DSP – Quick Reference Table (For EC 272: Digital Signal Processing)

DEPARTMENT OF ELECTRONICS & COMMUNICATION ENGINEERING

NATIONAL INSTITUTE OF TECHNOLOGY KARNATAKA, SURATHKAL

SRINIVASNAGAR 575025 KARNATAKA INDIA

January 2015

Dept. of E&C, NITK Surathkal 2

Z-Transform of Basic Signals

Definition: X(z) =

n

nznx ][ dzzzX

jnx n 1

2

1

Signal Transform ROC

1. [n] 1 z All

2. m][n mz All z except 0 if m> 0

or if m<0

3. u[n] 1z

z 1 z

4. 1]nu[ 1z

z 1z

5. u[n] a n az

z

a z

6. 1]nu[ an az

z

a z

7. u[n] an n)1( 2

az

z 2

a z

8. u[n]n sin 0 1 cosz z

z sin

0

0

22 1 z

9. u[n]n cos 0 1 cos2z z

cosz(z

0

0

2

) 1 z

Dept. of E&C, NITK Surathkal 3

Properties of z-Transform ________________________________________________________________________

Property Signal z-Transform

________________________________________________________________________

x[n] X(z)

x1[n] X1(z) x2[n] X2(z) ________________________________________________________________________

1. Linearity [n]xa[n]xa 2211 (z)Xa(z)Xa 2211

2. shifting Time k]x[n X(z)z k ; sided- twois x[n]

u[k] k]x[n

1-k

0m

kk ][z-X(z)z mzmx ;

sided-right is x[n]

3. expansion Time m of Multiplen ];mnx[ )X(zm

4. Scaling in the

z - domain x[n]a n ) a / X(z

5. Reversal Time x[-n] )X(z 1

6. Differentiation in

the z – domain nnx )(dz

dz zX

7. Convolution [n] x* [n]x 21 (z)(z)XX 21

8. Conjugation [n]x )(zX

________________________________________________________________________

Dept. of E&C, NITK Surathkal 4

Discrete-Time Fourier Series

Definition: x[n]= [ ] Ojk n

k N

X k e

1

[k] [ ]N

0jk n

n N

X x n e

/N 20 period lfundamentaN

Properties of DTFS

________________________________________________________________________

Property Signal DTFS coefficient

________________________________________________________________________

x[n] X[k]

y[n] Y[k]

________________________________________________________________________

1. Linearity Ax[n]+By[n] AX[k]+BY[k]

2. Time shifting x[n-m] ][kX emjk 0

3. Frequency shifting x[n] enjM 0 M]-X[k

4. Time expansion mnx ; n is multiple of m m

1X[k] ; period mN

5. Time Reversal x[-n] X[k]

6. Modulation x[n]y[n] [ ] [ ]m N

X m Y k m

7. Periodic Convolution nx ny kk baN

8. Conjugation x*[n] X*[-k]

________________________________________________________________________

nx ny = m N

x m y n m

Parsevaal’s Relation 2 21

P [ ] X[k]N n N k N

x n

Dept. of E&C, NITK Surathkal 5

Discrete-Time Fourier Transform

Definition: x[n]=j j n j

n -2

1( ) e d ( ) [ ]

2

j nX e X e x n e

DTFT of Basic Signals

____________________________________________________________

Signal Transform

____________________________________________________________

1) [n] 1

2) 1a ; u[n]a n j e a-1

1

3) 1a ; u[n] a )1( n n

2

1

j1-a e

4)

Mn

M

;0

n ; 1x[n]

)2/sin(

/2)1)((2M

sin

5)

M0 ; sin

n

Mn

2

M ;0

0 ; 1 )X(e j

withPeriodic

M

6) N

2 ; e ][ 0

)(

njk 0

Nk

kX

]k-[ ][2-k

0

kX tcoefficien DTFS X[k]

Dept. of E&C, NITK Surathkal 6

Properties of DTFT ____________________________________________________________________

Property Signal Transform

x[n] )X(e j

y[n] )Y(e j

________________________________________________________________________

1. Linearity By[n]Ax[n] )BY(e)AX(e jj

2. Time shifting m]-x[n )X(e e j-j m

3. Frequency shifting x[n]e j n ) e X( )( j

4. Time expansion ; ]mnx[ ) e X( j m

n is multiple of m

5. Time reversal x[-n] ) e X( -j

6. Differentiation in frequency n x[n] ) e X(d

dj j

7. Convolution y[n]*x[n] ) e Y( ) e X( jj

8. Modulation x[n]y[n] ) e Y(* ) e X(2

1 jj

9. Conjugation [n]*x ) e (X -j*

________________________________________________________________________

nconvolutio periodic indicates * :Note

) e Y(* ) e X( jj =

2

jjX e Y e d

Parsevaal’s Relation 22

2

1E [ ] ( )

2

j

n

x n X e d

Dept. of E&C, NITK Surathkal 7

Discrete Fourier Transform

Definition:

1

0

2

)(1

)(N

k

N

knj

ekXN

nx

; n=0,1,2……….N-1

1

0

2

)(][N

n

N

knj

enxkX

; k=0,1,2………..N-1

Where N is the number of samples in the frequency domain in the interval (0 to 2)

Properties of DFT

Property Time domain Signal Frequency domain Signal

x(n) & y(n) X(k) & Y(k)

1) Periodicity x(n)=x(n+N) X(k)=X(k+N)

for all n for all k

2) Linearity Ax(n)+By(n) AX(k)+BY(k)

3) Time reversal x(N-n) X(N-k)

4) Circular x((n-l))N X(k)e-j2kl/N

time shift

5) Circular x(n)ej2ln/N X((k-l))N

frequency shift

6) Complex x*(n) X*(N-k)

conjugate

7) Circular x(n) N y(n) X(k)Y(k)

convolution

8) Circular x(n) N y*(-n) X(k) Y*(k)

correlation

9) Multiplication x(n)y(n) N

1X(k) N Y(k)

of two sequences

10) Parsevaal’s

1

0

* )()(N

n

nynx

1

0

* )()(1 N

k

kYkXN

Theorem

Dept. of E&C, NITK Surathkal 8

Window Functions for FIR Filter Design

Name of window Time domain sequence

1) Rectangular window w(n)=1 ; 0nM-1

=0 ; otherwise

2) Bartlet (triangular) window 1

2

12

1

M

Mn

3) Blackman window 1

4cos08.0

1

2cos5.042.0

M

n

M

n

4) Hamming window 1

2cos46.054.0

M

n

5) Hanning window )1

2cos1(

2

1

M

n

Characteristics of commonly used window functions

Window function

Approximate width of main lobe

NormalisedTransition

width

Stop band attenuation As dB

Passband ripple Rp dB

Main lobe relative to sidelobe dB

Rectangular 4/M 1.8/M -21 0.7416 -13

Hanning 8/M 6.2/M -44 0.0546 -31

Hamming 8/M 6.6/M -53 0.0194 -41

Blackmann 12/M 11/M -74 0.0017 -57

Prototype Butterworth Low Pass Filter Transfer Functions

N HB (s)

1 1/( 1)s

2 21/( 2 1)s s

3 21/( 1)( 1)s s s

4 2 21/( 0.765 1)( 1.8477 1)s s s s

5 2 21/( 1)( 0.618 1)( 1.618 1)s s s s s

Dept. of E&C, NITK Surathkal 9

Filter Transformations for Analog Filters

Filter Type Transformation

1) Low pass sC

s

2) High pass ss

C

3) Band pass ss

sQ

O

O

)(22

4) Band reject s)(

22

O

O

sQ

s

where ΩC is cut-off frequency, 2

0 1 2

12

OQ 1 and 2 are lower and upper cut-off frequencies.

Prototype Low pass Filter has Band Edge Frequency p

Type of Transformation Band edge frequencies of

Transformation new filter

1) Low pass s sp

p

'

p

'

2) High pass ss

p p'

p'

3) Band pass s)(

)( 2

lu

ulp

s

s

ul ,

4) Band reject s)(

)(2

lu

cup

s

s

ul ,

Dept. of E&C, NITK Surathkal 10

Time domain and frequency domain relationships for sampled signals.

( )ax t

( )aX j

( )x n

( )jX e ( )X k

( ) ( )ax n x nT

TIME SAMPLING

sin ( ) /

( )( ) /

r

n

t nT Tx t x nT

t nT T

(

)(

)j

t

aa

Xj

xt

edt

FO

UR

IER

TR

AN

SF

OR

M

1(

)(

)2

jt

aa

xt

Xj

ed

( )

| | / 2

j Ta

s

X j TX e

1

2 /ja

k

X e X j k TT

()

jj

n

n

Xe

xn

e

1(

)(

)2

jj

nx

nX

ee

d

Dis

cret

e T

ime

Fouri

er T

ran

sfo

rm

2

1

0

1

()

() j

kn

N

N

k

xn

Xk

e

N

2

1

0

[]

() j

kn

N

N

n

Xk

xn

e

DISC

RETE

FOURIER

TRANSFO

RM

2 /( ) j k NX k X e

FREQUENCY SAMPLING

21

0

( 1) / 2

( ) ( )

sin( / 2)( )

sin( / 2)

kN jj N

k

j j N

X e X k P e

NP e e

N