Signals & Systems

Lecture 10 (Chapter 5)

Signals & Systems

The Discrete-Time Fourier Transform

Adapted from: Lecture notes from MIT, Purdue, Binghamton University, University of Rochester ,

University of Saskatchewan, Sonoma State University

Dr. Hamid R. Rabiee

Fall 2013


Sharif University of Technology, Department of Computer Engineering, Signals and Systems2

The Discrete-Time Fourier Transform

Derivation: (Analogous to CTFT) except ejωn = ej(ω+2π)n)

• x[n] – aperiodic sequence that is of finite duration

• N is large enough so that x[n] = 0 if |n| ≥ N/2

• = x[n] for [n] ≤ N/2 and periodic with period N][~ nx

[ ] [ ] for any as x n x n n N

n

][nx

2N1N 0

nN2N1N 0N

][~ nx



DTFT Derivation (Continued)

Define:

− Periodic in ω with period 2π

DTFS analysis eq.

DTFS synthesis eq.N

eanxNk

njk

k

2

,][~0

0

Nn

njk

k enxN

a 0][~1

n

njkN

Nn

njkenx

Nenx

N0

2

1

0 ][1

][~1

nj

n

j enxeX

][)(

)(1

0jk

k eXN

a



DTFT Derivation (Home Stretch)

for every n

The sum in (*) → an integral =>

The DTFT Pair

(*) )(2

1 )(

1][~

00000

njk

Nk

jknjk

Nk

jkeeXeeX

Nnx

][][~ : As nxnxN

d00 ,0

Synthesis equation

Analysis equation

deeXnx njj

2

)(2

1][

n

njj enxeX ][)(Any 2π

Interval in ω

ka



DT Fourier Transform Pair

– Synthesis Equation

– Inverse FT

deeXnx njj

2

)(2

1][

n

njj enxeX ][)( – Analysis Equation

– FT

)(][ jweXnx



Fourier Transforms

Transform Time Frequency

Continuous

Periodic

Discrete

Aperiodic

Discrete

Periodic

Discrete

Periodic

Continuous

Aperiodic

Continuous

Aperiodic

Discrete

Aperiodic

Continuous

Periodic

( ) kx t aFs

[ ] kx n aFs

( ) ( )x t X jF

[ ] ( )jx n X e F


DT vs. CT, Integral vs. Series


.2 of period lfundumenta a with offunction periodic a is )(

n andk integers ofpair any for , Since )2(

j

nkjnj

eX

ee

general).in periodicnot is )X(j ansformFourier tr CT that the(Recall

continuous is frequency theseries,Fourier DT the Unlike

.2length of interval

continuousany over taken becan integral synthesis DT theThus

equation. analysis seriesFourier CT theosimilar t is This


Frequency Concept for Discrete-Time Signals


k.integer any for that Recall )n2kj( njee

signals? time-discretefor mean sfrequenciehigh and

low do what n),cos( signals, time-discretefrequency -low toequal are

)n],2cos[( signals, time-discretefrequency -highvery seemingly If

k

sample".per radians" is ofunit that theNote

sample.per radian 2 0.1 offrequency a with one

as same theis sampleper radians 0.1 offrequency a with sinusoidA


Frequency Concept for Discrete-Time Signals(Cont'd)


.magnitudes opposite of samples

econsecutiv obetween tw faster"" oscillatecan signal DT No

sample.per radians 0 than slower"" oscillatecan signal DT No

frequency. eperceivabllowest is )k(2 0, (ii)

signals. DTfor frequency eperceivablhighest is )2 k( , (i)

Thus


Frequency Concept for Discrete-Time Signals(Cont'd)


0.near are that thoseare sfrequencie Low

.near thoseare sfrequencieHigh

between.in thoseare sfrequencie teIntermedia

sample.per cycles 0.5

toequal is sampleper radians frequency,highest that theNote



Convergence Issues

None, since integrating over a finite interval

Need conditions analogous to CTFT, e.g.

— Finite energy

— Absolutely summable

Synthesis Equation:

Analysis Equation:

n

nx2

][

n

nx ][



Examples

][][.1 nnx



Examples

][][.1 nnx

1][)(

n

njj eneX



Examples

1][.2 nx



Examples

1][.2 nx

• Does not satisfy existence conditions

• does not converge in ordinary sense

• Can not use reciprocity as we did for CT case.

n

nje

Consider inverse transform:

deeXnx njj

2

)(2

1][

What function would yield x[n]=1?)( jeX



Examples

Let ),(2)( jeX



Examples

Let

1)(2

1][

2

deeXnx njj

),(2)( jeX

Then

Note that must be periodic with period so we have:)( jeX 2

)2(2)( keXk

j



Examples

3. ?0 DFTnje

4. ?)cos( 0 DFTn



Examples

3. )](2[ 020

repe DFTnj

4. )]()([)cos( 0020 repn DFT



More Examples

1],[][ )4 anuanx n



More Examples

-Exponentially decaying function

Infinite sum formula

1],[][ )4 anuanx n

00

)()(n

nj

n

njnj aeeaeX

1 jae

sin)cos1(

1

1

1

jaaae j

aaaeX

aaaeX

j

j

1

1

21

1)(:

1

1

21

1)(:0

2

2

2cos21

1|)(|

aaeX j


Previous Example, A Better View…


][)5.0( nuofTransformFourier n

)( sampleradFrequency

)( jeX

)( jeX

][nx






Previous Example, A Better View…(Cont’d)




)( jeX

)( jeX

][nx

Negative

Constant



Still More

2for Drawn 1 N5) DT Rectangular Pulse



Still More

2for Drawn 1 N5) DT Rectangular Pulse

)()

2sin(

)2

1(sin

)()( )2(1

1

1

1

1

jN

Nn

njN

Nn

njj eX

N

eeeX




5N Pulser Rectangula DT 1 aofTransformFourier

)( jeX


Five Peaks

In a

Period2


Previous Example, A Better View…(Cont’d)


10N Pulser Rectangula DT 1 aofTransformFourier

)( jeX


Ten Peaks

In a

Period2



And So On…

6)



And So On…

n

Wndenx

W

W

nj

sin

2

1][

W

W

j WdeXx

)(2

1]0[

6)


Using MATLAB to Calculate DFT


1- N = 4;

2- x = [1,2,3,4]

3- for k1=1:N

4- X(k1)=0;

5- k=k1-1;

6- for n1=1:N;

7- n=n1-1;

8- X(k1)=X(k1)+x(n1)*exp(-j*2*pi*k*n/N);

9- end

10- end

11- x

12- X

n

njj enxeXFrom ][)(:



DFT examples

Applet 17


DTFTs of Sums of Complex Exponentials


Recall CT result:

What about DT:

)(2)()( 00

jXetxtj

?)(][ 0 jnjeXenx

factIn

2 period with periodic bemust )X(But b)

at )2 area (of impulsean expect Wea) 0

je

m

j meX )2(2)( 0

Thus, period. 2 onein )( needonly

period, 2over isequation synthesis in then integratio The:Note

jeX

njnj

m

π

πedem 2πnx 0 )2(

2

1][ 0

)( jeX

one


DTFT of Periodic Signals


DTFS

synthesis eq.

Linearity

of DTFT

From the last page:

][ ][ Nnxnx

Neanx

Nk

njk

k

2

,][ 00

m

njkmke )2(2 0

0

Nk m

k

j mkaeX )2(2)( 0

k

k

k

kN

kaka )

2(2)(2 0


Example #1: DT sine function


nnx 0sin][


Example #1: DT sine function


njnje

je

jnnx 00

2

1

2

1sin][ 0

mm

j

j mj

meX )2()2()( 00


Example #2: DT periodic impulse train


— Also periodic impulse train – in the frequency

domain!

NkNnnx

k

2 ][][ 0


Example #2: DT periodic impulse train


— Also periodic impulse train – in the frequency

domain!

NkNnnx

k

2 ][][ 0

Nn

njk

k enxN

a 0 ][1

1

0

1 ][

10

N

n

njk

Nenx

N

][n

k

j

N

k

NeX )

2(

2)(


Properties of the DT Fourier Transform


– Synthesis Equation

– Inverse FT

– Analysis Equation

– FT

— Different from CTFT1) Periodicity:

2) Linearity:

)()( )2( jj eXeX

)()(][][ 2121

jj ebXeaXnbxnax

deeXnx njj

2

)(2

1][

n

njj enxeX ][)(


More Properties


)(][ 0

0

jnjeXennx

)(][)( 00

jnj

eXnxe

Example

3) Time Shifting

4) Frequency Shifting

-Important Implications in DT because of periodicity


More Properties


)(][ 0

0

jnjeXennx

)(][)( 00

jnj

eXnxe

Example

3) Time Shifting

4) Frequency Shifting

-Important Implications in DT because of periodicity


Still More Properties


4) Time Reversal

5) Conjugate Symmetry

)(][ jeXnx

)()(real ][ * jj eXeXnx

functions odd are )( Im and )( jj eXeX

even and real )(even and real ][ jeXnx

functionseven are )( Re and )( jj eXeX

odd andimaginary purely )( odd and real ][ jeXnx

And


Conjugate Symmetry, A Better View…



Conjugate Symmetry, A Better View…



Yet Still More Properties


Time scale in CT is

infinitely fine

Insert two zeros in

this example

(k=3)

7) Time Expansion

Recall CT property: ))((1

)(a

jXa

atx

][ of valuesodd misses ]2[

sense no makes ]2[ :DTin But

nxnx

nx

But we can “slow” a DT signal down by inserting zeros:

k -an integer ≥ 1

- insert (k - 1) zeros between successive values][nxk


Yet Still More Properties


Time scale in CT is

infinitely fine

Insert two zeros in

this example

(k=3)

7) Time Expansion

Recall CT property: ))((1

)(a

jXa

atx

][ of valuesodd misses ]2[

sense no makes ]2[ :DTin But

nxnx

nx

But we can “slow” a DT signal down by inserting zeros:

k -an integer ≥ 1

- insert (k - 1) zeros between successive values][nxk


Time Expansion (continued)


-Stretched by a factor

of k in time domain

-compressed by a factor

of k in frequency domain

otherwise 0

of multipleinteger an is if ][][

knknxnxk

mkj

m

k

mknnj

n

k

j

k emkxenxeX

)( )()()(

)( )( jkmkj

m

eXemx


Is There No End to These Properties?


8) Differentiation in Frequency

Differentiation

in frequency

Multiplication

by n

Total energy in

frequency domain

Total energy in

time domain

multiply by j on both sides

n

njj

nj

n

j

ennxjeXd

d

enxeX

][)(

][)(

)(][

jeX

d

djnnx

deXnx j

n

2

2

2)(

2

1][

9) Parseval’s Relation


The Convolution Property


][nx ][*][][ nxnhny ][nh

)()()( jjj eXeHeY

Frequency response DTFT of the unit sample response)( jeH


Example #1

nj

enx 0][



Example #1

k

jPeriodicH

k

kj

k

jj

k

jnj

keH

keH

keHeY

keXenx

)2( 2)(

)2()( 2

)2( 2)( )(

)2(2 )(][

0

0

)2(

0

0

0

0

0

njjeeHny 00 )( ][



Example #2: Ideal Lowpass Filter



Example #2: Ideal Lowpass Filter

n

ndenh cnjc

c

sin

2

1][



Example #3


n

n

n

n

n

n

)4sin()2sin(*

)4sin(


Example #3


n

n

n

n

n

n

)4sin()2sin(*

)4sin(


DTFS Properties…


Applet 16

DTFS

Properties

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