznuazH

n

n

n

nn

n

n

nn

n

n If 11 ltminus za

azROC ltthere4 is 2

Re

Im

a[ ] [ ]

minusinfinrarr

lt

nnhnh

a



22times

the unit cycle

0


Z-Transform (cont)


[ ] [ ] [ ]121


⎠⎞

⎜⎝⎛minus⎟

⎠⎞


nn

[ ]

[ ]21

211

1 121

31

311

1 31

1

1

ltminus


⎜⎝⎛minus

gt+


⎜⎝⎛ minus

minus

minus

zz

nu

zz

nu

zn

zn

Re

Im

times

the unit cycle

times21

31



3

3

ROCnh

31 and


( )⎟⎠⎞


⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ minus

=minus

++

=minusminus

21

31

1212

211

1

311

1 11

3

zz

zz

zzzH

0


Z-Transform (cont)



=others 0

10 3 4

Nnanh

n

( ) ( ) ( )azaz

zazazazzazH

NN

N

NN

n

nN

n

nn

minusminus

=minus

minus=== minusminus

minusminus

=

minusminus

=

minus sumsum 11

11

0

11

04

11

1



Im

times

the unit cycle

31

minus


4π

( ) 11 2

minus== N kaez Nkj

k

π

If N=8

0 N-1

Re


Z-Transform (cont)






[ ]nh [ ] 1 0 nnnh le=

01 gen

0rz gtz

infin=z

[ ]nh [ ] 2 0 nnnh ge=

02 ltn 0=z[ ]nh


01 geninfin=zthere4








[ ]nh

[ ] [ ] [ ] [ ]

[ ]

( ) njj

kjnj

knj

eeH

ekhe

ekhnhnxny

ωω

ωω

ω

=

=

=lowast=

minusinfin

infin=

minusinfin

infin=

sum

sum

-k

)(

-k( )



[ ] njenx ω=[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


( )jωeH

( )jωeH

scalar


[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjωnjjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

ang++=

+=

+=

+=

+minusangminus+ang

+minus+

minusminusminus

φω

φωφω

φωφω

ωφωφ




[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0

ωφωφ

φω

minusminus+=

+=

[ ]nh

( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

minus

minus

+=rArr

minus=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω

jωjω

eeHeH

eHeHangminus

minus

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos

sincos

jje

jejyxz

jejyxz

j

j

j

minus=minus+minus=

minus=rArrminus=

+=rArr+=

minus






[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2





[ ] [ ] [ ]

[ ] [ ]

[ ] [ ] ( )

[ ] [ ]nxnh

knmhmnx

knhkx

nhnxny

m

k

mlet

=

minus=sum minus=

minussum=

=

infin

minusinfin=

infin

minusinfin=




[ ] [ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] ( )0000

00011

0

101

cosBut

coscos

suppose

nnωnnxnny

nnnxnnxny

nnynynnxnx

minusminus=minus

minus==

minus=rArrminus=

ωω





[ ] infinlesuminfin

minusinfin=kkh

[ ][ ] nBny

nBnx

y

x

forallinfinltle

forallinfinltle






[ ] [ ] [ ]sum sum= =

minus+minus=K

k

M

mkkk mnxknyny

1000 βα

z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα






[ ]nh( )ωjeH

π2( )ωjeH

ω( ) ( ) ( )

( ) ( )ωω

ωωω

jeHjj

ji

jr

j

eeH

ejHeHeHang=

+=

magnitudephase

( )ωjeH






[ ] infinltsuminfin

minusinfin=nnh

[ ]( )

( )

[ ]mnnmnm

mnmn

emnj

de

mnj

mnj

minus=⎩⎨⎧

ne=

=

minusminus

=

minus= minus

minus

minusminus

int

δ

πππ

ωπ

ππ

ω

ππ

ω

0 1

sin)(2

121

)(

)(

( ) [ ]

[ ] ( ) [ ]


deemhdeeHnh

enheH

m

mnj

m

nj

m

mjnjj

n

njj

=minus==

==

=

sumintsum

int sumint

sum

infin

minusinfin=minus

minusinfin

minusinfin=

minus

infin

minusinfin=

minusminus

infin

minusinfin=

minus

δωπ

ωπ

ωπ

ππ

ω

ππ

ωωππ

ωω

ωω

21

21

21


)(

( ) [ ]

[ ] ( )int

sum

minus

infin

minusinfin=

minus

=

=

ππ

ωω

ωω

ωπ

deeHnh

enheH

njj

n

njj

2

1

absolutely summable

DTFT

Inverse DTFT



( ) [ ]

[ ] [ ] [ ]

( ) [ ] [ ]

[ ] [ ]

( ) ( )

[ ] ( ) ( )ωω

ωω

ωω

ωω

ωω

jj

jj

nj

nk

kj

nj

n k

j

k

n

njj

eHeXnhnx

eHeX

enhekx

eknhkxeY

knhkxnhnxny

enheH

hArrlowastthere4

=

⎟⎠

⎞⎜⎝

⎛=

minus=

minus=lowast=

=

minusinfin

minusinfin=

infin

minusinfin=

minus

minusinfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus

sumsum

sum sum

sum

sum

][

][][

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )ωωω

ωωω

ωωω

jjj

jjj

jjj

eHeXeY

eHeXeY

eHeXeY

ang+ang=angrArr

=rArr

=

knnknnknn






[ ] ( )intsum minus

infin

minusinfin== π

πω ω

πdeXnx j

n

22 21

power spectrum

[ ]nx[ ]

( ) ][][][][

][][

nxnxlnxlx

mxnmxnR

l

mxx


+=

lowastinfin

minusinfin=

infin

minusinfin=

sum

sum

( ) ( ) ( ) ( ) 2 ωωωω XXXS xx ==

hArr


π

π

ωπ

π

ω ωωπ

ωωπ

deXdeSnR njnjxxxx

2

21

21


minusinfin=minus

infin

minusinfin=

===

=

mmxx dXmxmxmxR

nπ

πωω

π22

210

0Set


)(

lnnlmnml



conjugatecomplex





[ ] ( )[ ] [ ] [ ] ( )

[ ] ( ) [ ]

[ ] ( )( )

( ) [ ]( ) ( )( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( )( )[ ] ( ) ( ) ( )( )ωω

ωωωω

φωω

ωφω

ωφω

ωφω

φω

φω

φω

φω

φωφω

φω

ω

ω

jj

jjjj

eHjnjj

eHjjnj

jnj

kj

k

nj

knj

k

nj

nj

eHneHAny

eHneHjAeHneHA

eeHA

eeHAe

eHAe

ekhAe

Aekh

nhAeny

Aenxnxnx

njAnx

j

j

ang++=rArr

ang+++ang++=

=

=

=

sum=

sum=

=

=+=

+=

ang++

ang+

+

minusinfin

minusinfin=

+

+minusinfin

minusinfin=

+

+

cos

sincos

sin

0

10

1

[ ]ny [ ]ny1

( )( ) attenuate 1

amplify 1

lt

gt

ω

ω

j

j

eH

eH





Z-Transform





( ) [ ]suminfin

minusinfin=

minus=n

nznhzH

[ ] ( )zHnh

[ ] ( )ωjeHnh

[ ]nh

ωjrez =

( ) ( ) ωω

jezj zHeH

==

)1( == zez jω

complex plane

unit circle

Im

Re

ω

ωjez =


Z-Transform (cont)


stability



[ ] infinltsuminfin

minusinfin=

minus

n

nznh

complex plan

R1

R2

Re

Im

absolutely summable

1=z


Z-Transform (cont)





[ ] 0for 0 lt= nnh

[ ] 0for 0 gt= nnh


Left-sided sequence

[ ] infinltsuminfin

minusinfin=nnh

[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


Z-Transform (cont)


[ ] [ ] [ ]⎩⎨⎧

ltge

==0for 00for 1

ere wh 1 1 nn

nunuanh n

( ) ( ) 11

1 11

minus

infin

minusinfin=

minusminusinfin


azazzazH

n

nn

n

n

If 11 ltminusaz

azROC gtthere4 is 1

Re

Im

a[ ] 1 if exists

ofansformFourier tr

1 ltanhtimes


az =

the unit cycle

0


Z-Transform (cont)



( ) [ ]

( ) 11

1

10

1

1

12

11

11111

1

minusminus

minus

minus

infin

=

minusinfin

=

minus

minusminus

minusinfin=

minusinfin

minusinfin=

minus=

minusminus

minus=minus

minus=minus=minus=


sumsum

sumsum

azzaza

zazaza

zaznuazH

n

n

n

nn

n

n

nn

n

n If 11 ltminus za

azROC ltthere4 is 2

Re

Im

a[ ] [ ]

minusinfinrarr

lt

nnhnh

a



22times

the unit cycle

0


Z-Transform (cont)


[ ] [ ] [ ]121


⎠⎞

⎜⎝⎛minus⎟

⎠⎞


nn

[ ]

[ ]21

211

1 121

31

311

1 31

1

1

ltminus


⎜⎝⎛minus

gt+


⎜⎝⎛ minus

minus

minus

zz

nu

zz

nu

zn

zn

Re

Im

times

the unit cycle

times21

31



3

3

ROCnh

31 and


( )⎟⎠⎞


⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ minus

=minus

++

=minusminus

21

31

1212

211

1

311

1 11

3

zz

zz

zzzH

0


Z-Transform (cont)



=others 0

10 3 4

Nnanh

n

( ) ( ) ( )azaz

zazazazzazH

NN

N

NN

n

nN

n

nn

minusminus

=minus

minus=== minusminus

minusminus

=

minusminus

=

minus sumsum 11

11

0

11

04

11

1



Im

times

the unit cycle

31

minus


4π

( ) 11 2

minus== N kaez Nkj

k

π

If N=8

0 N-1

Re


Z-Transform (cont)






[ ]nh [ ] 1 0 nnnh le=

01 gen

0rz gtz

infin=z

[ ]nh [ ] 2 0 nnnh ge=

02 ltn 0=z[ ]nh


01 geninfin=zthere4








[ ]nh

[ ] [ ] [ ] [ ]

[ ]

( ) njj

kjnj

knj

eeH

ekhe

ekhnhnxny

ωω

ωω

ω

=

=

=lowast=

minusinfin

infin=

minusinfin

infin=

sum

sum

-k

)(

-k( )



[ ] njenx ω=[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


( )jωeH

( )jωeH

scalar


[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjωnjjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

ang++=

+=

+=

+=

+minusangminus+ang

+minus+

minusminusminus

φω

φωφω

φωφω

ωφωφ




[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0

ωφωφ

φω

minusminus+=

+=

[ ]nh

( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

minus

minus

+=rArr

minus=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω

jωjω

eeHeH

eHeHangminus

minus

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos

sincos

jje

jejyxz

jejyxz

j

j

j

minus=minus+minus=

minus=rArrminus=

+=rArr+=

minus






[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2





[ ] [ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] ( )0000

00011

0

101

cosBut

coscos

suppose

nnωnnxnny

nnnxnnxny

nnynynnxnx

minusminus=minus

minus==

minus=rArrminus=

ωω





[ ] infinlesuminfin

minusinfin=kkh

[ ][ ] nBny

nBnx

y

x

forallinfinltle

forallinfinltle






[ ] [ ] [ ]sum sum= =

minus+minus=K

k

M

mkkk mnxknyny

1000 βα

z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα






[ ]nh( )ωjeH

π2( )ωjeH

ω( ) ( ) ( )

( ) ( )ωω

ωωω

jeHjj

ji

jr

j

eeH

ejHeHeHang=

+=

magnitudephase

( )ωjeH






[ ] infinltsuminfin

minusinfin=nnh

[ ]( )

( )

[ ]mnnmnm

mnmn

emnj

de

mnj

mnj

minus=⎩⎨⎧

ne=

=

minusminus

=

minus= minus

minus

minusminus

int

δ

πππ

ωπ

ππ

ω

ππ

ω

0 1

sin)(2

121

)(

)(

( ) [ ]

[ ] ( ) [ ]


deemhdeeHnh

enheH

m

mnj

m

nj

m

mjnjj

n

njj

=minus==

==

=

sumintsum

int sumint

sum

infin

minusinfin=minus

minusinfin

minusinfin=

minus

infin

minusinfin=

minusminus

infin

minusinfin=

minus

δωπ

ωπ

ωπ

ππ

ω

ππ

ωωππ

ωω

ωω

21

21

21


)(

( ) [ ]

[ ] ( )int

sum

minus

infin

minusinfin=

minus

=

=

ππ

ωω

ωω

ωπ

deeHnh

enheH

njj

n

njj

2

1

absolutely summable

DTFT

Inverse DTFT



( ) [ ]

[ ] [ ] [ ]

( ) [ ] [ ]

[ ] [ ]

( ) ( )

[ ] ( ) ( )ωω

ωω

ωω

ωω

ωω

jj

jj

nj

nk

kj

nj

n k

j

k

n

njj

eHeXnhnx

eHeX

enhekx

eknhkxeY

knhkxnhnxny

enheH

hArrlowastthere4

=

⎟⎠

⎞⎜⎝

⎛=

minus=

minus=lowast=

=

minusinfin

minusinfin=

infin

minusinfin=

minus

minusinfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus

sumsum

sum sum

sum

sum

][

][][

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )ωωω

ωωω

ωωω

jjj

jjj

jjj

eHeXeY

eHeXeY

eHeXeY

ang+ang=angrArr

=rArr

=

knnknnknn






[ ] ( )intsum minus

infin

minusinfin== π

πω ω

πdeXnx j

n

22 21

power spectrum

[ ]nx[ ]

( ) ][][][][

][][

nxnxlnxlx

mxnmxnR

l

mxx


+=

lowastinfin

minusinfin=

infin

minusinfin=

sum

sum

( ) ( ) ( ) ( ) 2 ωωωω XXXS xx ==

hArr


π

π

ωπ

π

ω ωωπ

ωωπ

deXdeSnR njnjxxxx

2

21

21


minusinfin=minus

infin

minusinfin=

===

=

mmxx dXmxmxmxR

nπ

πωω

π22

210

0Set


)(

lnnlmnml



conjugatecomplex





[ ] ( )[ ] [ ] [ ] ( )

[ ] ( ) [ ]

[ ] ( )( )

( ) [ ]( ) ( )( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( )( )[ ] ( ) ( ) ( )( )ωω

ωωωω

φωω

ωφω

ωφω

ωφω

φω

φω

φω

φω

φωφω

φω

ω

ω

jj

jjjj

eHjnjj

eHjjnj

jnj

kj

k

nj

knj

k

nj

nj

eHneHAny

eHneHjAeHneHA

eeHA

eeHAe

eHAe

ekhAe

Aekh

nhAeny

Aenxnxnx

njAnx

j

j

ang++=rArr

ang+++ang++=

=

=

=

sum=

sum=

=

=+=

+=

ang++

ang+

+

minusinfin

minusinfin=

+

+minusinfin

minusinfin=

+

+

cos

sincos

sin

0

10

1

[ ]ny [ ]ny1

( )( ) attenuate 1

amplify 1

lt

gt

ω

ω

j

j

eH

eH





Z-Transform





( ) [ ]suminfin

minusinfin=

minus=n

nznhzH

[ ] ( )zHnh

[ ] ( )ωjeHnh

[ ]nh

ωjrez =

( ) ( ) ωω

jezj zHeH

==

)1( == zez jω

complex plane

unit circle

Im

Re

ω

ωjez =


Z-Transform (cont)


stability



[ ] infinltsuminfin

minusinfin=

minus

n

nznh

complex plan

R1

R2

Re

Im

absolutely summable

1=z


Z-Transform (cont)





[ ] 0for 0 lt= nnh

[ ] 0for 0 gt= nnh


Left-sided sequence

[ ] infinltsuminfin

minusinfin=nnh

[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


Z-Transform (cont)


[ ] [ ] [ ]⎩⎨⎧

ltge

==0for 00for 1

ere wh 1 1 nn

nunuanh n

( ) ( ) 11

1 11

minus

infin

minusinfin=

minusminusinfin


azazzazH

n

nn

n

n

If 11 ltminusaz

azROC gtthere4 is 1

Re

Im

a[ ] 1 if exists

ofansformFourier tr

1 ltanhtimes


az =

the unit cycle

0


Z-Transform (cont)



( ) [ ]

( ) 11

1

10

1

1

12

11

11111

1

minusminus

minus

minus

infin

=

minusinfin

=

minus

minusminus

minusinfin=

minusinfin

minusinfin=

minus=

minusminus

minus=minus

minus=minus=minus=


sumsum

sumsum

azzaza

zazaza

zaznuazH

n

n

n

nn

n

n

nn

n

n If 11 ltminus za

azROC ltthere4 is 2

Re

Im

a[ ] [ ]

minusinfinrarr

lt

nnhnh

a



22times

the unit cycle

0


Z-Transform (cont)


[ ] [ ] [ ]121


⎠⎞

⎜⎝⎛minus⎟

⎠⎞


nn

[ ]

[ ]21

211

1 121

31

311

1 31

1

1

ltminus


⎜⎝⎛minus

gt+


⎜⎝⎛ minus

minus

minus

zz

nu

zz

nu

zn

zn

Re

Im

times

the unit cycle

times21

31



3

3

ROCnh

31 and


( )⎟⎠⎞


⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ minus

=minus

++

=minusminus

21

31

1212

211

1

311

1 11

3

zz

zz

zzzH

0


Z-Transform (cont)



=others 0

10 3 4

Nnanh

n

( ) ( ) ( )azaz

zazazazzazH

NN

N

NN

n

nN

n

nn

minusminus

=minus

minus=== minusminus

minusminus

=

minusminus

=

minus sumsum 11

11

0

11

04

11

1



Im

times

the unit cycle

31

minus


4π

( ) 11 2

minus== N kaez Nkj

k

π

If N=8

0 N-1

Re


Z-Transform (cont)






[ ]nh [ ] 1 0 nnnh le=

01 gen

0rz gtz

infin=z

[ ]nh [ ] 2 0 nnnh ge=

02 ltn 0=z[ ]nh


01 geninfin=zthere4








[ ]nh

[ ] [ ] [ ] [ ]

[ ]

( ) njj

kjnj

knj

eeH

ekhe

ekhnhnxny

ωω

ωω

ω

=

=

=lowast=

minusinfin

infin=

minusinfin

infin=

sum

sum

-k

)(

-k( )



[ ] njenx ω=[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


( )jωeH

( )jωeH

scalar


[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjωnjjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

ang++=

+=

+=

+=

+minusangminus+ang

+minus+

minusminusminus

φω

φωφω

φωφω

ωφωφ




[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0

ωφωφ

φω

minusminus+=

+=

[ ]nh

( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

minus

minus

+=rArr

minus=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω

jωjω

eeHeH

eHeHangminus

minus

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos

sincos

jje

jejyxz

jejyxz

j

j

j

minus=minus+minus=

minus=rArrminus=

+=rArr+=

minus






[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2






[ ] infinlesuminfin

minusinfin=kkh

[ ][ ] nBny

nBnx

y

x

forallinfinltle

forallinfinltle






[ ] [ ] [ ]sum sum= =

minus+minus=K

k

M

mkkk mnxknyny

1000 βα

z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα






[ ]nh( )ωjeH

π2( )ωjeH

ω( ) ( ) ( )

( ) ( )ωω

ωωω

jeHjj

ji

jr

j

eeH

ejHeHeHang=

+=

magnitudephase

( )ωjeH






[ ] infinltsuminfin

minusinfin=nnh

[ ]( )

( )

[ ]mnnmnm

mnmn

emnj

de

mnj

mnj

minus=⎩⎨⎧

ne=

=

minusminus

=

minus= minus

minus

minusminus

int

δ

πππ

ωπ

ππ

ω

ππ

ω

0 1

sin)(2

121

)(

)(

( ) [ ]

[ ] ( ) [ ]


deemhdeeHnh

enheH

m

mnj

m

nj

m

mjnjj

n

njj

=minus==

==

=

sumintsum

int sumint

sum

infin

minusinfin=minus

minusinfin

minusinfin=

minus

infin

minusinfin=

minusminus

infin

minusinfin=

minus

δωπ

ωπ

ωπ

ππ

ω

ππ

ωωππ

ωω

ωω

21

21

21


)(

( ) [ ]

[ ] ( )int

sum

minus

infin

minusinfin=

minus

=

=

ππ

ωω

ωω

ωπ

deeHnh

enheH

njj

n

njj

2

1

absolutely summable

DTFT

Inverse DTFT



( ) [ ]

[ ] [ ] [ ]

( ) [ ] [ ]

[ ] [ ]

( ) ( )

[ ] ( ) ( )ωω

ωω

ωω

ωω

ωω

jj

jj

nj

nk

kj

nj

n k

j

k

n

njj

eHeXnhnx

eHeX

enhekx

eknhkxeY

knhkxnhnxny

enheH

hArrlowastthere4

=

⎟⎠

⎞⎜⎝

⎛=

minus=

minus=lowast=

=

minusinfin

minusinfin=

infin

minusinfin=

minus

minusinfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus

sumsum

sum sum

sum

sum

][

][][

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )ωωω

ωωω

ωωω

jjj

jjj

jjj

eHeXeY

eHeXeY

eHeXeY

ang+ang=angrArr

=rArr

=

knnknnknn






[ ] ( )intsum minus

infin

minusinfin== π

πω ω

πdeXnx j

n

22 21

power spectrum

[ ]nx[ ]

( ) ][][][][

][][

nxnxlnxlx

mxnmxnR

l

mxx


+=

lowastinfin

minusinfin=

infin

minusinfin=

sum

sum

( ) ( ) ( ) ( ) 2 ωωωω XXXS xx ==

hArr


π

π

ωπ

π

ω ωωπ

ωωπ

deXdeSnR njnjxxxx

2

21

21


minusinfin=minus

infin

minusinfin=

===

=

mmxx dXmxmxmxR

nπ

πωω

π22

210

0Set


)(

lnnlmnml



conjugatecomplex





[ ] ( )[ ] [ ] [ ] ( )

[ ] ( ) [ ]

[ ] ( )( )

( ) [ ]( ) ( )( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( )( )[ ] ( ) ( ) ( )( )ωω

ωωωω

φωω

ωφω

ωφω

ωφω

φω

φω

φω

φω

φωφω

φω

ω

ω

jj

jjjj

eHjnjj

eHjjnj

jnj

kj

k

nj

knj

k

nj

nj

eHneHAny

eHneHjAeHneHA

eeHA

eeHAe

eHAe

ekhAe

Aekh

nhAeny

Aenxnxnx

njAnx

j

j

ang++=rArr

ang+++ang++=

=

=

=

sum=

sum=

=

=+=

+=

ang++

ang+

+

minusinfin

minusinfin=

+

+minusinfin

minusinfin=

+

+

cos

sincos

sin

0

10

1

[ ]ny [ ]ny1

( )( ) attenuate 1

amplify 1

lt

gt

ω

ω

j

j

eH

eH





Z-Transform





( ) [ ]suminfin

minusinfin=

minus=n

nznhzH

[ ] ( )zHnh

[ ] ( )ωjeHnh

[ ]nh

ωjrez =

( ) ( ) ωω

jezj zHeH

==

)1( == zez jω

complex plane

unit circle

Im

Re

ω

ωjez =


Z-Transform (cont)


stability



[ ] infinltsuminfin

minusinfin=

minus

n

nznh

complex plan

R1

R2

Re

Im

absolutely summable

1=z


Z-Transform (cont)





[ ] 0for 0 lt= nnh

[ ] 0for 0 gt= nnh


Left-sided sequence

[ ] infinltsuminfin

minusinfin=nnh

[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


Z-Transform (cont)


[ ] [ ] [ ]⎩⎨⎧

ltge

==0for 00for 1

ere wh 1 1 nn

nunuanh n

( ) ( ) 11

1 11

minus

infin

minusinfin=

minusminusinfin


azazzazH

n

nn

n

n

If 11 ltminusaz

azROC gtthere4 is 1

Re

Im

a[ ] 1 if exists

ofansformFourier tr

1 ltanhtimes


az =

the unit cycle

0


Z-Transform (cont)



( ) [ ]

( ) 11

1

10

1

1

12

11

11111

1

minusminus

minus

minus

infin

=

minusinfin

=

minus

minusminus

minusinfin=

minusinfin

minusinfin=

minus=

minusminus

minus=minus

minus=minus=minus=


sumsum

sumsum

azzaza

zazaza

zaznuazH

n

n

n

nn

n

n

nn

n

n If 11 ltminus za

azROC ltthere4 is 2

Re

Im

a[ ] [ ]

minusinfinrarr

lt

nnhnh

a



22times

the unit cycle

0


Z-Transform (cont)


[ ] [ ] [ ]121


⎠⎞

⎜⎝⎛minus⎟

⎠⎞


nn

[ ]

[ ]21

211

1 121

31

311

1 31

1

1

ltminus


⎜⎝⎛minus

gt+


⎜⎝⎛ minus

minus

minus

zz

nu

zz

nu

zn

zn

Re

Im

times

the unit cycle

times21

31



3

3

ROCnh

31 and


( )⎟⎠⎞


⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ minus

=minus

++

=minusminus

21

31

1212

211

1

311

1 11

3

zz

zz

zzzH

0


Z-Transform (cont)



=others 0

10 3 4

Nnanh

n

( ) ( ) ( )azaz

zazazazzazH

NN

N

NN

n

nN

n

nn

minusminus

=minus

minus=== minusminus

minusminus

=

minusminus

=

minus sumsum 11

11

0

11

04

11

1



Im

times

the unit cycle

31

minus


4π

( ) 11 2

minus== N kaez Nkj

k

π

If N=8

0 N-1

Re


Z-Transform (cont)






[ ]nh [ ] 1 0 nnnh le=

01 gen

0rz gtz

infin=z

[ ]nh [ ] 2 0 nnnh ge=

02 ltn 0=z[ ]nh


01 geninfin=zthere4








[ ]nh

[ ] [ ] [ ] [ ]

[ ]

( ) njj

kjnj

knj

eeH

ekhe

ekhnhnxny

ωω

ωω

ω

=

=

=lowast=

minusinfin

infin=

minusinfin

infin=

sum

sum

-k

)(

-k( )



[ ] njenx ω=[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


( )jωeH

( )jωeH

scalar


[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjωnjjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

ang++=

+=

+=

+=

+minusangminus+ang

+minus+

minusminusminus

φω

φωφω

φωφω

ωφωφ




[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0

ωφωφ

φω

minusminus+=

+=

[ ]nh

( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

minus

minus

+=rArr

minus=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω

jωjω

eeHeH

eHeHangminus

minus

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos

sincos

jje

jejyxz

jejyxz

j

j

j

minus=minus+minus=

minus=rArrminus=

+=rArr+=

minus






[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2







[ ] [ ] [ ]sum sum= =

minus+minus=K

k

M

mkkk mnxknyny

1000 βα

z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα






[ ]nh( )ωjeH

π2( )ωjeH

ω( ) ( ) ( )

( ) ( )ωω

ωωω

jeHjj

ji

jr

j

eeH

ejHeHeHang=

+=

magnitudephase

( )ωjeH






[ ] infinltsuminfin

minusinfin=nnh

[ ]( )

( )

[ ]mnnmnm

mnmn

emnj

de

mnj

mnj

minus=⎩⎨⎧

ne=

=

minusminus

=

minus= minus

minus

minusminus

int

δ

πππ

ωπ

ππ

ω

ππ

ω

0 1

sin)(2

121

)(

)(

( ) [ ]

[ ] ( ) [ ]


deemhdeeHnh

enheH

m

mnj

m

nj

m

mjnjj

n

njj

=minus==

==

=

sumintsum

int sumint

sum

infin

minusinfin=minus

minusinfin

minusinfin=

minus

infin

minusinfin=

minusminus

infin

minusinfin=

minus

δωπ

ωπ

ωπ

ππ

ω

ππ

ωωππ

ωω

ωω

21

21

21


)(

( ) [ ]

[ ] ( )int

sum

minus

infin

minusinfin=

minus

=

=

ππ

ωω

ωω

ωπ

deeHnh

enheH

njj

n

njj

2

1

absolutely summable

DTFT

Inverse DTFT



( ) [ ]

[ ] [ ] [ ]

( ) [ ] [ ]

[ ] [ ]

( ) ( )

[ ] ( ) ( )ωω

ωω

ωω

ωω

ωω

jj

jj

nj

nk

kj

nj

n k

j

k

n

njj

eHeXnhnx

eHeX

enhekx

eknhkxeY

knhkxnhnxny

enheH

hArrlowastthere4

=

⎟⎠

⎞⎜⎝

⎛=

minus=

minus=lowast=

=

minusinfin

minusinfin=

infin

minusinfin=

minus

minusinfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus

sumsum

sum sum

sum

sum

][

][][

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )ωωω

ωωω

ωωω

jjj

jjj

jjj

eHeXeY

eHeXeY

eHeXeY

ang+ang=angrArr

=rArr

=

knnknnknn






[ ] ( )intsum minus

infin

minusinfin== π

πω ω

πdeXnx j

n

22 21

power spectrum

[ ]nx[ ]

( ) ][][][][

][][

nxnxlnxlx

mxnmxnR

l

mxx


+=

lowastinfin

minusinfin=

infin

minusinfin=

sum

sum

( ) ( ) ( ) ( ) 2 ωωωω XXXS xx ==

hArr


π

π

ωπ

π

ω ωωπ

ωωπ

deXdeSnR njnjxxxx

2

21

21


minusinfin=minus

infin

minusinfin=

===

=

mmxx dXmxmxmxR

nπ

πωω

π22

210

0Set


)(

lnnlmnml



conjugatecomplex





[ ] ( )[ ] [ ] [ ] ( )

[ ] ( ) [ ]

[ ] ( )( )

( ) [ ]( ) ( )( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( )( )[ ] ( ) ( ) ( )( )ωω

ωωωω

φωω

ωφω

ωφω

ωφω

φω

φω

φω

φω

φωφω

φω

ω

ω

jj

jjjj

eHjnjj

eHjjnj

jnj

kj

k

nj

knj

k

nj

nj

eHneHAny

eHneHjAeHneHA

eeHA

eeHAe

eHAe

ekhAe

Aekh

nhAeny

Aenxnxnx

njAnx

j

j

ang++=rArr

ang+++ang++=

=

=

=

sum=

sum=

=

=+=

+=

ang++

ang+

+

minusinfin

minusinfin=

+

+minusinfin

minusinfin=

+

+

cos

sincos

sin

0

10

1

[ ]ny [ ]ny1

( )( ) attenuate 1

amplify 1

lt

gt

ω

ω

j

j

eH

eH





Z-Transform





( ) [ ]suminfin

minusinfin=

minus=n

nznhzH

[ ] ( )zHnh

[ ] ( )ωjeHnh

[ ]nh

ωjrez =

( ) ( ) ωω

jezj zHeH

==

)1( == zez jω

complex plane

unit circle

Im

Re

ω

ωjez =


Z-Transform (cont)


stability



[ ] infinltsuminfin

minusinfin=

minus

n

nznh

complex plan

R1

R2

Re

Im

absolutely summable

1=z


Z-Transform (cont)





[ ] 0for 0 lt= nnh

[ ] 0for 0 gt= nnh


Left-sided sequence

[ ] infinltsuminfin

minusinfin=nnh

[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


Z-Transform (cont)


[ ] [ ] [ ]⎩⎨⎧

ltge

==0for 00for 1

ere wh 1 1 nn

nunuanh n

( ) ( ) 11

1 11

minus

infin

minusinfin=

minusminusinfin


azazzazH

n

nn

n

n

If 11 ltminusaz

azROC gtthere4 is 1

Re

Im

a[ ] 1 if exists

ofansformFourier tr

1 ltanhtimes


az =

the unit cycle

0


Z-Transform (cont)



( ) [ ]

( ) 11

1

10

1

1

12

11

11111

1

minusminus

minus

minus

infin

=

minusinfin

=

minus

minusminus

minusinfin=

minusinfin

minusinfin=

minus=

minusminus

minus=minus

minus=minus=minus=


sumsum

sumsum

azzaza

zazaza

zaznuazH

n

n

n

nn

n

n

nn

n

n If 11 ltminus za

azROC ltthere4 is 2

Re

Im

a[ ] [ ]

minusinfinrarr

lt

nnhnh

a



22times

the unit cycle

0


Z-Transform (cont)


[ ] [ ] [ ]121


⎠⎞

⎜⎝⎛minus⎟

⎠⎞


nn

[ ]

[ ]21

211

1 121

31

311

1 31

1

1

ltminus


⎜⎝⎛minus

gt+


⎜⎝⎛ minus

minus

minus

zz

nu

zz

nu

zn

zn

Re

Im

times

the unit cycle

times21

31



3

3

ROCnh

31 and


( )⎟⎠⎞


⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ minus

=minus

++

=minusminus

21

31

1212

211

1

311

1 11

3

zz

zz

zzzH

0


Z-Transform (cont)



=others 0

10 3 4

Nnanh

n

( ) ( ) ( )azaz

zazazazzazH

NN

N

NN

n

nN

n

nn

minusminus

=minus

minus=== minusminus

minusminus

=

minusminus

=

minus sumsum 11

11

0

11

04

11

1



Im

times

the unit cycle

31

minus


4π

( ) 11 2

minus== N kaez Nkj

k

π

If N=8

0 N-1

Re


Z-Transform (cont)






[ ]nh [ ] 1 0 nnnh le=

01 gen

0rz gtz

infin=z

[ ]nh [ ] 2 0 nnnh ge=

02 ltn 0=z[ ]nh


01 geninfin=zthere4








[ ]nh

[ ] [ ] [ ] [ ]

[ ]

( ) njj

kjnj

knj

eeH

ekhe

ekhnhnxny

ωω

ωω

ω

=

=

=lowast=

minusinfin

infin=

minusinfin

infin=

sum

sum

-k

)(

-k( )



[ ] njenx ω=[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


( )jωeH

( )jωeH

scalar


[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjωnjjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

ang++=

+=

+=

+=

+minusangminus+ang

+minus+

minusminusminus

φω

φωφω

φωφω

ωφωφ




[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0

ωφωφ

φω

minusminus+=

+=

[ ]nh

( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

minus

minus

+=rArr

minus=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω

jωjω

eeHeH

eHeHangminus

minus

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos

sincos

jje

jejyxz

jejyxz

j

j

j

minus=minus+minus=

minus=rArrminus=

+=rArr+=

minus






[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2







[ ]nh( )ωjeH

π2( )ωjeH

ω( ) ( ) ( )

( ) ( )ωω

ωωω

jeHjj

ji

jr

j

eeH

ejHeHeHang=

+=

magnitudephase

( )ωjeH






[ ] infinltsuminfin

minusinfin=nnh

[ ]( )

( )

[ ]mnnmnm

mnmn

emnj

de

mnj

mnj

minus=⎩⎨⎧

ne=

=

minusminus

=

minus= minus

minus

minusminus

int

δ

πππ

ωπ

ππ

ω

ππ

ω

0 1

sin)(2

121

)(

)(

( ) [ ]

[ ] ( ) [ ]


deemhdeeHnh

enheH

m

mnj

m

nj

m

mjnjj

n

njj

=minus==

==

=

sumintsum

int sumint

sum

infin

minusinfin=minus

minusinfin

minusinfin=

minus

infin

minusinfin=

minusminus

infin

minusinfin=

minus

δωπ

ωπ

ωπ

ππ

ω

ππ

ωωππ

ωω

ωω

21

21

21


)(

( ) [ ]

[ ] ( )int

sum

minus

infin

minusinfin=

minus

=

=

ππ

ωω

ωω

ωπ

deeHnh

enheH

njj

n

njj

2

1

absolutely summable

DTFT

Inverse DTFT



( ) [ ]

[ ] [ ] [ ]

( ) [ ] [ ]

[ ] [ ]

( ) ( )

[ ] ( ) ( )ωω

ωω

ωω

ωω

ωω

jj

jj

nj

nk

kj

nj

n k

j

k

n

njj

eHeXnhnx

eHeX

enhekx

eknhkxeY

knhkxnhnxny

enheH

hArrlowastthere4

=

⎟⎠

⎞⎜⎝

⎛=

minus=

minus=lowast=

=

minusinfin

minusinfin=

infin

minusinfin=

minus

minusinfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus

sumsum

sum sum

sum

sum

][

][][

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )ωωω

ωωω

ωωω

jjj

jjj

jjj

eHeXeY

eHeXeY

eHeXeY

ang+ang=angrArr

=rArr

=

knnknnknn






[ ] ( )intsum minus

infin

minusinfin== π

πω ω

πdeXnx j

n

22 21

power spectrum

[ ]nx[ ]

( ) ][][][][

][][

nxnxlnxlx

mxnmxnR

l

mxx


+=

lowastinfin

minusinfin=

infin

minusinfin=

sum

sum

( ) ( ) ( ) ( ) 2 ωωωω XXXS xx ==

hArr


π

π

ωπ

π

ω ωωπ

ωωπ

deXdeSnR njnjxxxx

2

21

21


minusinfin=minus

infin

minusinfin=

===

=

mmxx dXmxmxmxR

nπ

πωω

π22

210

0Set


)(

lnnlmnml



conjugatecomplex





[ ] ( )[ ] [ ] [ ] ( )

[ ] ( ) [ ]

[ ] ( )( )

( ) [ ]( ) ( )( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( )( )[ ] ( ) ( ) ( )( )ωω

ωωωω

φωω

ωφω

ωφω

ωφω

φω

φω

φω

φω

φωφω

φω

ω

ω

jj

jjjj

eHjnjj

eHjjnj

jnj

kj

k

nj

knj

k

nj

nj

eHneHAny

eHneHjAeHneHA

eeHA

eeHAe

eHAe

ekhAe

Aekh

nhAeny

Aenxnxnx

njAnx

j

j

ang++=rArr

ang+++ang++=

=

=

=

sum=

sum=

=

=+=

+=

ang++

ang+

+

minusinfin

minusinfin=

+

+minusinfin

minusinfin=

+

+

cos

sincos

sin

0

10

1

[ ]ny [ ]ny1

( )( ) attenuate 1

amplify 1

lt

gt

ω

ω

j

j

eH

eH





Z-Transform





( ) [ ]suminfin

minusinfin=

minus=n

nznhzH

[ ] ( )zHnh

[ ] ( )ωjeHnh

[ ]nh

ωjrez =

( ) ( ) ωω

jezj zHeH

==

)1( == zez jω

complex plane

unit circle

Im

Re

ω

ωjez =


Z-Transform (cont)


stability



[ ] infinltsuminfin

minusinfin=

minus

n

nznh

complex plan

R1

R2

Re

Im

absolutely summable

1=z


Z-Transform (cont)





[ ] 0for 0 lt= nnh

[ ] 0for 0 gt= nnh


Left-sided sequence

[ ] infinltsuminfin

minusinfin=nnh

[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


Z-Transform (cont)


[ ] [ ] [ ]⎩⎨⎧

ltge

==0for 00for 1

ere wh 1 1 nn

nunuanh n

( ) ( ) 11

1 11

minus

infin

minusinfin=

minusminusinfin


azazzazH

n

nn

n

n

If 11 ltminusaz

azROC gtthere4 is 1

Re

Im

a[ ] 1 if exists

ofansformFourier tr

1 ltanhtimes


az =

the unit cycle

0


Z-Transform (cont)



( ) [ ]

( ) 11

1

10

1

1

12

11

11111

1

minusminus

minus

minus

infin

=

minusinfin

=

minus

minusminus

minusinfin=

minusinfin

minusinfin=

minus=

minusminus

minus=minus

minus=minus=minus=


sumsum

sumsum

azzaza

zazaza

zaznuazH

n

n

n

nn

n

n

nn

n

n If 11 ltminus za

azROC ltthere4 is 2

Re

Im

a[ ] [ ]

minusinfinrarr

lt

nnhnh

a



22times

the unit cycle

0


Z-Transform (cont)


[ ] [ ] [ ]121


⎠⎞

⎜⎝⎛minus⎟

⎠⎞


nn

[ ]

[ ]21

211

1 121

31

311

1 31

1

1

ltminus


⎜⎝⎛minus

gt+


⎜⎝⎛ minus

minus

minus

zz

nu

zz

nu

zn

zn

Re

Im

times

the unit cycle

times21

31



3

3

ROCnh

31 and


( )⎟⎠⎞


⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ minus

=minus

++

=minusminus

21

31

1212

211

1

311

1 11

3

zz

zz

zzzH

0


Z-Transform (cont)



=others 0

10 3 4

Nnanh

n

( ) ( ) ( )azaz

zazazazzazH

NN

N

NN

n

nN

n

nn

minusminus

=minus

minus=== minusminus

minusminus

=

minusminus

=

minus sumsum 11

11

0

11

04

11

1



Im

times

the unit cycle

31

minus


4π

( ) 11 2

minus== N kaez Nkj

k

π

If N=8

0 N-1

Re


Z-Transform (cont)






[ ]nh [ ] 1 0 nnnh le=

01 gen

0rz gtz

infin=z

[ ]nh [ ] 2 0 nnnh ge=

02 ltn 0=z[ ]nh


01 geninfin=zthere4








[ ]nh

[ ] [ ] [ ] [ ]

[ ]

( ) njj

kjnj

knj

eeH

ekhe

ekhnhnxny

ωω

ωω

ω

=

=

=lowast=

minusinfin

infin=

minusinfin

infin=

sum

sum

-k

)(

-k( )



[ ] njenx ω=[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


( )jωeH

( )jωeH

scalar


[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjωnjjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

ang++=

+=

+=

+=

+minusangminus+ang

+minus+

minusminusminus

φω

φωφω

φωφω

ωφωφ




[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0

ωφωφ

φω

minusminus+=

+=

[ ]nh

( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

minus

minus

+=rArr

minus=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω

jωjω

eeHeH

eHeHangminus

minus

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos

sincos

jje

jejyxz

jejyxz

j

j

j

minus=minus+minus=

minus=rArrminus=

+=rArr+=

minus






[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2






[ ] infinltsuminfin

minusinfin=nnh

[ ]( )

( )

[ ]mnnmnm

mnmn

emnj

de

mnj

mnj

minus=⎩⎨⎧

ne=

=

minusminus

=

minus= minus

minus

minusminus

int

δ

πππ

ωπ

ππ

ω

ππ

ω

0 1

sin)(2

121

)(

)(

( ) [ ]

[ ] ( ) [ ]


deemhdeeHnh

enheH

m

mnj

m

nj

m

mjnjj

n

njj

=minus==

==

=

sumintsum

int sumint

sum

infin

minusinfin=minus

minusinfin

minusinfin=

minus

infin

minusinfin=

minusminus

infin

minusinfin=

minus

δωπ

ωπ

ωπ

ππ

ω

ππ

ωωππ

ωω

ωω

21

21

21


)(

( ) [ ]

[ ] ( )int

sum

minus

infin

minusinfin=

minus

=

=

ππ

ωω

ωω

ωπ

deeHnh

enheH

njj

n

njj

2

1

absolutely summable

DTFT

Inverse DTFT



( ) [ ]

[ ] [ ] [ ]

( ) [ ] [ ]

[ ] [ ]

( ) ( )

[ ] ( ) ( )ωω

ωω

ωω

ωω

ωω

jj

jj

nj

nk

kj

nj

n k

j

k

n

njj

eHeXnhnx

eHeX

enhekx

eknhkxeY

knhkxnhnxny

enheH

hArrlowastthere4

=

⎟⎠

⎞⎜⎝

⎛=

minus=

minus=lowast=

=

minusinfin

minusinfin=

infin

minusinfin=

minus

minusinfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus

sumsum

sum sum

sum

sum

][

][][

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )ωωω

ωωω

ωωω

jjj

jjj

jjj

eHeXeY

eHeXeY

eHeXeY

ang+ang=angrArr

=rArr

=

knnknnknn






[ ] ( )intsum minus

infin

minusinfin== π

πω ω

πdeXnx j

n

22 21

power spectrum

[ ]nx[ ]

( ) ][][][][

][][

nxnxlnxlx

mxnmxnR

l

mxx


+=

lowastinfin

minusinfin=

infin

minusinfin=

sum

sum

( ) ( ) ( ) ( ) 2 ωωωω XXXS xx ==

hArr


π

π

ωπ

π

ω ωωπ

ωωπ

deXdeSnR njnjxxxx

2

21

21


minusinfin=minus

infin

minusinfin=

===

=

mmxx dXmxmxmxR

nπ

πωω

π22

210

0Set


)(

lnnlmnml



conjugatecomplex





[ ] ( )[ ] [ ] [ ] ( )

[ ] ( ) [ ]

[ ] ( )( )

( ) [ ]( ) ( )( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( )( )[ ] ( ) ( ) ( )( )ωω

ωωωω

φωω

ωφω

ωφω

ωφω

φω

φω

φω

φω

φωφω

φω

ω

ω

jj

jjjj

eHjnjj

eHjjnj

jnj

kj

k

nj

knj

k

nj

nj

eHneHAny

eHneHjAeHneHA

eeHA

eeHAe

eHAe

ekhAe

Aekh

nhAeny

Aenxnxnx

njAnx

j

j

ang++=rArr

ang+++ang++=

=

=

=

sum=

sum=

=

=+=

+=

ang++

ang+

+

minusinfin

minusinfin=

+

+minusinfin

minusinfin=

+

+

cos

sincos

sin

0

10

1

[ ]ny [ ]ny1

( )( ) attenuate 1

amplify 1

lt

gt

ω

ω

j

j

eH

eH





Z-Transform





( ) [ ]suminfin

minusinfin=

minus=n

nznhzH

[ ] ( )zHnh

[ ] ( )ωjeHnh

[ ]nh

ωjrez =

( ) ( ) ωω

jezj zHeH

==

)1( == zez jω

complex plane

unit circle

Im

Re

ω

ωjez =


Z-Transform (cont)


stability



[ ] infinltsuminfin

minusinfin=

minus

n

nznh

complex plan

R1

R2

Re

Im

absolutely summable

1=z


Z-Transform (cont)





[ ] 0for 0 lt= nnh

[ ] 0for 0 gt= nnh


Left-sided sequence

[ ] infinltsuminfin

minusinfin=nnh

[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


Z-Transform (cont)


[ ] [ ] [ ]⎩⎨⎧

ltge

==0for 00for 1

ere wh 1 1 nn

nunuanh n

( ) ( ) 11

1 11

minus

infin

minusinfin=

minusminusinfin


azazzazH

n

nn

n

n

If 11 ltminusaz

azROC gtthere4 is 1

Re

Im

a[ ] 1 if exists

ofansformFourier tr

1 ltanhtimes


az =

the unit cycle

0


Z-Transform (cont)



( ) [ ]

( ) 11

1

10

1

1

12

11

11111

1

minusminus

minus

minus

infin

=

minusinfin

=

minus

minusminus

minusinfin=

minusinfin

minusinfin=

minus=

minusminus

minus=minus

minus=minus=minus=


sumsum

sumsum

azzaza

zazaza

zaznuazH

n

n

n

nn

n

n

nn

n

n If 11 ltminus za

azROC ltthere4 is 2

Re

Im

a[ ] [ ]

minusinfinrarr

lt

nnhnh

a



22times

the unit cycle

0


Z-Transform (cont)


[ ] [ ] [ ]121


⎠⎞

⎜⎝⎛minus⎟

⎠⎞


nn

[ ]

[ ]21

211

1 121

31

311

1 31

1

1

ltminus


⎜⎝⎛minus

gt+


⎜⎝⎛ minus

minus

minus

zz

nu

zz

nu

zn

zn

Re

Im

times

the unit cycle

times21

31



3

3

ROCnh

31 and


( )⎟⎠⎞


⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ minus

=minus

++

=minusminus

21

31

1212

211

1

311

1 11

3

zz

zz

zzzH

0


Z-Transform (cont)



=others 0

10 3 4

Nnanh

n

( ) ( ) ( )azaz

zazazazzazH

NN

N

NN

n

nN

n

nn

minusminus

=minus

minus=== minusminus

minusminus

=

minusminus

=

minus sumsum 11

11

0

11

04

11

1



Im

times

the unit cycle

31

minus


4π

( ) 11 2

minus== N kaez Nkj

k

π

If N=8

0 N-1

Re


Z-Transform (cont)






[ ]nh [ ] 1 0 nnnh le=

01 gen

0rz gtz

infin=z

[ ]nh [ ] 2 0 nnnh ge=

02 ltn 0=z[ ]nh


01 geninfin=zthere4








[ ]nh

[ ] [ ] [ ] [ ]

[ ]

( ) njj

kjnj

knj

eeH

ekhe

ekhnhnxny

ωω

ωω

ω

=

=

=lowast=

minusinfin

infin=

minusinfin

infin=

sum

sum

-k

)(

-k( )



[ ] njenx ω=[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


( )jωeH

( )jωeH

scalar


[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjωnjjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

ang++=

+=

+=

+=

+minusangminus+ang

+minus+

minusminusminus

φω

φωφω

φωφω

ωφωφ




[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0

ωφωφ

φω

minusminus+=

+=

[ ]nh

( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

minus

minus

+=rArr

minus=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω

jωjω

eeHeH

eHeHangminus

minus

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos

sincos

jje

jejyxz

jejyxz

j

j

j

minus=minus+minus=

minus=rArrminus=

+=rArr+=

minus






[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2




( ) [ ]

[ ] [ ] [ ]

( ) [ ] [ ]

[ ] [ ]

( ) ( )

[ ] ( ) ( )ωω

ωω

ωω

ωω

ωω

jj

jj

nj

nk

kj

nj

n k

j

k

n

njj

eHeXnhnx

eHeX

enhekx

eknhkxeY

knhkxnhnxny

enheH

hArrlowastthere4

=

⎟⎠

⎞⎜⎝

⎛=

minus=

minus=lowast=

=

minusinfin

minusinfin=

infin

minusinfin=

minus

minusinfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus

sumsum

sum sum

sum

sum

][

][][

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )ωωω

ωωω

ωωω

jjj

jjj

jjj

eHeXeY

eHeXeY

eHeXeY

ang+ang=angrArr

=rArr

=

knnknnknn






[ ] ( )intsum minus

infin

minusinfin== π

πω ω

πdeXnx j

n

22 21

power spectrum

[ ]nx[ ]

( ) ][][][][

][][

nxnxlnxlx

mxnmxnR

l

mxx


+=

lowastinfin

minusinfin=

infin

minusinfin=

sum

sum

( ) ( ) ( ) ( ) 2 ωωωω XXXS xx ==

hArr


π

π

ωπ

π

ω ωωπ

ωωπ

deXdeSnR njnjxxxx

2

21

21


minusinfin=minus

infin

minusinfin=

===

=

mmxx dXmxmxmxR

nπ

πωω

π22

210

0Set


)(

lnnlmnml



conjugatecomplex





[ ] ( )[ ] [ ] [ ] ( )

[ ] ( ) [ ]

[ ] ( )( )

( ) [ ]( ) ( )( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( )( )[ ] ( ) ( ) ( )( )ωω

ωωωω

φωω

ωφω

ωφω

ωφω

φω

φω

φω

φω

φωφω

φω

ω

ω

jj

jjjj

eHjnjj

eHjjnj

jnj

kj

k

nj

knj

k

nj

nj

eHneHAny

eHneHjAeHneHA

eeHA

eeHAe

eHAe

ekhAe

Aekh

nhAeny

Aenxnxnx

njAnx

j

j

ang++=rArr

ang+++ang++=

=

=

=

sum=

sum=

=

=+=

+=

ang++

ang+

+

minusinfin

minusinfin=

+

+minusinfin

minusinfin=

+

+

cos

sincos

sin

0

10

1

[ ]ny [ ]ny1

( )( ) attenuate 1

amplify 1

lt

gt

ω

ω

j

j

eH

eH





Z-Transform





( ) [ ]suminfin

minusinfin=

minus=n

nznhzH

[ ] ( )zHnh

[ ] ( )ωjeHnh

[ ]nh

ωjrez =

( ) ( ) ωω

jezj zHeH

==

)1( == zez jω

complex plane

unit circle

Im

Re

ω

ωjez =


Z-Transform (cont)


stability



[ ] infinltsuminfin

minusinfin=

minus

n

nznh

complex plan

R1

R2

Re

Im

absolutely summable

1=z


Z-Transform (cont)





[ ] 0for 0 lt= nnh

[ ] 0for 0 gt= nnh


Left-sided sequence

[ ] infinltsuminfin

minusinfin=nnh

[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


Z-Transform (cont)


[ ] [ ] [ ]⎩⎨⎧

ltge

==0for 00for 1

ere wh 1 1 nn

nunuanh n

( ) ( ) 11

1 11

minus

infin

minusinfin=

minusminusinfin


azazzazH

n

nn

n

n

If 11 ltminusaz

azROC gtthere4 is 1

Re

Im

a[ ] 1 if exists

ofansformFourier tr

1 ltanhtimes


az =

the unit cycle

0


Z-Transform (cont)



( ) [ ]

( ) 11

1

10

1

1

12

11

11111

1

minusminus

minus

minus

infin

=

minusinfin

=

minus

minusminus

minusinfin=

minusinfin

minusinfin=

minus=

minusminus

minus=minus

minus=minus=minus=


sumsum

sumsum

azzaza

zazaza

zaznuazH

n

n

n

nn

n

n

nn

n

n If 11 ltminus za

azROC ltthere4 is 2

Re

Im

a[ ] [ ]

minusinfinrarr

lt

nnhnh

a



22times

the unit cycle

0


Z-Transform (cont)


[ ] [ ] [ ]121


⎠⎞

⎜⎝⎛minus⎟

⎠⎞


nn

[ ]

[ ]21

211

1 121

31

311

1 31

1

1

ltminus


⎜⎝⎛minus

gt+


⎜⎝⎛ minus

minus

minus

zz

nu

zz

nu

zn

zn

Re

Im

times

the unit cycle

times21

31



3

3

ROCnh

31 and


( )⎟⎠⎞


⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ minus

=minus

++

=minusminus

21

31

1212

211

1

311

1 11

3

zz

zz

zzzH

0


Z-Transform (cont)



=others 0

10 3 4

Nnanh

n

( ) ( ) ( )azaz

zazazazzazH

NN

N

NN

n

nN

n

nn

minusminus

=minus

minus=== minusminus

minusminus

=

minusminus

=

minus sumsum 11

11

0

11

04

11

1



Im

times

the unit cycle

31

minus


4π

( ) 11 2

minus== N kaez Nkj

k

π

If N=8

0 N-1

Re


Z-Transform (cont)






[ ]nh [ ] 1 0 nnnh le=

01 gen

0rz gtz

infin=z

[ ]nh [ ] 2 0 nnnh ge=

02 ltn 0=z[ ]nh


01 geninfin=zthere4








[ ]nh

[ ] [ ] [ ] [ ]

[ ]

( ) njj

kjnj

knj

eeH

ekhe

ekhnhnxny

ωω

ωω

ω

=

=

=lowast=

minusinfin

infin=

minusinfin

infin=

sum

sum

-k

)(

-k( )



[ ] njenx ω=[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


( )jωeH

( )jωeH

scalar


[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjωnjjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

ang++=

+=

+=

+=

+minusangminus+ang

+minus+

minusminusminus

φω

φωφω

φωφω

ωφωφ




[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0

ωφωφ

φω

minusminus+=

+=

[ ]nh

( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

minus

minus

+=rArr

minus=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω

jωjω

eeHeH

eHeHangminus

minus

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos

sincos

jje

jejyxz

jejyxz

j

j

j

minus=minus+minus=

minus=rArrminus=

+=rArr+=

minus






[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2






[ ] ( )intsum minus

infin

minusinfin== π

πω ω

πdeXnx j

n

22 21

power spectrum

[ ]nx[ ]

( ) ][][][][

][][

nxnxlnxlx

mxnmxnR

l

mxx


+=

lowastinfin

minusinfin=

infin

minusinfin=

sum

sum

( ) ( ) ( ) ( ) 2 ωωωω XXXS xx ==

hArr


π

π

ωπ

π

ω ωωπ

ωωπ

deXdeSnR njnjxxxx

2

21

21


minusinfin=minus

infin

minusinfin=

===

=

mmxx dXmxmxmxR

nπ

πωω

π22

210

0Set


)(

lnnlmnml



conjugatecomplex





[ ] ( )[ ] [ ] [ ] ( )

[ ] ( ) [ ]

[ ] ( )( )

( ) [ ]( ) ( )( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( )( )[ ] ( ) ( ) ( )( )ωω

ωωωω

φωω

ωφω

ωφω

ωφω

φω

φω

φω

φω

φωφω

φω

ω

ω

jj

jjjj

eHjnjj

eHjjnj

jnj

kj

k

nj

knj

k

nj

nj

eHneHAny

eHneHjAeHneHA

eeHA

eeHAe

eHAe

ekhAe

Aekh

nhAeny

Aenxnxnx

njAnx

j

j

ang++=rArr

ang+++ang++=

=

=

=

sum=

sum=

=

=+=

+=

ang++

ang+

+

minusinfin

minusinfin=

+

+minusinfin

minusinfin=

+

+

cos

sincos

sin

0

10

1

[ ]ny [ ]ny1

( )( ) attenuate 1

amplify 1

lt

gt

ω

ω

j

j

eH

eH





Z-Transform





( ) [ ]suminfin

minusinfin=

minus=n

nznhzH

[ ] ( )zHnh

[ ] ( )ωjeHnh

[ ]nh

ωjrez =

( ) ( ) ωω

jezj zHeH

==

)1( == zez jω

complex plane

unit circle

Im

Re

ω

ωjez =


Z-Transform (cont)


stability



[ ] infinltsuminfin

minusinfin=

minus

n

nznh

complex plan

R1

R2

Re

Im

absolutely summable

1=z


Z-Transform (cont)





[ ] 0for 0 lt= nnh

[ ] 0for 0 gt= nnh


Left-sided sequence

[ ] infinltsuminfin

minusinfin=nnh

[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


Z-Transform (cont)


[ ] [ ] [ ]⎩⎨⎧

ltge

==0for 00for 1

ere wh 1 1 nn

nunuanh n

( ) ( ) 11

1 11

minus

infin

minusinfin=

minusminusinfin


azazzazH

n

nn

n

n

If 11 ltminusaz

azROC gtthere4 is 1

Re

Im

a[ ] 1 if exists

ofansformFourier tr

1 ltanhtimes


az =

the unit cycle

0


Z-Transform (cont)



( ) [ ]

( ) 11

1

10

1

1

12

11

11111

1

minusminus

minus

minus

infin

=

minusinfin

=

minus

minusminus

minusinfin=

minusinfin

minusinfin=

minus=

minusminus

minus=minus

minus=minus=minus=


sumsum

sumsum

azzaza

zazaza

zaznuazH

n

n

n

nn

n

n

nn

n

n If 11 ltminus za

azROC ltthere4 is 2

Re

Im

a[ ] [ ]

minusinfinrarr

lt

nnhnh

a



22times

the unit cycle

0


Z-Transform (cont)


[ ] [ ] [ ]121


⎠⎞

⎜⎝⎛minus⎟

⎠⎞


nn

[ ]

[ ]21

211

1 121

31

311

1 31

1

1

ltminus


⎜⎝⎛minus

gt+


⎜⎝⎛ minus

minus

minus

zz

nu

zz

nu

zn

zn

Re

Im

times

the unit cycle

times21

31



3

3

ROCnh

31 and


( )⎟⎠⎞


⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ minus

=minus

++

=minusminus

21

31

1212

211

1

311

1 11

3

zz

zz

zzzH

0


Z-Transform (cont)



=others 0

10 3 4

Nnanh

n

( ) ( ) ( )azaz

zazazazzazH

NN

N

NN

n

nN

n

nn

minusminus

=minus

minus=== minusminus

minusminus

=

minusminus

=

minus sumsum 11

11

0

11

04

11

1



Im

times

the unit cycle

31

minus


4π

( ) 11 2

minus== N kaez Nkj

k

π

If N=8

0 N-1

Re


Z-Transform (cont)






[ ]nh [ ] 1 0 nnnh le=

01 gen

0rz gtz

infin=z

[ ]nh [ ] 2 0 nnnh ge=

02 ltn 0=z[ ]nh


01 geninfin=zthere4








[ ]nh

[ ] [ ] [ ] [ ]

[ ]

( ) njj

kjnj

knj

eeH

ekhe

ekhnhnxny

ωω

ωω

ω

=

=

=lowast=

minusinfin

infin=

minusinfin

infin=

sum

sum

-k

)(

-k( )



[ ] njenx ω=[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


( )jωeH

( )jωeH

scalar


[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjωnjjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

ang++=

+=

+=

+=

+minusangminus+ang

+minus+

minusminusminus

φω

φωφω

φωφω

ωφωφ




[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0

ωφωφ

φω

minusminus+=

+=

[ ]nh

( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

minus

minus

+=rArr

minus=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω

jωjω

eeHeH

eHeHangminus

minus

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos

sincos

jje

jejyxz

jejyxz

j

j

j

minus=minus+minus=

minus=rArrminus=

+=rArr+=

minus






[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2






[ ] ( )[ ] [ ] [ ] ( )

[ ] ( ) [ ]

[ ] ( )( )

( ) [ ]( ) ( )( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( )( )[ ] ( ) ( ) ( )( )ωω

ωωωω

φωω

ωφω

ωφω

ωφω

φω

φω

φω

φω

φωφω

φω

ω

ω

jj

jjjj

eHjnjj

eHjjnj

jnj

kj

k

nj

knj

k

nj

nj

eHneHAny

eHneHjAeHneHA

eeHA

eeHAe

eHAe

ekhAe

Aekh

nhAeny

Aenxnxnx

njAnx

j

j

ang++=rArr

ang+++ang++=

=

=

=

sum=

sum=

=

=+=

+=

ang++

ang+

+

minusinfin

minusinfin=

+

+minusinfin

minusinfin=

+

+

cos

sincos

sin

0

10

1

[ ]ny [ ]ny1

( )( ) attenuate 1

amplify 1

lt

gt

ω

ω

j

j

eH

eH





Z-Transform





( ) [ ]suminfin

minusinfin=

minus=n

nznhzH

[ ] ( )zHnh

[ ] ( )ωjeHnh

[ ]nh

ωjrez =

( ) ( ) ωω

jezj zHeH

==

)1( == zez jω

complex plane

unit circle

Im

Re

ω

ωjez =


Z-Transform (cont)


stability



[ ] infinltsuminfin

minusinfin=

minus

n

nznh

complex plan

R1

R2

Re

Im

absolutely summable

1=z


Z-Transform (cont)





[ ] 0for 0 lt= nnh

[ ] 0for 0 gt= nnh


Left-sided sequence

[ ] infinltsuminfin

minusinfin=nnh

[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


Z-Transform (cont)


[ ] [ ] [ ]⎩⎨⎧

ltge

==0for 00for 1

ere wh 1 1 nn

nunuanh n

( ) ( ) 11

1 11

minus

infin

minusinfin=

minusminusinfin


azazzazH

n

nn

n

n

If 11 ltminusaz

azROC gtthere4 is 1

Re

Im

a[ ] 1 if exists

ofansformFourier tr

1 ltanhtimes


az =

the unit cycle

0


Z-Transform (cont)



( ) [ ]

( ) 11

1

10

1

1

12

11

11111

1

minusminus

minus

minus

infin

=

minusinfin

=

minus

minusminus

minusinfin=

minusinfin

minusinfin=

minus=

minusminus

minus=minus

minus=minus=minus=


sumsum

sumsum

azzaza

zazaza

zaznuazH

n

n

n

nn

n

n

nn

n

n If 11 ltminus za

azROC ltthere4 is 2

Re

Im

a[ ] [ ]

minusinfinrarr

lt

nnhnh

a



22times

the unit cycle

0


Z-Transform (cont)


[ ] [ ] [ ]121


⎠⎞

⎜⎝⎛minus⎟

⎠⎞


nn

[ ]

[ ]21

211

1 121

31

311

1 31

1

1

ltminus


⎜⎝⎛minus

gt+


⎜⎝⎛ minus

minus

minus

zz

nu

zz

nu

zn

zn

Re

Im

times

the unit cycle

times21

31



3

3

ROCnh

31 and


( )⎟⎠⎞


⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ minus

=minus

++

=minusminus

21

31

1212

211

1

311

1 11

3

zz

zz

zzzH

0


Z-Transform (cont)



=others 0

10 3 4

Nnanh

n

( ) ( ) ( )azaz

zazazazzazH

NN

N

NN

n

nN

n

nn

minusminus

=minus

minus=== minusminus

minusminus

=

minusminus

=

minus sumsum 11

11

0

11

04

11

1



Im

times

the unit cycle

31

minus


4π

( ) 11 2

minus== N kaez Nkj

k

π

If N=8

0 N-1

Re


Z-Transform (cont)






[ ]nh [ ] 1 0 nnnh le=

01 gen

0rz gtz

infin=z

[ ]nh [ ] 2 0 nnnh ge=

02 ltn 0=z[ ]nh


01 geninfin=zthere4








[ ]nh

[ ] [ ] [ ] [ ]

[ ]

( ) njj

kjnj

knj

eeH

ekhe

ekhnhnxny

ωω

ωω

ω

=

=

=lowast=

minusinfin

infin=

minusinfin

infin=

sum

sum

-k

)(

-k( )



[ ] njenx ω=[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


( )jωeH

( )jωeH

scalar


[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjωnjjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

ang++=

+=

+=

+=

+minusangminus+ang

+minus+

minusminusminus

φω

φωφω

φωφω

ωφωφ




[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0

ωφωφ

φω

minusminus+=

+=

[ ]nh

( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

minus

minus

+=rArr

minus=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω

jωjω

eeHeH

eHeHangminus

minus

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos

sincos

jje

jejyxz

jejyxz

j

j

j

minus=minus+minus=

minus=rArrminus=

+=rArr+=

minus






[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2





Z-Transform





( ) [ ]suminfin

minusinfin=

minus=n

nznhzH

[ ] ( )zHnh

[ ] ( )ωjeHnh

[ ]nh

ωjrez =

( ) ( ) ωω

jezj zHeH

==

)1( == zez jω

complex plane

unit circle

Im

Re

ω

ωjez =


Z-Transform (cont)


stability



[ ] infinltsuminfin

minusinfin=

minus

n

nznh

complex plan

R1

R2

Re

Im

absolutely summable

1=z


Z-Transform (cont)





[ ] 0for 0 lt= nnh

[ ] 0for 0 gt= nnh


Left-sided sequence

[ ] infinltsuminfin

minusinfin=nnh

[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


Z-Transform (cont)


[ ] [ ] [ ]⎩⎨⎧

ltge

==0for 00for 1

ere wh 1 1 nn

nunuanh n

( ) ( ) 11

1 11

minus

infin

minusinfin=

minusminusinfin


azazzazH

n

nn

n

n

If 11 ltminusaz

azROC gtthere4 is 1

Re

Im

a[ ] 1 if exists

ofansformFourier tr

1 ltanhtimes


az =

the unit cycle

0


Z-Transform (cont)



( ) [ ]

( ) 11

1

10

1

1

12

11

11111

1

minusminus

minus

minus

infin

=

minusinfin

=

minus

minusminus

minusinfin=

minusinfin

minusinfin=

minus=

minusminus

minus=minus

minus=minus=minus=


sumsum

sumsum

azzaza

zazaza

zaznuazH

n

n

n

nn

n

n

nn

n

n If 11 ltminus za

azROC ltthere4 is 2

Re

Im

a[ ] [ ]

minusinfinrarr

lt

nnhnh

a



22times

the unit cycle

0


Z-Transform (cont)


[ ] [ ] [ ]121


⎠⎞

⎜⎝⎛minus⎟

⎠⎞


nn

[ ]

[ ]21

211

1 121

31

311

1 31

1

1

ltminus


⎜⎝⎛minus

gt+


⎜⎝⎛ minus

minus

minus

zz

nu

zz

nu

zn

zn

Re

Im

times

the unit cycle

times21

31



3

3

ROCnh

31 and


( )⎟⎠⎞


⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ minus

=minus

++

=minusminus

21

31

1212

211

1

311

1 11

3

zz

zz

zzzH

0


Z-Transform (cont)



=others 0

10 3 4

Nnanh

n

( ) ( ) ( )azaz

zazazazzazH

NN

N

NN

n

nN

n

nn

minusminus

=minus

minus=== minusminus

minusminus

=

minusminus

=

minus sumsum 11

11

0

11

04

11

1



Im

times

the unit cycle

31

minus


4π

( ) 11 2

minus== N kaez Nkj

k

π

If N=8

0 N-1

Re


Z-Transform (cont)






[ ]nh [ ] 1 0 nnnh le=

01 gen

0rz gtz

infin=z

[ ]nh [ ] 2 0 nnnh ge=

02 ltn 0=z[ ]nh


01 geninfin=zthere4








[ ]nh

[ ] [ ] [ ] [ ]

[ ]

( ) njj

kjnj

knj

eeH

ekhe

ekhnhnxny

ωω

ωω

ω

=

=

=lowast=

minusinfin

infin=

minusinfin

infin=

sum

sum

-k

)(

-k( )



[ ] njenx ω=[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


( )jωeH

( )jωeH

scalar


[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjωnjjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

ang++=

+=

+=

+=

+minusangminus+ang

+minus+

minusminusminus

φω

φωφω

φωφω

ωφωφ




[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0

ωφωφ

φω

minusminus+=

+=

[ ]nh

( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

minus

minus

+=rArr

minus=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω

jωjω

eeHeH

eHeHangminus

minus

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos

sincos

jje

jejyxz

jejyxz

j

j

j

minus=minus+minus=

minus=rArrminus=

+=rArr+=

minus






[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2



Z-Transform





( ) [ ]suminfin

minusinfin=

minus=n

nznhzH

[ ] ( )zHnh

[ ] ( )ωjeHnh

[ ]nh

ωjrez =

( ) ( ) ωω

jezj zHeH

==

)1( == zez jω

complex plane

unit circle

Im

Re

ω

ωjez =


Z-Transform (cont)


stability



[ ] infinltsuminfin

minusinfin=

minus

n

nznh

complex plan

R1

R2

Re

Im

absolutely summable

1=z


Z-Transform (cont)





[ ] 0for 0 lt= nnh

[ ] 0for 0 gt= nnh


Left-sided sequence

[ ] infinltsuminfin

minusinfin=nnh

[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


Z-Transform (cont)


[ ] [ ] [ ]⎩⎨⎧

ltge

==0for 00for 1

ere wh 1 1 nn

nunuanh n

( ) ( ) 11

1 11

minus

infin

minusinfin=

minusminusinfin


azazzazH

n

nn

n

n

If 11 ltminusaz

azROC gtthere4 is 1

Re

Im

a[ ] 1 if exists

ofansformFourier tr

1 ltanhtimes


az =

the unit cycle

0


Z-Transform (cont)



( ) [ ]

( ) 11

1

10

1

1

12

11

11111

1

minusminus

minus

minus

infin

=

minusinfin

=

minus

minusminus

minusinfin=

minusinfin

minusinfin=

minus=

minusminus

minus=minus

minus=minus=minus=


sumsum

sumsum

azzaza

zazaza

zaznuazH

n

n

n

nn

n

n

nn

n

n If 11 ltminus za

azROC ltthere4 is 2

Re

Im

a[ ] [ ]

minusinfinrarr

lt

nnhnh

a



22times

the unit cycle

0


Z-Transform (cont)


[ ] [ ] [ ]121


⎠⎞

⎜⎝⎛minus⎟

⎠⎞


nn

[ ]

[ ]21

211

1 121

31

311

1 31

1

1

ltminus


⎜⎝⎛minus

gt+


⎜⎝⎛ minus

minus

minus

zz

nu

zz

nu

zn

zn

Re

Im

times

the unit cycle

times21

31



3

3

ROCnh

31 and


( )⎟⎠⎞


⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ minus

=minus

++

=minusminus

21

31

1212

211

1

311

1 11

3

zz

zz

zzzH

0


Z-Transform (cont)



=others 0

10 3 4

Nnanh

n

( ) ( ) ( )azaz

zazazazzazH

NN

N

NN

n

nN

n

nn

minusminus

=minus

minus=== minusminus

minusminus

=

minusminus

=

minus sumsum 11

11

0

11

04

11

1



Im

times

the unit cycle

31

minus


4π

( ) 11 2

minus== N kaez Nkj

k

π

If N=8

0 N-1

Re


Z-Transform (cont)






[ ]nh [ ] 1 0 nnnh le=

01 gen

0rz gtz

infin=z

[ ]nh [ ] 2 0 nnnh ge=

02 ltn 0=z[ ]nh


01 geninfin=zthere4








[ ]nh

[ ] [ ] [ ] [ ]

[ ]

( ) njj

kjnj

knj

eeH

ekhe

ekhnhnxny

ωω

ωω

ω

=

=

=lowast=

minusinfin

infin=

minusinfin

infin=

sum

sum

-k

)(

-k( )



[ ] njenx ω=[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


( )jωeH

( )jωeH

scalar


[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjωnjjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

ang++=

+=

+=

+=

+minusangminus+ang

+minus+

minusminusminus

φω

φωφω

φωφω

ωφωφ




[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0

ωφωφ

φω

minusminus+=

+=

[ ]nh

( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

minus

minus

+=rArr

minus=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω

jωjω

eeHeH

eHeHangminus

minus

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos

sincos

jje

jejyxz

jejyxz

j

j

j

minus=minus+minus=

minus=rArrminus=

+=rArr+=

minus






[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2



Z-Transform (cont)


stability



[ ] infinltsuminfin

minusinfin=

minus

n

nznh

complex plan

R1

R2

Re

Im

absolutely summable

1=z


Z-Transform (cont)





[ ] 0for 0 lt= nnh

[ ] 0for 0 gt= nnh


Left-sided sequence

[ ] infinltsuminfin

minusinfin=nnh

[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


Z-Transform (cont)


[ ] [ ] [ ]⎩⎨⎧

ltge

==0for 00for 1

ere wh 1 1 nn

nunuanh n

( ) ( ) 11

1 11

minus

infin

minusinfin=

minusminusinfin


azazzazH

n

nn

n

n

If 11 ltminusaz

azROC gtthere4 is 1

Re

Im

a[ ] 1 if exists

ofansformFourier tr

1 ltanhtimes


az =

the unit cycle

0


Z-Transform (cont)



( ) [ ]

( ) 11

1

10

1

1

12

11

11111

1

minusminus

minus

minus

infin

=

minusinfin

=

minus

minusminus

minusinfin=

minusinfin

minusinfin=

minus=

minusminus

minus=minus

minus=minus=minus=


sumsum

sumsum

azzaza

zazaza

zaznuazH

n

n

n

nn

n

n

nn

n

n If 11 ltminus za

azROC ltthere4 is 2

Re

Im

a[ ] [ ]

minusinfinrarr

lt

nnhnh

a



22times

the unit cycle

0


Z-Transform (cont)


[ ] [ ] [ ]121


⎠⎞

⎜⎝⎛minus⎟

⎠⎞


nn

[ ]

[ ]21

211

1 121

31

311

1 31

1

1

ltminus


⎜⎝⎛minus

gt+


⎜⎝⎛ minus

minus

minus

zz

nu

zz

nu

zn

zn

Re

Im

times

the unit cycle

times21

31



3

3

ROCnh

31 and


( )⎟⎠⎞


⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ minus

=minus

++

=minusminus

21

31

1212

211

1

311

1 11

3

zz

zz

zzzH

0


Z-Transform (cont)



=others 0

10 3 4

Nnanh

n

( ) ( ) ( )azaz

zazazazzazH

NN

N

NN

n

nN

n

nn

minusminus

=minus

minus=== minusminus

minusminus

=

minusminus

=

minus sumsum 11

11

0

11

04

11

1



Im

times

the unit cycle

31

minus


4π

( ) 11 2

minus== N kaez Nkj

k

π

If N=8

0 N-1

Re


Z-Transform (cont)






[ ]nh [ ] 1 0 nnnh le=

01 gen

0rz gtz

infin=z

[ ]nh [ ] 2 0 nnnh ge=

02 ltn 0=z[ ]nh


01 geninfin=zthere4








[ ]nh

[ ] [ ] [ ] [ ]

[ ]

( ) njj

kjnj

knj

eeH

ekhe

ekhnhnxny

ωω

ωω

ω

=

=

=lowast=

minusinfin

infin=

minusinfin

infin=

sum

sum

-k

)(

-k( )



[ ] njenx ω=[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


( )jωeH

( )jωeH

scalar


[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjωnjjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

ang++=

+=

+=

+=

+minusangminus+ang

+minus+

minusminusminus

φω

φωφω

φωφω

ωφωφ




[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0

ωφωφ

φω

minusminus+=

+=

[ ]nh

( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

minus

minus

+=rArr

minus=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω

jωjω

eeHeH

eHeHangminus

minus

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos

sincos

jje

jejyxz

jejyxz

j

j

j

minus=minus+minus=

minus=rArrminus=

+=rArr+=

minus






[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2



Z-Transform (cont)





[ ] 0for 0 lt= nnh

[ ] 0for 0 gt= nnh


Left-sided sequence

[ ] infinltsuminfin

minusinfin=nnh

[ ] [ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


Z-Transform (cont)


[ ] [ ] [ ]⎩⎨⎧

ltge

==0for 00for 1

ere wh 1 1 nn

nunuanh n

( ) ( ) 11

1 11

minus

infin

minusinfin=

minusminusinfin


azazzazH

n

nn

n

n

If 11 ltminusaz

azROC gtthere4 is 1

Re

Im

a[ ] 1 if exists

ofansformFourier tr

1 ltanhtimes


az =

the unit cycle

0


Z-Transform (cont)



( ) [ ]

( ) 11

1

10

1

1

12

11

11111

1

minusminus

minus

minus

infin

=

minusinfin

=

minus

minusminus

minusinfin=

minusinfin

minusinfin=

minus=

minusminus

minus=minus

minus=minus=minus=


sumsum

sumsum

azzaza

zazaza

zaznuazH

n

n

n

nn

n

n

nn

n

n If 11 ltminus za

azROC ltthere4 is 2

Re

Im

a[ ] [ ]

minusinfinrarr

lt

nnhnh

a



22times

the unit cycle

0


Z-Transform (cont)


[ ] [ ] [ ]121


⎠⎞

⎜⎝⎛minus⎟

⎠⎞


nn

[ ]

[ ]21

211

1 121

31

311

1 31

1

1

ltminus


⎜⎝⎛minus

gt+


⎜⎝⎛ minus

minus

minus

zz

nu

zz

nu

zn

zn

Re

Im

times

the unit cycle

times21

31



3

3

ROCnh

31 and


( )⎟⎠⎞


⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ minus

=minus

++

=minusminus

21

31

1212

211

1

311

1 11

3

zz

zz

zzzH

0


Z-Transform (cont)



=others 0

10 3 4

Nnanh

n

( ) ( ) ( )azaz

zazazazzazH

NN

N

NN

n

nN

n

nn

minusminus

=minus

minus=== minusminus

minusminus

=

minusminus

=

minus sumsum 11

11

0

11

04

11

1



Im

times

the unit cycle

31

minus


4π

( ) 11 2

minus== N kaez Nkj

k

π

If N=8

0 N-1

Re


Z-Transform (cont)






[ ]nh [ ] 1 0 nnnh le=

01 gen

0rz gtz

infin=z

[ ]nh [ ] 2 0 nnnh ge=

02 ltn 0=z[ ]nh


01 geninfin=zthere4








[ ]nh

[ ] [ ] [ ] [ ]

[ ]

( ) njj

kjnj

knj

eeH

ekhe

ekhnhnxny

ωω

ωω

ω

=

=

=lowast=

minusinfin

infin=

minusinfin

infin=

sum

sum

-k

)(

-k( )



[ ] njenx ω=[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


( )jωeH

( )jωeH

scalar


[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjωnjjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

ang++=

+=

+=

+=

+minusangminus+ang

+minus+

minusminusminus

φω

φωφω

φωφω

ωφωφ




[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0

ωφωφ

φω

minusminus+=

+=

[ ]nh

( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

minus

minus

+=rArr

minus=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω

jωjω

eeHeH

eHeHangminus

minus

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos

sincos

jje

jejyxz

jejyxz

j

j

j

minus=minus+minus=

minus=rArrminus=

+=rArr+=

minus






[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2



Z-Transform (cont)


[ ] [ ] [ ]⎩⎨⎧

ltge

==0for 00for 1

ere wh 1 1 nn

nunuanh n

( ) ( ) 11

1 11

minus

infin

minusinfin=

minusminusinfin


azazzazH

n

nn

n

n

If 11 ltminusaz

azROC gtthere4 is 1

Re

Im

a[ ] 1 if exists

ofansformFourier tr

1 ltanhtimes


az =

the unit cycle

0


Z-Transform (cont)



( ) [ ]

( ) 11

1

10

1

1

12

11

11111

1

minusminus

minus

minus

infin

=

minusinfin

=

minus

minusminus

minusinfin=

minusinfin

minusinfin=

minus=

minusminus

minus=minus

minus=minus=minus=


sumsum

sumsum

azzaza

zazaza

zaznuazH

n

n

n

nn

n

n

nn

n

n If 11 ltminus za

azROC ltthere4 is 2

Re

Im

a[ ] [ ]

minusinfinrarr

lt

nnhnh

a



22times

the unit cycle

0


Z-Transform (cont)


[ ] [ ] [ ]121


⎠⎞

⎜⎝⎛minus⎟

⎠⎞


nn

[ ]

[ ]21

211

1 121

31

311

1 31

1

1

ltminus


⎜⎝⎛minus

gt+


⎜⎝⎛ minus

minus

minus

zz

nu

zz

nu

zn

zn

Re

Im

times

the unit cycle

times21

31



3

3

ROCnh

31 and


( )⎟⎠⎞


⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ minus

=minus

++

=minusminus

21

31

1212

211

1

311

1 11

3

zz

zz

zzzH

0


Z-Transform (cont)



=others 0

10 3 4

Nnanh

n

( ) ( ) ( )azaz

zazazazzazH

NN

N

NN

n

nN

n

nn

minusminus

=minus

minus=== minusminus

minusminus

=

minusminus

=

minus sumsum 11

11

0

11

04

11

1



Im

times

the unit cycle

31

minus


4π

( ) 11 2

minus== N kaez Nkj

k

π

If N=8

0 N-1

Re


Z-Transform (cont)






[ ]nh [ ] 1 0 nnnh le=

01 gen

0rz gtz

infin=z

[ ]nh [ ] 2 0 nnnh ge=

02 ltn 0=z[ ]nh


01 geninfin=zthere4








[ ]nh

[ ] [ ] [ ] [ ]

[ ]

( ) njj

kjnj

knj

eeH

ekhe

ekhnhnxny

ωω

ωω

ω

=

=

=lowast=

minusinfin

infin=

minusinfin

infin=

sum

sum

-k

)(

-k( )



[ ] njenx ω=[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


( )jωeH

( )jωeH

scalar


[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjωnjjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

ang++=

+=

+=

+=

+minusangminus+ang

+minus+

minusminusminus

φω

φωφω

φωφω

ωφωφ




[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0

ωφωφ

φω

minusminus+=

+=

[ ]nh

( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

minus

minus

+=rArr

minus=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω

jωjω

eeHeH

eHeHangminus

minus

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos

sincos

jje

jejyxz

jejyxz

j

j

j

minus=minus+minus=

minus=rArrminus=

+=rArr+=

minus






[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2



Z-Transform (cont)



( ) [ ]

( ) 11

1

10

1

1

12

11

11111

1

minusminus

minus

minus

infin

=

minusinfin

=

minus

minusminus

minusinfin=

minusinfin

minusinfin=

minus=

minusminus

minus=minus

minus=minus=minus=


sumsum

sumsum

azzaza

zazaza

zaznuazH

n

n

n

nn

n

n

nn

n

n If 11 ltminus za

azROC ltthere4 is 2

Re

Im

a[ ] [ ]

minusinfinrarr

lt

nnhnh

a



22times

the unit cycle

0


Z-Transform (cont)


[ ] [ ] [ ]121


⎠⎞

⎜⎝⎛minus⎟

⎠⎞


nn

[ ]

[ ]21

211

1 121

31

311

1 31

1

1

ltminus


⎜⎝⎛minus

gt+


⎜⎝⎛ minus

minus

minus

zz

nu

zz

nu

zn

zn

Re

Im

times

the unit cycle

times21

31



3

3

ROCnh

31 and


( )⎟⎠⎞


⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ minus

=minus

++

=minusminus

21

31

1212

211

1

311

1 11

3

zz

zz

zzzH

0


Z-Transform (cont)



=others 0

10 3 4

Nnanh

n

( ) ( ) ( )azaz

zazazazzazH

NN

N

NN

n

nN

n

nn

minusminus

=minus

minus=== minusminus

minusminus

=

minusminus

=

minus sumsum 11

11

0

11

04

11

1



Im

times

the unit cycle

31

minus


4π

( ) 11 2

minus== N kaez Nkj

k

π

If N=8

0 N-1

Re


Z-Transform (cont)






[ ]nh [ ] 1 0 nnnh le=

01 gen

0rz gtz

infin=z

[ ]nh [ ] 2 0 nnnh ge=

02 ltn 0=z[ ]nh


01 geninfin=zthere4








[ ]nh

[ ] [ ] [ ] [ ]

[ ]

( ) njj

kjnj

knj

eeH

ekhe

ekhnhnxny

ωω

ωω

ω

=

=

=lowast=

minusinfin

infin=

minusinfin

infin=

sum

sum

-k

)(

-k( )



[ ] njenx ω=[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


( )jωeH

( )jωeH

scalar


[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjωnjjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

ang++=

+=

+=

+=

+minusangminus+ang

+minus+

minusminusminus

φω

φωφω

φωφω

ωφωφ




[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0

ωφωφ

φω

minusminus+=

+=

[ ]nh

( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

minus

minus

+=rArr

minus=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω

jωjω

eeHeH

eHeHangminus

minus

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos

sincos

jje

jejyxz

jejyxz

j

j

j

minus=minus+minus=

minus=rArrminus=

+=rArr+=

minus






[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2



Z-Transform (cont)


[ ] [ ] [ ]121


⎠⎞

⎜⎝⎛minus⎟

⎠⎞


nn

[ ]

[ ]21

211

1 121

31

311

1 31

1

1

ltminus


⎜⎝⎛minus

gt+


⎜⎝⎛ minus

minus

minus

zz

nu

zz

nu

zn

zn

Re

Im

times

the unit cycle

times21

31



3

3

ROCnh

31 and


( )⎟⎠⎞


⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ minus

=minus

++

=minusminus

21

31

1212

211

1

311

1 11

3

zz

zz

zzzH

0


Z-Transform (cont)



=others 0

10 3 4

Nnanh

n

( ) ( ) ( )azaz

zazazazzazH

NN

N

NN

n

nN

n

nn

minusminus

=minus

minus=== minusminus

minusminus

=

minusminus

=

minus sumsum 11

11

0

11

04

11

1



Im

times

the unit cycle

31

minus


4π

( ) 11 2

minus== N kaez Nkj

k

π

If N=8

0 N-1

Re


Z-Transform (cont)






[ ]nh [ ] 1 0 nnnh le=

01 gen

0rz gtz

infin=z

[ ]nh [ ] 2 0 nnnh ge=

02 ltn 0=z[ ]nh


01 geninfin=zthere4








[ ]nh

[ ] [ ] [ ] [ ]

[ ]

( ) njj

kjnj

knj

eeH

ekhe

ekhnhnxny

ωω

ωω

ω

=

=

=lowast=

minusinfin

infin=

minusinfin

infin=

sum

sum

-k

)(

-k( )



[ ] njenx ω=[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


( )jωeH

( )jωeH

scalar


[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjωnjjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

ang++=

+=

+=

+=

+minusangminus+ang

+minus+

minusminusminus

φω

φωφω

φωφω

ωφωφ




[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0

ωφωφ

φω

minusminus+=

+=

[ ]nh

( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

minus

minus

+=rArr

minus=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω

jωjω

eeHeH

eHeHangminus

minus

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos

sincos

jje

jejyxz

jejyxz

j

j

j

minus=minus+minus=

minus=rArrminus=

+=rArr+=

minus






[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2



Z-Transform (cont)



=others 0

10 3 4

Nnanh

n

( ) ( ) ( )azaz

zazazazzazH

NN

N

NN

n

nN

n

nn

minusminus

=minus

minus=== minusminus

minusminus

=

minusminus

=

minus sumsum 11

11

0

11

04

11

1



Im

times

the unit cycle

31

minus


4π

( ) 11 2

minus== N kaez Nkj

k

π

If N=8

0 N-1

Re


Z-Transform (cont)






[ ]nh [ ] 1 0 nnnh le=

01 gen

0rz gtz

infin=z

[ ]nh [ ] 2 0 nnnh ge=

02 ltn 0=z[ ]nh


01 geninfin=zthere4








[ ]nh

[ ] [ ] [ ] [ ]

[ ]

( ) njj

kjnj

knj

eeH

ekhe

ekhnhnxny

ωω

ωω

ω

=

=

=lowast=

minusinfin

infin=

minusinfin

infin=

sum

sum

-k

)(

-k( )



[ ] njenx ω=[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


( )jωeH

( )jωeH

scalar


[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjωnjjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

ang++=

+=

+=

+=

+minusangminus+ang

+minus+

minusminusminus

φω

φωφω

φωφω

ωφωφ




[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0

ωφωφ

φω

minusminus+=

+=

[ ]nh

( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

minus

minus

+=rArr

minus=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω

jωjω

eeHeH

eHeHangminus

minus

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos

sincos

jje

jejyxz

jejyxz

j

j

j

minus=minus+minus=

minus=rArrminus=

+=rArr+=

minus






[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2



Z-Transform (cont)






[ ]nh [ ] 1 0 nnnh le=

01 gen

0rz gtz

infin=z

[ ]nh [ ] 2 0 nnnh ge=

02 ltn 0=z[ ]nh


01 geninfin=zthere4








[ ]nh

[ ] [ ] [ ] [ ]

[ ]

( ) njj

kjnj

knj

eeH

ekhe

ekhnhnxny

ωω

ωω

ω

=

=

=lowast=

minusinfin

infin=

minusinfin

infin=

sum

sum

-k

)(

-k( )



[ ] njenx ω=[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


( )jωeH

( )jωeH

scalar


[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjωnjjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

ang++=

+=

+=

+=

+minusangminus+ang

+minus+

minusminusminus

φω

φωφω

φωφω

ωφωφ




[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0

ωφωφ

φω

minusminus+=

+=

[ ]nh

( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

minus

minus

+=rArr

minus=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω

jωjω

eeHeH

eHeHangminus

minus

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos

sincos

jje

jejyxz

jejyxz

j

j

j

minus=minus+minus=

minus=rArrminus=

+=rArr+=

minus






[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2









[ ]nh

[ ] [ ] [ ] [ ]

[ ]

( ) njj

kjnj

knj

eeH

ekhe

ekhnhnxny

ωω

ωω

ω

=

=

=lowast=

minusinfin

infin=

minusinfin

infin=

sum

sum

-k

)(

-k( )



[ ] njenx ω=[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


( )jωeH

( )jωeH

scalar


[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjωnjjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

ang++=

+=

+=

+=

+minusangminus+ang

+minus+

minusminusminus

φω

φωφω

φωφω

ωφωφ




[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0

ωφωφ

φω

minusminus+=

+=

[ ]nh

( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

minus

minus

+=rArr

minus=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω

jωjω

eeHeH

eHeHangminus

minus

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos

sincos

jje

jejyxz

jejyxz

j

j

j

minus=minus+minus=

minus=rArrminus=

+=rArr+=

minus






[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2







[ ]nh

[ ] [ ] [ ] [ ]

[ ]

( ) njj

kjnj

knj

eeH

ekhe

ekhnhnxny

ωω

ωω

ω

=

=

=lowast=

minusinfin

infin=

minusinfin

infin=

sum

sum

-k

)(

-k( )



[ ] njenx ω=[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ]knxkh

knhkx

nxnhnhnxny

k

k

minus=

minus=

==

sum

sum

infin

minusinfin=

infin

minusinfin=


( )jωeH

( )jωeH

scalar


[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjωnjjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

ang++=

+=

+=

+=

+minusangminus+ang

+minus+

minusminusminus

φω

φωφω

φωφω

ωφωφ




[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0

ωφωφ

φω

minusminus+=

+=

[ ]nh

( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

minus

minus

+=rArr

minus=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω

jωjω

eeHeH

eHeHangminus

minus

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos

sincos

jje

jejyxz

jejyxz

j

j

j

minus=minus+minus=

minus=rArrminus=

+=rArr+=

minus






[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2



[ ] ( ) ( )

( ) ( )[ ]

( ) ( ) ( ) ( )[ ]( ) ( )[ ]00

00

000

0

0000

0000

0

)()(

)()(

cos 2

2

22

jωjω

njeHjjωnjeHjjω

njjωnjjω

njjjωnjjjω

eHneHA

eeeHeeeHA

eeHeeHA

eeAeHeeAeHny

jωjω

ang++=

+=

+=

+=

+minusangminus+ang

+minus+

minusminusminus

φω

φωφω

φωφω

ωφωφ




[ ] ( )φ+= nwAnx 0cos[ ] ( )

njjnjj eeAeeAnAnx

00

22

cos 0

ωφωφ

φω

minusminus+=

+=

[ ]nh

( )θθ

θ

θ

θ

θθ

θθ

jj

j

j

ee

ieie

minus

minus

+=rArr

minus=

+=

21cos

sincossincos

( ) ( )( ) ( ) ( )0

00

00

jωeHjjωjω

jωjω

eeHeH

eHeHangminus

minus

=

=

( )( ) ( )

ωωωω

ωω

ωω

ω

ω

ω

sincos sincos

sincos

sincos

jje

jejyxz

jejyxz

j

j

j

minus=minus+minus=

minus=rArrminus=

+=rArr+=

minus






[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2






[ ] ( )sum=

+=K

kkkk nAnx

1cos φω

[ ] ( ) ( )[ ]sum=

ang++=K

k

jkk

jk

kk eHneHAny1





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2





[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ


minusinfin=

anuanhk

n

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞


infin

minusinfin=

kPP

eXk

j πωδπω 22

( ) ωω

jj

aeeH minusminus

=1

1

DTFT

DTFT

( ) ( ) ( )

ω

ω

ωωω

π

πωδπ

j

kj

jjj

aeP

kPPae

eXeHeY

minus

infin

minusinfin=minus

minus=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ minussum

minus=

=

112

221

1



πω2

Pk =





21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2






21

[ ] [ ]suminfin

minusinfin=minus=

kkPnnx δ

[ ]⎪⎩

⎪⎨⎧

minus=⎟⎠⎞

⎜⎝⎛

minusminus=

otherwise 0

110 1

2cos460540 NnN

nnw

πHamming window

( ) ( ) ( )

( )

( )

( )

sum

sum sum

sum

sum

infin

minusinfin=

⎟⎠⎞

⎜⎝⎛ minus

infin

minusinfin=

infin=

minusinfin=

infin

minusinfin=

infin

minusinfin=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎭⎬⎫

⎩⎨⎧

⎟⎠⎞


⎟⎠⎞


lowast=

k

kP

j

k

m

m

jm

k

j

k

j

jjj

eWP

mkP

eWP

kP

eWP

kPP

eW

eXeWeY

πω

ω

ω

ωωω

πωδ

πωδ

πωδππ

π

21

2 1

21

2221

21


kP

m πω 2minus=




[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2





[ ] [ ] [ ]sum sum= =

minus+minus=N

k

M

kkk knxknyny

1 0βα

( ) ( ) ( ) kN

k

M

kk

kk zzXzzYzY minus

= =

minussum sum+=1 0

βα

delay property

( ) ( )( ) sum

sum

=

minus

=

minus

minus== N

k

kk

M

k

kk

z

z

zXzYzH

1

0

1 α

β

[ ] ( )[ ] ( ) 0

0nzzXnnx

zXnxminusrarrminus



z-1

z-1

z-1

Mβ

2β

0β [ ]ny[ ]nx

1β

z-1

z-1

z-1

1α

2α

Nα










1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2











1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2









1

111

plusmn

minus ⎥⎦

⎤⎢⎣

⎡

minus azz-a


111

1 =minus minusaz

z-a





( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2






( )( )

( ) ( )( ) ( )

( )( ) ( )( )



where

111




1

11

1

1

1

1

1

minus

minus

minusminus

minusminus

minus

minusminus

minus=

minus=

lt

azza

azzH

azzaazzH

zHzazHzH

zH

aazH


FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2



FIR Filters




( )zH

[ ] [ ]sum=

minus=M

rr rnxny

0β

z-1

z-1

z-1

Mβ

1β0β

[ ]ny[ ]nx

[ ]⎩⎨⎧ lele

=otherwise 00 Mn

nh nβ

( ) ( )( ) sum

=

minus==M

k

kk z

zXzYzH

0β





( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2






( ) ωω α jj eeH minus+= 1( ) ( )( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

+minus=

+=++=

minus+=

ωαωαθ

ωαωαωα

ωωα

ω

ω

cos1sinarctan

cos21sincos1

sincos122

22

j

j

e

jeH

( ) 2log10 ωjeH







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2







( ) [ ]

[ ] ( ) 10 1

10

21

0

21

0

minuslelesum=

minuslelesum=

minus

=

minusminus

=

NnekXN

nx

NnenxkX

knN

jN

k

knN

jN

nπ

π


DFT Analysis



[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2




[ ] [ ]

( )

( ) ( ) ( )

[ ][ ]

[ ]

[ ][ ]

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

minus

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡


⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

minuslele=

minusleleforall



minusminus

=sum

1

10

1

10

1

1

111

10

10

112112

112112

21

0

NX

XX

Nx

xx

ee

ee

NnenxkX

Nk

NNN

jNN

j

NN

jN

j

knN

jN

n

MM

L

MLMM

L

L

ππ

ππ

π




⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2





⎩⎨⎧ =

=summinus

=

minus

otherwise 0 if 11 1

0

2 mNk-re

N

N

n

nrkN

j π

[ ] [ ]

[ ] [ ]( )

[ ]( )

[ ]

[ ] [ ]kn

Nj

nrkN

j

nrkN

jrnN

j

knN

j

enxkX

rX

eN

kX

ekXN

enx

ekXN

nx

N

r

N

n

N

k

N

n

N

k

N

n

N

k

π

π

ππ

π

2

2

22

2

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

1

1

minus

minus

minusminus

sum

sumsum

sum sumsum

sum

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

minus

=

=rArr

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

=rArr

=

[ ] [ ][ ]rX

mNrXkX=

+=




[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2





[ ] ( )sumsumminus

=

minus

=

=1

0

21

0

2 1 N

k

N

nkX

Nnx Energy density





Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2






Analog Signal


nTt =






Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2




Impulse TrainTo


Sampling

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ

( )tx a


Discrete-TimeSignal

Digital Signal


amplitude

[ ] nx


( ) ( )suminfin

minusinfin=minus=

nnTtts δ


switch

( )tx a

( ) ( )suminfin

minusinfin=minus=

nnTtts δ

( )

( )intinfin

infinminus

=

neforall=

1

0 0

dtt

tt

δ

δ1

T0 2T 3T 4T 5T 6T 7T 8T-T-2T




T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2





T

1

( )tx a ( )tx a

( )

( ) ( ) ( ) ( )

( ) ( ) [ ] ( )sumsum

suminfin

minusinfin=

infin

minusinfin=

infin

minusinfin=

minus=minus=

minus==

nna

naa

s

nTtnxnTtnTx

nTttxtstx

tx

δδ

δ



( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2




( )ΩjXa


Tππ

==Ω

⎟⎠⎞

⎜⎝⎛=ΩltΩ

TsNπ

21

⎟⎠⎞

⎜⎝⎛=ΩgtΩ

TsNπ

21

aliasing distortion

( ) ( )suminfin


ksk

TjS δπ2

( ) ( ) ( )

( ) ( )( )suminfin

minusinfin=

ΩminusΩ=Ω

ΩlowastΩ=Ω

ksas

as

kjXT

jX

jSjXjX

121π

NΩ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎭⎬⎫

⎩⎨⎧


2 Ns

NNsQ

( )⎩⎨⎧ ΩltΩ


jRs



Low-pass filter


bull Spectra







( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2








( ) tth sΩ= sinc

( ) ( ) ( )

( ) ( )sum

suminfin

minusinfin=

infin

minusinfin=

minusΩ=

minus=

nsa

naa

nTtnTx

nTthnTxtx

sinc

Ns ΩgtΩ 2


Documents

SP2004F Lecture07-01 Digital Signal Processingberlin.csie.ntnu.edu.tw/Courses/2004F-SpeechRecognition/Slides... · 2004 SP- Berlin Chen 3 Two Main Approaches to Digital Signal Processing