Topic 5: Discrete-Time Fourier Transform (DTFT) o DT Fourier - - PowerPoint PPT Presentation

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Topic 5: Discrete-Time Fourier Transform (DTFT) o DT Fourier - - PowerPoint PPT Presentation

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ELEC361: Signals And Systems Topic 5: Discrete-Time Fourier Transform (DTFT) o DT Fourier Transform o Overview of Fourier methods o DT Fourier Transform of Periodic Signals o Properties of DT Fourier Transform o Relations among Fourier Methods o

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SLIDE 1
  • DT Fourier Transform
  • Overview of Fourier methods
  • DT Fourier Transform of Periodic Signals
  • Properties of DT Fourier Transform
  • Relations among Fourier Methods
  • Summary
  • Appendix:
  • Transition from DT Fourier Series to DT Fourier

Transform

ELEC361: Signals And Systems

Topic 5: Discrete-Time Fourier Transform (DTFT)

  • Dr. 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
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2

DT Fourier Transform

(Note: a Fourier transform 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 cannot represent 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

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3

DT Fourier Transform

  • 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

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4

DT Fourier Transform

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

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5

DT Fourier Transform

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

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6

DT Fourier Transform

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

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7

DT Fourier Transform: Forms

∫ ∑

= ⇒ + ⊗ = + ⇒ ⊗

∞ −∞ = − π ω ω π ω ω π

ω π 2

2 2

) ( 2 1 ] [ DT P CT : Transform Fourier DT Inverse ] [ ) ( P CT DT : Transform Fourier DT d e e X n x e n x e X

n j j n n j j

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SLIDE 8

8

DT Fourier Transform: Examples

1 1

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SLIDE 9

9

DT Fourier Transform: Example

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10

Outline

  • DT Fourier Transform
  • Overview of Fourier methods
  • DT Fourier Transform of Periodic Signals
  • Properties of DT Fourier Transform
  • Relations among Fourier Methods
  • DTFT: Summary
  • Appendix:
  • Transition from DT Fourier Series to DT Fourier

Transform

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11

Overview of Fourier Analysis Methods: Types of signals

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12

Overview of Fourier Analysis Methods: Types of signals

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13

Overview of Fourier Analysis Methods: Continuous-Value and

Continuous-Time Signals

  • All continuous signals are

CT but not all CT signals are continuous

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14

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 t jk k T t jk k

e a t x dt e t x T a ) ( P

  • CT

DT : Series Fourier Inverse CT ) ( 1 DT P

  • CT

: Series Fourier CT

T T ω ω

∫ ∑

= ⇒ + ⊗ = + ⇒ ⊗

∞ −∞ = − π ω ω π ω ω π

ω π 2

2 2

) ( 2 1 ] [ DT P CT : Transform Fourier DT Inverse ] [ ) ( P CT DT : Transform Fourier DT d e e X n x e n x e X

n j j n n j j

∑ ∑

− = − = −

= ⇒ ⊗ = ⇒ ⊗

1 N N 1 N N

] [ 1 ] [ P

  • DT

P

  • DT

Series Fourier DT Inverse ] [ ] [ P

  • DT

P

  • DT

Series Fourier DT

N k kn j N n kn j

e k X N n x e n x k X

ω ω

∫ ∫

∞ ∞ − ∞ ∞ − −

= ⇒ ⊗ = ⇒ ⊗ ω ω π ω

ω ω

d e j X t x dt e t x j X

t j t j

) ( 2 1 ) ( CT CT : Transform Fourier CT Inverse ) ( ) ( CT CT : Transform Fourier CT

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SLIDE 15

15

Outline

  • DT Fourier Transform
  • Overview of Fourier methods
  • DT Fourier Transform of Periodic Signals
  • Properties of DT Fourier Transform
  • Relations among Fourier Methods
  • DTFT: Summary
  • Appendix:
  • Transition from DT Fourier Series to DT Fourier

Transform

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SLIDE 16

16

Fourier Transform of Periodic DT Signals

Consider the continuous time signal This signal is periodic Furthermore, the Fourier series representation 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, which agrees with the properties of DT Fourier Transform

In particular, the DT Fourier Transform for this signal is

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SLIDE 17

17

Fourier Transform of Periodic DT Signals: Example

1 The signal can be expressed as We can immediately write Or equivalently

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

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SLIDE 18

18

Outline

  • DT Fourier Transform
  • Overview of Fourier methods
  • DT Fourier Transform of Periodic Signals
  • Properties of DT Fourier Transform
  • Relations among Fourier Methods
  • DTFT: Summary
  • Appendix:
  • Transition from DT Fourier Series to DT Fourier

Transform

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SLIDE 19

19

Properties of the DT Fourier Transform

Note: the function ejω is periodic with

N=2π

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20

Properties of the DT Fourier Transform

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21

Properties of the DT Fourier Transform

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22

Properties of the DT Fourier Transform

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23

Properties of the DT Fourier Transform

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24

Properties of the DT Fourier Transform

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25

Properties of the DT Fourier Transform

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26

Properties of the DT Fourier Transform

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SLIDE 27

27

Properties of the DT Fourier Transform

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28

Properties of the DT Fourier Transform

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29

Properties of the DT Fourier Transform

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30

Properties of the DT Fourier Transform

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31

Properties of the DT Fourier Transform

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32

Properties of the DT Fourier Transform

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33

Properties of the DT Fourier Transform: Example

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34

Properties of the DT Fourier Transform

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Properties of the DT Fourier Transform: 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 Fourier transform to both sides of the

above equation, we have

But we know that the input and the output are related to each

  • ther through the impulse response of the system, denoted

by h[n], i.e.,

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36

Properties of the DT Fourier Transform : Difference equation

Applying the convolution property

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

Fourier transform of the impulse response = Frequency response Inverse Fourier transform of the frequency response = Impulse

response

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37

Properties of the DT Fourier Transform: Example

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

us characterized by the difference equation

From the discussion, it is easy to see that the

frequency response of the system is

From tables (or by applying inverse Fourier

transform), one can easily find that

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SLIDE 38

38

Outline

  • DT Fourier Transform
  • Overview of Fourier methods
  • DT Fourier Transform of Periodic Signals
  • Properties of DT Fourier Transform
  • Relations among Fourier Methods
  • DTFT: Summary
  • Appendix:
  • Transition from DT Fourier Series to DT Fourier

Transform

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SLIDE 39

39

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 t jk k T t jk k

e a t x dt e t x T a ) ( P

  • CT

DT : Series Fourier Inverse CT ) ( 1 DT P

  • CT

: Series Fourier CT

T T ω ω

∫ ∑

= ⇒ + ⊗ = + ⇒ ⊗

∞ −∞ = − π ω ω π ω ω π

ω π 2

2 2

) ( 2 1 ] [ DT P CT : Transform Fourier DT Inverse ] [ ) ( P CT DT : Transform Fourier DT d e e X n x e n x e X

n j j n n j j

∑ ∑

− = − = −

= ⇒ ⊗ = ⇒ ⊗

1 N N 1 N N

] [ 1 ] [ P

  • DT

P

  • DT

Series Fourier DT Inverse ] [ ] [ P

  • DT

P

  • DT

Series Fourier DT

N k kn j N n kn j

e k X N n x e n x k X

ω ω

∫ ∫

∞ ∞ − ∞ ∞ − −

= ⇒ ⊗ = ⇒ ⊗ ω ω π ω

ω ω

d e j X t x dt e t x j X

t j t j

) ( 2 1 ) ( CT CT : Transform Fourier CT Inverse ) ( ) ( CT CT : Transform Fourier CT

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40

Relations Among Fourier Methods

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CT Fourier Transform - CT Fourier Series

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CT Fourier Transform - CT Fourier Series

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CT Fourier Transform - DT Fourier Transform

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CT Fourier Transform - DT Fourier Transform

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DT Fourier Series - DT Fourier Transform

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46

DT Fourier Series - DT Fourier Transform

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47

Outline

  • DT Fourier Transform
  • Overview of Fourier methods
  • DT Fourier Transform of Periodic Signals
  • Properties of DT Fourier Transform
  • Relations among Fourier Methods
  • DTFT: Summary
  • Appendix:
  • Transition from DT Fourier Series to DT Fourier

Transform

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48

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

  • nly need to determine it for
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49

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, …

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50

DT-FT Summary: a quiz

  • A discrete-time LTI system has impulse response
  • Find the output y[n] due to input
  • (Suggestion: work with and using the convolution property)
  • Solution
  • This can be solved using convolution of h[n] and x[n].
  • However, the point was to use the convolution in time

multiplication in frequency property.

  • Therefore,
  • It can be readily shown that
  • Therefore,

] [ 2 1 ] [ n u n h

n

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ] [ 7 1 ] [ n u n x

n

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ =

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

ω ω ω j j j

e X e H e Y n x n h n y = ⇒ =

( )

1 , 1 1 ) ( ] [ ] [ < − = ⇒ =

−

a ae e M n u a n m

j j n ω ω

ω ω ω ω j j j j

e e X e e H

− −

− = − = 7 1 1 1 ) ( and 2 1 1 1 ) (

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SLIDE 51

51

DT-FT Summary: a quiz

Exploiting the convolution in time multiplication in

frequency property gives:

Using partial fraction expansion method of finding inverse

Fourier transform gives:

Therefore, since a Fourier transform is unique, (i.e. no two same signals in

time give the same function in frequency) and

since

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

Therefore, the inverse Fourier transform of is the inverse transform of is Thus the complete output ) 2 1 1 1 )( 7 1 1 1 ( ) (

ω ω ω j j j

e e e Y

− −

− − = + − − =

− ω ω j j

e e Y 7 1 1 5 / 2 ) (

ω j

e− − 2 1 1 5 / 7

( )

ω ω j j n

ae e M n u a n m

−

− = ⇒ = 1 1 ) ( ] [ ] [

ω j

ae− − 1 1 ] [n u a n

ω j

e− − − 7 1 1 5 / 2 ] [ 7 1 5 2 n u

n

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −

ω j

e− − 2 1 1 5 / 7 ] [ 2 1 5 7 n u

n

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = ] [ 7 1 5 2 ] [ n u n y

n

] [ 2 1 5 7 n u

n

⎟ ⎠ ⎞ ⎜ ⎝ ⎛

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52

Outline

  • DT Fourier Transform
  • Overview of Fourier methods
  • DT Fourier Transform of Periodic Signals
  • Properties of DT Fourier Transform
  • Relations among Fourier Methods
  • DTFT: Summary
  • Appendix:
  • Transition from DT Fourier Series to DT Fourier

Transform

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SLIDE 53

53

Transition: DT Fourier Series to DT Fourier Transform

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

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Transition: DT Fourier Series to DT Fourier Transform

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

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55

Transition: DT Fourier Series to DT Fourier Transform

Normalized DT Fourier

Series of DT Pulse Train

By multiplying the DT

Fourier Series by its period and plotting versus instead

  • f 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

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Transition: DT Fourier Series to DT Fourier Transform

The normalized DT Fourier Series approaches

this limit as the DT period approaches infinity