ELEC264: Signals And Systems Topic 5:Discrete …amer/teach/elec264/notes/topic5...ELEC264: Signals...

oIntroduction

o DT Fourier Transform

o Sufficient condition for the DTFT

o DT Fourier Transform of Periodic Signals

o DTFT and LTI systems: Frequency response

o Properties of DT Fourier Transform

o Summary

o Appendix: Transition from DT Fourier Series to DT Fourier Transform

o Appendix: Relations among Fourier Methods

ELEC264: Signals And Systems

Topic 5:Discrete-Time Fourier

Transform (DTFT)

Aishy Amer

Concordia University

Electrical and Computer Engineering

Figures and examples in these course slides are taken from the following sources:

•A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997

•M.J. Roberts, Signals and Systems, McGraw Hill, 2004

•J. McClellan, R. Schafer, M. Yoder, Signal Processing First, Prentice Hall, 2003

2

Fourier representation

A Fourier function is unique, i.e., no two same signals in time give the same function in frequency

The DT Fourier Series is a good analysis tool for systems with periodic excitation but cannotrepresent an aperiodic DT signal for all time

The DT Fourier Transform can represent an aperiodic discrete-time signal for all time Its development follows exactly the same as that of the

Fourier transform for continuous-time aperiodic signals

3

Overview of Frequency Analysis

Methods

4

Overview of Fourier Analysis

Methods

Periodic in Time

Discrete in Frequency

Aperiodic in Time

Continuous in Frequency

Continuous

in Time

Aperiodic in

Frequency

Discrete in

Time

Periodic in

Frequency

k

tjk

k

T

tjk

k

eatx

dtetxT

a

0

0

)(

P-CTDT :SeriesFourier Inverse CT

)(1

DTP-CT :SeriesFourier CT

T

0

T

2

2

2

)(2

1][

DT PCT :TransformFourier DT Inverse

][)(

PCTDT :TransformFourier DT

deeXnx

enxeX

njj

n

njj

1

0

NN

1

0

NN

0

0

][1

][

P-DTP-DT SeriesFourier DT Inverse

][][

P-DTP-DT SeriesFourier DT

N

k

knj

N

n

knj

ekXN

nx

enxkX

dejXtx

dtetxjX

tj

tj

)(2

1)(

CTCT :TransformFourier CT Inverse

)()(

CTCT :TransformFourier CT

5

Overview of Fourier symbols

Variable Period Continuous

Frequency

Discrete

Frequency

DT x[n] n N k

CT x(t) t T k

Nkk /2

Tkk /2

•DT-FT: Discrete in time; Aperiodic in time; Continous in Frequency; Periodic in Frequency

•DT-FS: Discrete in time; Periodic in time; Discrete in Frequency; Periodic in Frequency

•CT-FS: Continuous in time; Periodic in time; Discrete in Frequency; Aperiodic in Frequency

• CT-FT: Continuous in time; Aperiodic in time; Continous in Frequency; Aperiodic in Frequency

6

Outline

o Introduction


o Sufficient condition for DTFT



o DTFT & LTI systems: Frequency response

o DTFT: Summary

o Appendix: Transition from DT Fourier Series to DT

Fourier Transform


7

DT Fourier Transform

DT Fourier transform and the inverse FT

FT describes which frequencies are present in the original function

The original signal can be recovered from knowing the Fourier

transform, and vice versa

The function X(ejω) is periodic in ω with period 2π

(The function ejω is periodic with N=2π)

deeXnxenxeX njj

n

njj )(2

1][,][)(

n

fnj

n

njj enxfXenxeX 2][)( ][)(:Forms

8


DT signal representations:

A sum of scaled, delayed impulse

A linear combination of weighted sinusoidal signals

k

knkxnx ][][][

deeXnx njj )(2

1][

9

DT Fourier Transform: Derivation

Let x[n] be the aperiodic DT signal

We construct a periodic signal ˜x[n] for which x[n] is one period

˜x[n] is comprised of infinite number of replicas of x[n]

Each replica is centered at an integer multiple of N

N is the period of ˜x[n]

Consider the following figure which illustrates an example of x[n] and the construction of

Clearly, x[n] is defined between −N1 and N2

Consequently, N has to be chosen such that N > N1 + N2 + 1 so that adjacent replicas do not overlap

Clearly, as we let

as desired

10


Let us now examine the FS representation of

Since x[n] is defined between −N1 and N2

ak in the above expression simplifies to

ω = 2π/N

11


Now defining the function

We can see that the coefficients ak are related to

X(ejω) as

where ω0 = 2π/N is the spacing of the samples in

the frequency domain

Therefore

As N increases ω0 decreases, and as N → ∞ the

above equation becomes an integral

12


One important observation here is that the

function X(ejω) is periodic in ω with period 2π

Therefore, as N → ∞,

(Note: the function ejω is periodic with N=2π)

This leads us to the DT-FT pair of equations

13

DT Fourier Transform: Examples

)2(2)( 1][ reXnxr

j

The periodic impulse train

j

jn

aeeXanuanx

1

1 )( 1|| ][][Let

14

DT Fourier Transform: Examples

Periodic

Aperiodic

15

Fourier transform pairs

16

Outline

o Introduction






o DTFT: Summary


Fourier Transform


17

Sufficient condition for DTFT

Condition for the convergence of the infinite sum

If x[n] is absolutely summable, its FT exists

(sufficient condition)

|][| |||][|

|][| |)(|

nxenx

enxeX

nj

njj

18

Example: Exponential sequence

existnot does DTFT :1||

)2(1

1 )( :1

1

1 )( :1|| ][][

a

ke

eXa

aeeXanuanx

kj

j

j

jn

19

Outline

o Introduction






o DTFT: Summary


Fourier Transform


20

FT of Periodic DT Signals

Consider the continuous time signal

This signal is periodic

Furthermore, the Fourier series of this signal is just an impulse of weight one centered at ω= ω0

Now consider this signal

It is also periodic and there is one impulse per period

However, the separation between adjacent impulses is 2π

In particular, the DT Fourier Transform for this signal is

DTFT of a periodic signal with period N

N

kkXeX k

k

k

j 2 );(][2)(

21

DTFT: Periodic signal

1

The signal can be expressed as

We can immediately write

Equivalently

period 2π

22

DT FT of periodic signals

FS vs. FT

23

Outline

o Introduction






o DTFT: Summary


Fourier Transform


24 realarezzandzzjyxz

zjzezz

x

yz

yxz

jyxz

j

)()( ;

:ConjugateComplex

sincos :tionrepresentaPolar *

used) being are degreesnot (radians

angle same thegive still and 2 of multipleany by changecan

axis positive real the toangle theis

tan z of (argument) Phase

origin thefrom zpoint a of distance theisIt

||r z of Magnitude

:tionrepresentaCartesian *

***

1

22

Complex numbers

25

Properties of the DTFT

The function ejω is periodic with N=2π

26


27


28

Example: Time shift

Determining the DTFT of

Solution

j

jj

n

j

jjjj

n

j

jF

n

ae

eaeX

nuanxanx

ae

eeXeeX

nuanxnx

aeeXnuanx

1)(

])5[(i.e. ][][

1)()(

])5[(i.e. ]5[][

1

1)(][][

55

2

5

5

1

5

2

5

12

11

]5[][ nuanx n

29


30


Symmetry properties of the DTFT

Symmetry properties of the DTFTDuality property

34


35

Properties of the DT FT

k

kcomb )2()( impulses ofTrain

summation thefrom

result may that component) dc(or valueaverage the

reflects side hand-right on the train impulse thewhere

)()02

1

1

1

21

1

:onAccumulati

2

0

fcombX(X(f)-e

x[m]

πm)δ(ω) πX(e)X(e-e

x[m]

fj

n

-m

m

jjω

jω

n

-m

36


37


38


39


40


41


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

:h[n] response impulse with system LTIan for :followsIt

)((][][

)((2

1][][

:dualityn Convolutio &tion Multiplica

jjj

jj

jj

e)XeHeYnxnhny

e)Ye Xnynx

eY)eX nynx

42


43


44

Properties of the DT FT:

Difference equation DT LTI Systems are characterized by Linear Constant-Coefficient

Difference Equations

A general linear constant-coefficient difference equation for an LTI system with input x[n] and output y[n] is of the form

Now applying the FT to both sides of the above equation, we have

But we know that the input and the output are related to each other through the impulse response of the system, denoted by h[n], i.e.,

45

Properties of the DT FT :

Difference equation

Applying the convolution property

if one is given a difference equation corresponding to some system, the FT of the impulse response of the system can found directly from the difference equation by applying the Fourier transform

FT of the impulse response = Frequency response

Inverse FT of the frequency response = Impulse response

46

Properties of the DT FT:

Example

With |a| < 1 , consider the causal LTI system that

is characterized by the difference equation

The frequency response of the system is

From tables (or by applying inverse FT), we get

47

Table 2.2

48

Outline

o Introduction






o DTFT: Summary


Fourier Transform


49

Frequency response of LTI

systems

If input is complex exponentials

Define

(h[n] & H(): Frequency and impulse responses are a FT pair)

nj

k

kj

k

knjnj

nj

eekhekheTny

enx

)][( ][}{][

][

)(

njj

k

kjj eeHnyekheH )(][][)(

deeHeXnydeeXnx njjjnjj )()(2

1][ )(

2

1][ if

50

Frequency response

The frequency response of discrete-time LTI systems

is always a periodic function of the frequency variable

w with period

Only specify over the interval

The ‘low frequencies’ are close to 0

The ‘high frequencies’ are close to

Frequency response is generally complex

)(][][)( 2)2()2( j

n

njnj

n

njj eHeenhenheH

2

)(|)(|

)()()(jeHjj

j

I

j

R

j

eeH

ejHeHeHdescribes changes to x[n] in magnitude and phase

51

Frequency response: Example

Frequency response of the ideal delay

system

d

jj

d

j

Id

j

R

nj

n

nj

d

j

dd

neHeH

neHneH

eenneH

nnnhnnxny

d

)(,1|)(|

)sin()(),cos()(

][)(

][][ ][][ :delay Ideal

52

Frequency response

pair FT a are responses impulse &Frequency

responsefrequency the)( with

; )()( :Response )(input If

:n theoremConvolutio

][][][ ][][][

response impulse the with ; ][][ ][][

:systemsLTI of Response

eH

eHeXeX

knhkxnyknkxnx

h[n] nhnynnx

j

jjj

kk

53

Frequency response: Example

54

Ideal frequency-selective LTI-

systems (or filters)

Ideal frequency-selective filter have unity frequency response over a

certain range of frequencies, and is zero at the remaining frequencies

Example: Ideal low-pass filter: passes only low frequencies and

rejects high frequencies of an input signal x[n]

55

Example : ideal lowpass filter

Frequency response

h[n] is not absolutely summable Filter noncausal

|| ,0

|| ,1)(

c

cj

lp eH

nn

ndenh cnj

lp

c

c

,sin

2

1][

56

Outline

o Introduction






o DTFT: Summary


Fourier Transform


57

DTFT: Summary

DT Fourier Transform represents a discrete

time aperiodic signal as a sum of infinitely

many complex exponentials, with the

frequency varying continuously in (-π, π)

DTFT is periodic

only need to determine it for

58

Summary: Signal & System

representations

Signal: A sum of scaled, delayed impulse

Signal: A linear combination of weighted sinusoidal signals

LTI system: Convolution

LTI system: Difference equation:

k

knkxnx ][][][

deeXnx njj )(2

1][

)()()( ][][][*][][y jjj

k

eHeXeYknhkxnhnxn

59

Real-world application: Image compression

Energy Distribution of transform (DCT)

Coefficients in Typical Images

Waveform-based video coding60

Real-world applications: Image compression

Images Approximated by Different Number of

transform (DCT) Coefficients

Original

With 8/64

Coefficients

With 16/64

Coefficients

With 4/64

Coefficients

61

DTFT: Summary

Know how to calculate the DTFT of simple functions

Know the geometric sum:

Know Fourier transforms of special functions, e.g. δ[n], exponential

Know how to calculate the inverse transform of rational functions using partial fraction expansion

Properties of DT Fourier transform

Linearity, Time-shift, Frequency-shift, …

62

DT-FT Summary: a quiz

A discrete-time LTI system has impulse response

Find the output y[n] due to input

Solution : Use the convolution property:

][2

1][ nunh

n

][7

1][ nunx

n

)()()(][*][][ jjj eXeHeYnxnhny

1,1

1)(][][ a

aeeMnuanm

j

jn

j

j

j

j

e

eX

e

eH

7

11

1)( and

2

11

1)(

63

DT-FT Summary: a quiz (cont.)

Using partial fraction expansion method of finding inverse FT gives:

Therefore, since a FT is unique, (i.e. no two same signals in time give the same

function in frequency) and since

It can be seen that a FT of the type should correspond to a signal .

Therefore, the inverse FT of is

the inverse FT of is

Thus the complete output

)

2

11

1)(

7

11

1()(

jj

j

ee

eY

j

j

e

eY

7

11

5/2)(

je2

11

5/7

j

jn

aeeMnuanm

1

1)(][][

jae1

1

][nuan

je7

11

5/2

][7

1

5

2nu

n

je2

11

5/7

][2

1

5

7nu

n

][2

1

5

7][

7

1

5

2][ nununy

nn

64

Outline

o Introduction






o DTFT: Summary


Fourier Transform


65

Transition: DT Fourier Series to


DT Pulse Train Signal

This DT periodic rectangular-wave signal is analogous to the CT periodic rectangular-wave signal used to illustrate the transition from the CT Fourier Series to the CT Fourier Transform

66



DTFS of DT Pulse Train

As the period of the rectangular wave increases, the period of the DT Fourier Series increases and the amplitude of the DT Fourier Series decreases

67



Normalized DT Fourier Series of DT Pulse Train

By multiplying the DT Fourier Series by its period and plotting versus instead of k, the amplitude of the DT Fourier Series stays the same as the period increases and the period of the normalized DT Fourier Series stays at one

68



The normalized DT Fourier Series approaches

this limit as the DT period approaches infinity

69

Outline

o Introduction






o DTFT: Summary


Fourier Transform


70

Relations Among Fourier

Methods

Periodic in Time

Discrete in Frequency

Aperiodic in Time

Continuous in Frequency

Continuous

in Time

Aperiodic in

Frequency

Discrete in

Time

Periodic in

Frequency

k

tjk

k

T

tjk

k

eatx

dtetxT

a

0

0

)(

P-CTDT :SeriesFourier Inverse CT

)(1

DTP-CT :SeriesFourier CT

T

0

T

2

2

2

)(2

1][

DT PCT :TransformFourier DT Inverse

][)(

PCTDT :TransformFourier DT

deeXnx

enxeX

njj

n

njj

1

0

NN

1

0

NN

0

0

][1

][

P-DTP-DT SeriesFourier DT Inverse

][][

P-DTP-DT SeriesFourier DT

N

k

knj

N

n

knj

ekXN

nx

enxkX

dejXtx

dtetxjX

tj

tj

)(2

1)(

CTCT :TransformFourier CT Inverse

)()(

CTCT :TransformFourier CT

71

Relations Among Fourier

Methods

72

CT Fourier Transform - CT Fourier Series

73

CT Fourier Transform - CT Fourier Series

74

CT Fourier Transform - DT Fourier Transform

75

CT Fourier Transform - DT Fourier Transform

76

DT Fourier Series - DT Fourier Transform

77

DT Fourier Series - DT Fourier Transform

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