elec361: signals and systems topic 5: discrete …amer/teach/elec361/slides/topic5... · •a....

o DT Fourier Transformo Overview of Fourier methodso DT Fourier Transform of Periodic Signalso Properties of DT Fourier Transform o Relations among Fourier Methods o Summary o Appendix:

oTransition from DT Fourier Series to DT Fourier Transform

ELEC361: Signals And Systems

Topic 5: Discrete-Time Fourier Transform (DTFT)

Dr. Aishy AmerConcordia UniversityElectrical 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

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 timeThe DT Fourier Transform can represent an aperiodic discrete-time signal for all timeIts development follows exactly the same as that of the Fourier transform for continuous-time aperiodic signals

3

DT Fourier TransformLet x[n] be the aperiodic DT signalWe 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 NN 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 N2Consequently, N has to be chosen such that N > N1 + N2 + 1 so that adjacent replicas do not overlap Clearly, as we let

as desired

4


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

5


Now defining the functionWe can see that the coefficients ak are related to X(ejω) aswhere ω0 = 2π/N is the spacing of the samples in the frequency domainTherefore

As N increases ω0 decreases, and as N → ∞ the above equation becomes an integral

6


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

7

DT Fourier Transform: Forms

∫

∑

=

⇒+⊗

=

+⇒⊗∞

−∞=

−

π

ωω

π

ωω

π

ωπ 2

2

2

)(21][

DT PCT :TransformFourier DT Inverse

][)(

PCTDT :TransformFourier DT

deeXnx

enxeX

njj

n

njj

8

DT Fourier Transform:Examples

11

9

DT Fourier Transform: Example

10

Outline

o DT Fourier Transformo Overview of Fourier methodso DT Fourier Transform of Periodic Signalso Properties of DT Fourier Transform o Relations among Fourier Methods o DTFT: Summary o Appendix:

o Transition from DT Fourier Series to DT Fourier Transform

11

Overview of Fourier Analysis Methods: Types of signals

12

Overview of Fourier Analysis Methods: Types of signals

13

Overview of Fourier Analysis Methods: Continuous-Value and Continuous-Time Signals

All continuous signals are CT but not all CT signals are continuous

14

Overview of Fourier Analysis MethodsPeriodic in TimeDiscrete in Frequency

Aperiodic in TimeContinuous in Frequency

Continuous in Time

Aperiodic in Frequency

Discrete in Time

Periodic in Frequency

∑

∫

∞

−∞=

−

=

⇒⊗

=

⇒⊗

k

tjkk

Ttjk

k

eatx

dtetxT

a

0

0

)(

P-CTDT :SeriesFourier Inverse CT

)(1

DTP-CT :SeriesFourier CT

T

0

T

ω

ω

∫

∑

=

⇒+⊗

=

+⇒⊗∞

−∞=

−

π

ωω

π

ωω

π

ωπ 2

2

2

)(21][


][)(


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

ω

ω

∫

∫

∞

∞−

∞

∞−

−

=

⇒⊗

=

⇒⊗

ωωπ

ω

ω

ω

dejXtx

dtetxjX

tj

tj

)(21)(

CTCT :TransformFourier CT Inverse

)()(

CTCT :TransformFourier CT

15

Outline



16

Fourier Transform of Periodic DT Signals

Consider the continuous time signalThis signal is periodicFurthermore, the Fourier series representation of this signal is just an impulse of weight one centered at ω= ω0Now consider this signalIt 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

17

Fourier Transform of Periodic DT Signals: Example

1The signal can be expressed asWe can immediately write

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

18

Outline

o DT Fourier Transformo Overview of Fourier methodso DT Fourier Transform of Periodic Signalso Properties of DT Fourier Transformo Relations among Fourier Methods o DTFT: Summary o Appendix:


19

Properties of the DT Fourier Transform

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

20


21


22


23


24


25


26


27


28


29


30


31


32


33

Properties of the DT Fourier Transform: Example

34


35

Properties of the DT Fourier Transform: Difference equation

DT LTI Systems are characterized by Linear Constant-Coefficient Difference EquationsA 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 other through the impulse response of the system, denoted by h[n], i.e.,

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 responseInverse Fourier transform of the frequency response = Impulse response

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

38

Outline

o DT Fourier Transformo Overview of Fourier methodso DT Fourier Transform of Periodic Signalso Properties of DT Fourier Transform o Relations among Fourier Methodso DTFT: Summary o Appendix:


39

Relations Among Fourier MethodsPeriodic in TimeDiscrete in Frequency

Aperiodic in TimeContinuous in Frequency

Continuous in Time

Aperiodic in Frequency

Discrete in Time

Periodic in Frequency

∑

∫

∞

−∞=

−

=

⇒⊗

=

⇒⊗

k

tjkk

Ttjk

k

eatx

dtetxT

a

0

0

)(

P-CTDT :SeriesFourier Inverse CT

)(1

DTP-CT :SeriesFourier CT

T

0

T

ω

ω

∫

∑

=

⇒+⊗

=

+⇒⊗∞

−∞=

−

π

ωω

π

ωω

π

ωπ 2

2

2

)(21][


][)(


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

ω

ω

∫

∫

∞

∞−

∞

∞−

−

=

⇒⊗

=

⇒⊗

ωωπ

ω

ω

ω

dejXtx

dtetxjX

tj

tj

)(21)(

CTCT :TransformFourier CT Inverse

)()(

CTCT :TransformFourier CT

40

Relations Among Fourier Methods

41

CT Fourier Transform - CT Fourier Series

42

CT Fourier Transform - CT Fourier Series

43

CT Fourier Transform - DT Fourier Transform

44

CT Fourier Transform - DT Fourier Transform

45

DT Fourier Series - DT Fourier Transform

46

DT Fourier Series - DT Fourier Transform

47

Outline



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 only need to determine it for

49

DTFT: SummaryKnow how to calculate the DTFT of simple functions

Know the geometric sum:

Know Fourier transforms of special functions, e.g. δ[n], exponentialKnow how to calculate the inverse transform of rational functions using partial fraction expansionProperties of DT Fourier transform

Linearity, Time-shift, Frequency-shift, …

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,

][21][ nunh

n

⎟⎠⎞

⎜⎝⎛=

][71][ nunx

n

⎟⎠⎞

⎜⎝⎛=

)()()(][*][][ ωωω jjj eXeHeYnxnhny =⇒=

( ) 1,1

1)(][][ <−

=⇒= − aae

eMnuanm jjn

ωω

ω

ω

ω

ω

j

j

j

j

eeX

eeH

−− −=

−=

711

1)( and

211

1)(

51

DT-FT Summary: a quizExploiting 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 isThus the complete output

)

211

1)(

711

1()(ωω

ω

jj

j

eeeY

−− −−=

+−

−=

− ω

ω

j

j

eeY

711

5/2)(ωje−−

211

5/7

( ) ωω

jjn

aeeMnuanm −−

=⇒=1

1)(][][

ωjae−−11

][nuan

ωje−−

−

711

5/2][

71

52 nu

n

⎟⎠⎞

⎜⎝⎛−

ωje−−211

5/7][

21

57 nu

n

⎟⎠⎞

⎜⎝⎛

+⎟⎠⎞

⎜⎝⎛−= ][

71

52][ nuny

n

][21

57 nu

n

⎟⎠⎞

⎜⎝⎛

52

Outline



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

54


DTFS of DT Pulse TrainAs the period of the rectangular wave increases, the period of the DT Fourier Series increases and the amplitude of the DT Fourier Series decreases

55


Normalized DT Fourier Series of DT Pulse TrainBy 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

56


The normalized DT Fourier Series approaches this limit as the DT period approaches infinity

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