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

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

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

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

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

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

DT Fourier Transform: Forms

⇒+⊗

+⇒⊗∞

−∞=

ωπ 2

)(21][

DT PCT :TransformFourier DT Inverse

PCTDT :TransformFourier DT

deeXnx

DT Fourier Transform:Examples

DT Fourier Transform: Example

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

Overview of Fourier Analysis Methods: Types of signals

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

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

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

−∞=

⇒⊗

dtetxT

P-CTDT :SeriesFourier Inverse CT

DTP-CT :SeriesFourier CT

⇒+⊗

+⇒⊗∞

−∞=

ωπ 2

)(21][

deeXnx

⇒⊗

P-DTP-DT SeriesFourier DT Inverse

P-DTP-DT SeriesFourier DT

∞−

⇒⊗

ωωπ

dejXtx

dtetxjX

)(21)(

CTCT :TransformFourier CT Inverse

CTCT :TransformFourier CT

Outline

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

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

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:

Properties of the DT Fourier Transform

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

Properties of the DT Fourier Transform: Example

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.,

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

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

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:

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

−∞=

⇒⊗

dtetxT

P-CTDT :SeriesFourier Inverse CT

DTP-CT :SeriesFourier CT

⇒+⊗

+⇒⊗∞

−∞=

ωπ 2

)(21][

deeXnx

⇒⊗

P-DTP-DT SeriesFourier DT Inverse

P-DTP-DT SeriesFourier DT

∞−

⇒⊗

ωωπ

dejXtx

dtetxjX

)(21)(

CTCT :TransformFourier CT Inverse

CTCT :TransformFourier CT

Relations Among Fourier Methods

CT Fourier Transform - CT Fourier Series

CT Fourier Transform - DT Fourier Transform

DT Fourier Series - DT Fourier Transform

Outline

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

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, …

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

⎟⎠⎞

⎜⎝⎛=

][71][ nunx

⎟⎠⎞

⎜⎝⎛=

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

( ) 1,1

1)(][][ <−

=⇒= − aae

eMnuanm jjn

−− −=

1)( and

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

1()(ωω

−− −−=

− ω

5/2)(ωje−−

( ) ωω

aeeMnuanm −−

=⇒=1

1)(][][

ωjae−−11

][nuan

ωje−−

⎟⎠⎞

⎜⎝⎛−

ωje−−211

⎟⎠⎞

⎜⎝⎛

+⎟⎠⎞

⎜⎝⎛−= ][

52][ nuny

⎟⎠⎞

⎜⎝⎛

Outline

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

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

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

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

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