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
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
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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 DT Fourier Transform
o Sufficient condition for DTFT
o DT Fourier Transform of Periodic Signals
o Properties of DT Fourier Transform
o DTFT & LTI systems: Frequency response
o DTFT: Summary
o Appendix: Transition from DT Fourier Series to DT
Fourier Transform
o Appendix: Relations among Fourier Methods
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)
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8
DT Fourier Transform
DT signal representations:
A sum of scaled, delayed impulse
A linear combination of weighted sinusoidal signals
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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
DT Fourier Transform: Derivation
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
DT Fourier Transform: Derivation
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
DT Fourier Transform: Derivation
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
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16
Outline
o Introduction
o DT Fourier Transform
o Sufficient condition for DTFT
o DT Fourier Transform of Periodic Signals
o Properties of DT Fourier Transform
o DTFT & LTI systems: Frequency response
o DTFT: Summary
o Appendix: Transition from DT Fourier Series to DT
Fourier Transform
o Appendix: Relations among Fourier Methods
17
Sufficient condition for DTFT
Condition for the convergence of the infinite sum
If x[n] is absolutely summable, its FT exists
(sufficient condition)
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18
Example: Exponential sequence
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19
Outline
o Introduction
o DT Fourier Transform
o Sufficient condition for DTFT
o DT Fourier Transform of Periodic Signals
o Properties of DT Fourier Transform
o DTFT & LTI systems: Frequency response
o DTFT: Summary
o Appendix: Transition from DT Fourier Series to DT
Fourier Transform
o Appendix: Relations among Fourier Methods
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
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21
DTFT: Periodic signal
1
The signal can be expressed as
We can immediately write
Equivalently
period 2π
23
Outline
o Introduction
o DT Fourier Transform
o Sufficient condition for DTFT
o DT Fourier Transform of Periodic Signals
o Properties of DT Fourier Transform
o DTFT & LTI systems: Frequency response
o DTFT: Summary
o Appendix: Transition from DT Fourier Series to DT
Fourier Transform
o Appendix: Relations among Fourier Methods
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28
Example: Time shift
Determining the DTFT of
Solution
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Properties of the DT FT
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41
Properties of the DT FT
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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
48
Outline
o Introduction
o DT Fourier Transform
o Sufficient condition for DTFT
o DT Fourier Transform of Periodic Signals
o Properties of DT Fourier Transform
o DTFT & LTI systems: Frequency response
o DTFT: Summary
o Appendix: Transition from DT Fourier Series to DT
Fourier Transform
o Appendix: Relations among Fourier Methods
49
Frequency response of LTI
systems
If input is complex exponentials
Define
(h[n] & H(): Frequency and impulse responses are a FT pair)
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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
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51
Frequency response: Example
Frequency response of the ideal delay
system
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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
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56
Outline
o Introduction
o DT Fourier Transform
o Sufficient condition for DTFT
o DT Fourier Transform of Periodic Signals
o Properties of DT Fourier Transform
o DTFT & LTI systems: Frequency response
o DTFT: Summary
o Appendix: Transition from DT Fourier Series to DT
Fourier Transform
o Appendix: Relations among Fourier Methods
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:
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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:
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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
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64
Outline
o Introduction
o DT Fourier Transform
o Sufficient condition for DTFT
o DT Fourier Transform of Periodic Signals
o Properties of DT Fourier Transform
o DTFT & LTI systems: Frequency response
o DTFT: Summary
o Appendix: Transition from DT Fourier Series to DT
Fourier Transform
o Appendix: Relations among Fourier Methods
65
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
66
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
67
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 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
Transition: DT Fourier Series to
DT Fourier Transform
The normalized DT Fourier Series approaches
this limit as the DT period approaches infinity
69
Outline
o Introduction
o DT Fourier Transform
o Sufficient condition for DTFT
o DT Fourier Transform of Periodic Signals
o Properties of DT Fourier Transform
o DTFT & LTI systems: Frequency response
o DTFT: Summary
o Appendix: Transition from DT Fourier Series to DT
Fourier Transform
o Appendix: Relations among Fourier Methods
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
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