ELEC264: Signals And Systems
Topic 5:Discrete-Time Fourier
Transform (DTFT)
oIntroduction
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
o Appendix:
Relations among Fourier Methods
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
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 cannot
represent an aperiodic DT signal for all time

The DT Fourier Transform can represent an
aperiodic discrete-time signal for all time

2
Its development follows exactly the same as that of the
Fourier transform for continuous-time aperiodic signals
Overview of Frequency Analysis
Methods
3
Overview of Fourier Analysis
Methods
Periodic in Time
Discrete in Frequency
Continuous
in Time
Aperiodic in
Frequency
CT Fourier Series :
Aperiodic in Time
Continuous in Frequency
CT - PT
CT Fourier Transform:
DT
T
ak
1
x(t )e
T 0
jk
0t
X(j )
dt
CT InverseFourier Series : DT
ak e jk
x(t )
CT - PT
1
2
k
Discrete in
Time
DT Fourier Series DT - PN
DT - PN
CT
X ( j )e j t d
CT P
2
InverseDT Fourier Transform: CT P
2
N 1
X [k ]
x[n]e
j
0 kn
X (e j )
x[n]
1
N
DT - PN
N 1
X [k ]e j
k 0
x[n]e
j n
n
InverseDT Fourier Series DT - PN
4
dt
DT Fourier Transform: DT
n 0
Periodic in
Frequency
j t
CT
InverseCT Fourier Transform: CT
x(t )
0t
x(t )e
CT
0 kn
x[n]
1
2
X (e j )e j n d
2
DT
Overview of Fourier symbols
DT x[n]
Variable
Period
n
N
Continuous
Frequency
Discrete
Frequency
k
2 k/N
k
CT x(t)
t
T
k
k
2 k /T
•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
5
•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
Outline
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Introduction
DT Fourier Transform
Sufficient condition for DTFT
DT Fourier Transform of Periodic Signals
Properties of DT Fourier Transform
DTFT & LTI systems: Frequency response
DTFT: Summary
Appendix: Transition from DT Fourier Series to DT
Fourier Transform
Appendix: Relations among Fourier Methods
DT Fourier Transform

DT Fourier transform and the inverse FT
j
X (e )
x[n]e
j n
,
n
X (e j )e j n d

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

7
x[n]
1
2
(The function ejω is periodic with N=2π)
 Forms : X (e j )
x[n]e
n
j n
X(f )
x[n]e
n
j 2 fn
DT Fourier Transform

DT signal representations:

A sum of scaled, delayed impulse
x[n]
x[k ] [n k ]
k

A linear combination of weighted sinusoidal signals
x[n]
8
1
2
j
X ( e )e
j n
d
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
9
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
10
DT Fourier Transform: Derivation





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
DT Fourier Transform: Derivation

One important observation here is that the
function X(ejω) is periodic in ω with period 2π

12
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: Examples
x[n] 1
X (e j )
2
(
2 r)
r
The periodic impulse train
n
Let x[n] a u[n]
13
|a| 1
1
X (e )
1 ae
j
j
DT Fourier Transform: Examples
Aperiodic
Periodic
14
Fourier transform pairs
15
Outline
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Introduction
DT Fourier Transform
Sufficient condition for DTFT
DT Fourier Transform of Periodic Signals
Properties of DT Fourier Transform
DTFT & LTI systems: Frequency response
DTFT: Summary
Appendix: Transition from DT Fourier Series to DT
Fourier Transform
Appendix: Relations among Fourier Methods
Sufficient condition for DTFT

Condition for the convergence of the infinite sum
| X (e j ) | |
x[n]e
j n
| x[n] ||e
|
j n
|
| x[n] |
If x[n] is absolutely summable, its FT exists
(sufficient condition)
17
Example: Exponential sequence
n
x[n] a u[n]
18
1
| a | 1 : X (e )
1 ae j
1
j
a 1 : X (e )
1 e j
k
| a | 1 : DTFT does not exist
j
(
2 k)
Outline
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Introduction
DT Fourier Transform
Sufficient condition for DTFT
DT Fourier Transform of Periodic Signals
Properties of DT Fourier Transform
DTFT & LTI systems: Frequency response
DTFT: Summary
Appendix: Transition from DT Fourier Series to DT
Fourier Transform
Appendix: Relations among Fourier Methods
FT of Periodic DT Signals

Consider the continuous time signal



Now consider this signal




This signal is periodic
Furthermore, the Fourier series of this signal is just an impulse of
weight one centered at ω= ω0
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
j
20
X (e )
2
X [k ] (
k
k
);
k
2 k
N
DTFT: Periodic signal
1
 The signal can be expressed as
 We can immediately write


21
Equivalently
period 2π
DT FT of periodic signals
FS vs. FT
22
Outline
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Introduction
DT Fourier Transform
Sufficient condition for DTFT
DT Fourier Transform of Periodic Signals
Properties of DT Fourier Transform
DTFT & LTI systems: Frequency response
DTFT: Summary
Appendix: Transition from DT Fourier Series to DT
Fourier Transform
Appendix: Relations among Fourier Methods
Complex numbers
* Cartesian representation : z
Magnitude of z r | z |
x
x2
jy
y2
It is the distance of a point z from the origin
y
Phase (argument)of z
z tan 1
x
is the angle to the real positive axis
can change by any multiple of 2 and still give the same angle
(radians not degrees are being used)
* Polar representation : z
zej
z cos
 Complex Conjugate :
24
z*
x
jy ; ( z z * ) and ( zz * ) are real
j z sin
Properties of the DTFT

25
The function ejω is periodic with N=2π
Properties of the DTFT
26
Properties of the DTFT
27
Example: Time shift

Determining the DTFT of
a n u[n 5]
x[n]

Solution
x1[n]
a u[n]
x 2 [ n]
x1[n 5]
j
X 2 (e )
e
j5
X 1 (e j )
(i.e.
j
X (e )
j
X 1 (e )
a 5 x 2 [ n]
x[n]
28
F
n
a 5e j5
1 ae j
1
1 ae j
a n 5u[n 5])
e
j5
1 ae j
(i.e. a n u[n 5])
Properties of the DTFT
29
Properties of the DTFT
30
31
Symmetry properties of the DTFT
Symmetry properties of the DTFT
Duality property
Properties of the DTFT
34
Properties of the DT FT
Accumulation :
n
x[m]
m n
1
X(e
jω
1-e
jω
) πX(e
j0
δ(ω 2πm)
)
m
1
1
x[m]
X(f)
X( 0)comb( f )
j2 f
1-e
2
m where the impulse train on the right - hand side reflects
the average value (or dc component) that may result
from the summation
35
Train of impulses comb ( )
(
k
2 k)
Properties of the DT FT
36
Properties of the DT FT
37
Properties of the DT FT
38
Properties of the DT FT
39
Properties of the DT FT
40
Properties of the DT FT
Multiplica tion & Convolution duality :
x[n] y[n]
x[n] y[n]
1
X (e j ) Y (e j )
2
X (e j )Y (e j )
It follows : for an LTI system with impulse responseh[n] :
y[n] h[n] * x[n]
41
Y (e j )
H (e j )X (e j )
Properties of the DT FT
42
Properties of the DT FT
43
Properties of the DT FT:
Difference equation
44

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.,
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
45

FT of the impulse response = Frequency response

Inverse FT of the frequency response = Impulse response
Properties of the DT FT:
Example
46

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
Outline
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Introduction
DT Fourier Transform
Sufficient condition for DTFT
DT Fourier Transform of Periodic Signals
Properties of DT Fourier Transform
DTFT & LTI systems: Frequency response
DTFT: Summary
Appendix: Transition from DT Fourier Series to DT
Fourier Transform
Appendix: Relations among Fourier Methods
Frequency response of LTI
systems

If input is complex exponentials
x[n] e j
n
y[n] T {e j n }
h[k ]e j
(n k )
(
k

Define H (e j )
h[k ]e
j k
)ej
n
k
h[k ]e
j k
y[n]
H (e j )e j
n
k

(h[n] & H(): Frequency and impulse responses are a FT pair)
if x[n]
49
1
2
j
X (e ) e
j n
d
y[n]
1
2
X (e j ) H (e j ) e j n d
Frequency response

The frequency response of discrete-time LTI systems
is always a periodic function of the frequency variable
2
w with period
H (e j (
2 )
)
h[n]e
n
j(
2 )n
h[n]e
j n
e
j2 n
H (e j )
n
Only specify over the interval
The ‘low frequencies’ are close to 0
The ‘high frequencies’ are close to

Frequency response is generally complex
H (e j )
50
H R (e j )
| H (e j ) | e j
jH I (e j )
H (e
j
)
describes changes to x[n] in
magnitude and phase
Frequency response: Example

Frequency response of the ideal delay
system
 Ideal delay : y[n]
H (e j )
x[n nd ]
[n nd ]e
j n
h[n]
e
[n nd ]
j nd
n
j
 H R (e ) cos( nd ),
 | H (e j ) | 1,
51
j
H I (e )
H (e j )
nd
sin( nd )
Frequency response
 Response of LTI systems:
x[n] [n]
y[n] h[n] ; with h[n] the impulse response
x[n]
x[k ] [n k ]
k
y[n]
x[k ]h[n k ]
k
 Convolution theorem:
If input X (e j )
Response: X (e j ) H (e j ) ;
with H (e j ) the frequencyresponse
Frequency& impulse responsesare a FT pair
52
Frequency response: Example
53
Ideal frequency-selective LTIsystems (or filters)

54
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]
Example : ideal lowpass filter

Frequency response
H lp (e
j
)
1, | |
0, c | |
hlp [n]

55
1
2
c
c
e
c
j n
d
sin
c
n
n
,
n
h[n] is not absolutely summable  Filter noncausal
Outline
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Introduction
DT Fourier Transform
Sufficient condition for DTFT
DT Fourier Transform of Periodic Signals
Properties of DT Fourier Transform
DTFT & LTI systems: Frequency response
DTFT: Summary
Appendix: Transition from DT Fourier Series to DT
Fourier Transform
Appendix: Relations among Fourier Methods
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)

57
DTFT is periodic
 only need to determine it for
Summary: Signal & System
representations


Signal: A sum of scaled, delayed impulse x[n]
k
Signal: A linear combination of weighted sinusoidal signals
x[n]

1
2
x[n] * h[n]
x[k ]h[n k ]
k
58
X ( e j )e j n d
LTI system: Convolution
y[n]

x[k ] [n k ]
LTI system: Difference equation:
Y (e j )
X (e j ) H (e j )
Real-world application: Image compression
Energy Distribution of transform (DCT)
Coefficients in Typical Images
59
Real-world applications: Image compression
Images Approximated by Different Number of
transform (DCT) Coefficients
With 16/64
Coefficients
Original
With 8/64
Coefficients
60
With 4/64
Coefficients
Waveform-based video coding
DTFT: Summary

Know how to calculate the DTFT of simple functions




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

61
Know the geometric sum:
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

m[n]
n
a u[n]
1
H (e )
1
1
e
2
j
Y (e )
u[n]
u[n]
j
j
H (e ) X ( e )
1
M (e )
1 ae
j
1
and X (e )
1
1
e
7
j
j
j
n
n
1
7
Solution : Use the convolution property:
y[n] h[n] * x[n]
62
h[n]
x[n]
j
1
2
j
,
a 1
DT-FT Summary: a quiz (cont.)
1
Y (e j ) (
1
1
e
7


j
1
)(
1
1
e
2
)
j
Using partial fraction expansion method of finding inverse FT gives:
2/5
1 j
1
e
7
Y (e j )

Therefore,

It can be seen that a FT of the type
.
Therefore,


the inverse FT of
the inverse FT of
2/5
1 j
1
e
7
7/5
1 j
1
e
2
is
2 1
5 7
1
1 ae
n
a u[n]
63
Thus the complete output
1
1 ae
M (e j )
a n u[n]
n
u[n]
n
is
7 1
u[n]
5 2
y[n]
j
should correspond to a signal
j
n

j
since a FT is unique, (i.e. no two same signals in time give the same
function in frequency) and since
m[n]

7/5
1
1
e
2
n
2 1
7 1
u[n]
u[n]
5 7
5 2
Outline
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Introduction
DT Fourier Transform
Sufficient condition for DTFT
DT Fourier Transform of Periodic Signals
Properties of DT Fourier Transform
DTFT & LTI systems: Frequency response
DTFT: Summary
Appendix: Transition from DT Fourier Series to DT
Fourier Transform
Appendix: Relations among Fourier Methods
Transition: DT Fourier Series to
DT Fourier Transform
65

DT Pulse Train Signal

This DT periodic rectangular-wave signal is
analogous to the CT periodic rectangularwave signal used to illustrate the transition
from the CT Fourier Series to the CT
Fourier Transform
Transition: DT Fourier Series to
DT 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
Transition: DT Fourier Series to
DT Fourier Transform


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
Transition: DT Fourier Series to
DT Fourier Transform

68
The normalized DT Fourier Series approaches
this limit as the DT period approaches infinity
Outline
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Introduction
DT Fourier Transform
Sufficient condition for DTFT
DT Fourier Transform of Periodic Signals
Properties of DT Fourier Transform
DTFT & LTI systems: Frequency response
DTFT: Summary
Appendix: Transition from DT Fourier Series to DT
Fourier Transform
Appendix: Relations among Fourier Methods
Relations Among Fourier
Methods
Periodic in Time
Discrete in Frequency
Continuous
in Time
Aperiodic in
Frequency
CT Fourier Series :
Aperiodic in Time
Continuous in Frequency
CT - PT
CT Fourier Transform:
DT
T
ak
1
x(t )e
T 0
jk
0t
X(j )
dt
CT InverseFourier Series : DT
ak e jk
x(t )
CT - PT
1
2
k
Discrete in
Time
DT Fourier Series DT - PN
DT - PN
CT
X ( j )e j t d
CT P
2
InverseDT Fourier Transform: CT P
2
N 1
X [k ]
x[n]e
j
0 kn
X (e j )
x[n]
1
N
DT - PN
N 1
X [k ]e j
k 0
x[n]e
j n
n
InverseDT Fourier Series DT - PN
70
dt
DT Fourier Transform: DT
n 0
Periodic in
Frequency
j t
CT
InverseCT Fourier Transform: CT
x(t )
0t
x(t )e
CT
0 kn
x[n]
1
2
X (e j )e j n d
2
DT
Relations Among Fourier
Methods
71
CT Fourier Transform - CT Fourier Series
72
CT Fourier Transform - CT Fourier Series
73
CT Fourier Transform - DT Fourier Transform
74
CT Fourier Transform - DT Fourier Transform
75
DT Fourier Series - DT Fourier Transform
76
DT Fourier Series - DT Fourier Transform
77