CHAPTER 3
Discrete-Time Signals in the
Transform-Domain
Wang Weilian
wlwang@ynu.edu.cn
School of Information Science and Technology
Yunnan University
Outline
• The Discrete-Time Fourier Transform
• The Discrete Fourier Transform
• Relation between the DTFT and the DFT, and
Their Inverses
• Discrete Fourier Transform Properties
• Computation of the DFT of Real Sequences
• Linear Convolution Using the DFT
• The z-Transform
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Discrete-Time Signals in the Transform-Domain 2
Outline
• Region of Convergence of a Rational z-Transform
• Inverse z-Transform
• z-Transform Properties
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Discrete-Time Signals in the Transform-Domain 3
The Discrete-Time Fourier Transform
• The discrete-time Fourier transform (DTFT) or,
simply, the Fourier transform of a discrete–time
sequence x[n] is a representation of the
sequence in terms of the complex exponential
sequence  e  j x  where  is the real frequency
variable.
• The discrete-time Fourier transform
sequence x[n] is defined by
X e
j

 
x [ n ]e
X e
j
 of a
 j n
n  
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Discrete-Time Signals in the Transform-Domain 4
The Discrete-Time Fourier Transform
• In general X  e j  is a complex function of the real
variable and can be written in rectangular form as
X e
j

 X re  e
where X re  e j  and
imaginary parts of
X im  e
X e
j
j
j
  jX im  e
j

 are, respectively, the real and
 , and are real functions of  .
• Polar form
X  e j   X  e j
w h ere
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
e
     arg X  e
j  
j


Discrete-Time Signals in the Transform-Domain 5
The Discrete-Time Fourier Transform
• Convergence Condition:
If x[n] is an absolutely summable sequence, i.e.,


if
x n  
n  
th en
X e
j




n  
x ne
 j n



x n  
n  
Thus the equation is a sufficient condition for the
existence of the DTFT.
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Discrete-Time Signals in the Transform-Domain 6
The Discrete-Time Fourier Transform
• Bandlimited Signals:
– A full-band discrete-time signal has a spectrum
occupying the whole frequency rang 0     .
– If the spectrum is limited to a portion of the frequency
range 0     , it is called a bandlimited signal.
– A lowpass discrete-time signal has a spectrum
occupying the frequency range 0     p   , where  p
is called the bandwidth of the signal.
– A bandpass discrete-time signal has a spectrum
occupying the frequency range 0   L     H   ,
where  H   L is its bandwidth.
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Discrete-Time Signals in the Transform-Domain 7
The Discrete-Time Fourier Transform
• Discrete-Time Fourier Transform Properties
There are a number of important properties of the
discrete-time Fourier transform which are useful in
digital signal processing applications. We list the
general properties in Table 3.2, and the symmetry
properties in Tables 3.3 and 3.4.
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Discrete-Time Signals in the Transform-Domain 8
The Discrete-Time Fourier Transform
• Energy Density Spectrum
P arseval's relation :


g [ n ]h [ n ] 
*
n -

1

2

G (e
j
*
) H (e
j
)d 
T otal en ergy of a fin ite-en ergy seq u en ce g [ n ] :


g 
g n
2
n  
If
h[ n ]  g [ n ],
th en from P arseval's relation w e ob ser ve

g 

| g[ n ] | 
2
n -
T h e q u an tity : S g g  e
j

1
2
G e

j



| G (e
j
) | d
2
2
is called th e en ergy d en sity sp ectru m of th e seq u en ce g [ n ].
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Discrete-Time Signals in the Transform-Domain 9
The Discrete Fourier Transform
• DTFT Computation Using MATLAB
– The Signal Processing Toolbox in MATLAB
– Functions:
• freqz
• abs
• Angle
– The forms of freqz:
• H = freqz(num, den, w)
• [H, w] = freqz(num, den, k, ’whole’)
– Example 3.8:
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Program 3_1
Discrete-Time Signals in the Transform-Domain 10
The Discrete Fourier Transform
• Definition
The simplest relation between a finite-length
sequence x[n], defined for 0  n  N  1 , and its
DTFT X  e j  is obtained by uniformly sampling
X e
j
 on the  -axis between
 k  2 k / N ,
F rom X  e
j
0    2
at
0  k  N 1 .

 
x [ n ]e
 j n
n  
X k   X e
j
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
N 1
  2 k / N


x [ n ]e
 j 2  kn / N
, 0 k  N 1
n0
Discrete-Time Signals in the Transform-Domain 11
The Discrete Fourier Transform
• The sequence X[k] is called the discrete Fourier
transform (DFT) of the sequence x[n].
• Using the commonly used notation
WN  e
 j 2 / N
• We can rewrite as
N 1
X [k ] 

0  k  N 1
kn
x [ n ]W N ,
n0
• Inverse discrete Fourier transform (IDFT)
N 1
x[ n ] 

X [ k ]e
n0
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j 2  kn / N
N 1


 kn
X [ k ]W N
,
0  n  N 1
n0
Discrete-Time Signals in the Transform-Domain 12
The Discrete Fourier Transform
• Matrix Relations
N 1
The DFT samples defined in
X [k ] 

kn
x [ n ]W N
can
n0
be expressed in matrix form as
X  DN x
where X is the vector composed of the N DFT
samples,
T
X   X  0  X 1
X  N  1 
x is the vector of N input samples,
x   x [0] x [1]
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x[ N  1] 
T
Discrete-Time Signals in the Transform-Domain 13
The Discrete Fourier Transform
•
D N is the N  N
DN
DFT matrix given by
1
1 1　1 


1
2
N- 1
1
W　W
W
N
N
N


2
4
2( N- 1)

 1 WN WN
WN





N- 1
2( N- 1)
( N- 1) ( N- 1) 
WN
1 WN WN

• IDFT relations
1
N
x  D X 
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1
N
*
DN X
Discrete-Time Signals in the Transform-Domain 14
The Discrete Fourier Transform
• DFT computation Using MATLAB
– MATLAB functions:
fft(x), fft(x,N), ifft(X), ifft(X,N)
– X = fft(x, N)
If N < R=length(x), truncate (截短) to the first N
samples.
If N > R=length(x), zero-padded (补零) at the end.
– Example 3.11, 3.12, 3.13, Program 3_2, 3_3, 3_4.
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Discrete-Time Signals in the Transform-Domain 15
Relation between the DTFT and the
DFT, and their Inverses
• DTFT from DFT by Interpolation
We could express
X e
j


1
N
1
N

1
N
N 1

x [ n ]e
 j n

k0
N 1

k0

1
N
n0
N 1
X e
N 1
X [ k ] e
j
 in terms of X[k]:
N 1

j 2 kn / N
n0
e
 N 1
 j n
 kn 
e
X
[
k
]
W

N


 k0

 j n
n0
  N  2 k
sin 
2

X k 
  N  2 k
sin 
2N

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

  j     2  k / N     N  1  / 2 
e



Discrete-Time Signals in the Transform-Domain 16
Relation between the DTFT and the DFT, and
their Inverses
• Sampling the DTFT
– Consider the following question
 
DTFT
x[ n ]
X (e
j
)
  k  2 k / N , 0  k  N - 1
?
y [ n ], 0  n  N - 1   Y [ k ]  X ( e
DFT
j k
), 0  k  N - 1
– We obtain the relation

y[ n ] 

x [ n  m N ],
0  n  N -1
m  
– Example 3.14
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Discrete-Time Signals in the Transform-Domain 17
Relation between the DTFT and the DFT, and
their Inverses
• Numerical Computation of the DTFT Using the DFT
– Let X ( e j ) be the DTFT of length-N sequence x[n]. We
wish to evaluate X ( e j ) at a dense grid of frequencies:
 k  2  k / M , 0  k  M  1, w h ere M

X e
j k

 X e
j

N 1
 k


x [ n ]e
 j k n
N
N 1

n0

x [ n ]e
 j 2  kn / M
n0
 x[n ] 0  n  N  1
D efin e a n ew seq u en ce x e [ n ]  
N  n  M -1
0

th en X e
j k
  DFT { x
M 1
[ n ]} 
e

x e [ n ]e
 j 2  kn / M
n0
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Discrete-Time Signals in the Transform-Domain 18
Discrete Fourier Transform Properties
• Discrete Fourier Transform Properties
Like the DTFT, the DFT also satisfies a number
of properties that are useful in signal processing
application. A summary of the DFT properties are
included in Tables 3.5, 3.6, and 3.7.
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Discrete-Time Signals in the Transform-Domain 19
Discrete Fourier Transform Properties
• Circular Shift of a Sequence
– Time-shifting property of the DTFT
x 1 [ n ]  x [ n  n 0 ]    X 1 ( e
DTFT
j
) e
 j n 0
X (e
j
)
– Circular shifting property of the DFT
x [ n ], 0  n  N - 1   
 X [ k ], 0  k  N - 1
DFT

?
x c [ n ], 0  n  N - 1   
 X c [ k ]  W N 0 X [ k ], 0  k  N - 1
DFT
W e ob tain
kn
x c [ n ]  x [  n  n 0  N ]  x [( n  n 0 )% N ]
n0  n  N  1
 x[ n  n0 ]
F or n 0 > 0, x c [ n ]  
0  n  n0
 x[n  n0  N ]
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Discrete-Time Signals in the Transform-Domain 20
Computation of the DFT of Real
Sequences
• Computation of the DFT of Real Sequences
Tow N-point DFTs can be computed efficiently
using a single N-point DFT X[k] of a complex
length-N sequence x[n] defined by
x  n   g  n   jh  n 
where, g  n   R e{ x [ n ]} and h[ n ]  Im { x[ n ]}
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Discrete-Time Signals in the Transform-Domain 21
Computation of the DFT of Real
Sequences
we arrive at:
Note that
G k  
1
H k  
X k   X

2j
*
{ X [ k ]  X   k
2
*
X   k
云南大学滇池学院课程：数字信号处理
1
N
*
N
 },

 k

 X* N k


N
N





Discrete-Time Signals in the Transform-Domain 22
Linear Convolution Using the DFT
• Linear Convolution of Two Finite-Length
Sequences
Let g[n] and h[n] be finite-length sequences of
lengths N and M, respectively. Denote L=M+N-1.
Define two length-L sequences,
 g  n  , 0  n  N  1
ge n  
 0. N  n  L  1
 h  n  , 0  n  M  1
he  n   
 0. M  n  L  1
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Discrete-Time Signals in the Transform-Domain 23
Linear Convolution Using the DFT
obtained by appending g[n] and h[n] with
zero-valued samples. Then
yL  n   g  n   h  n   yc  n 
 g e  n  linear convolution h e  n 
• Linear Convolution of a Finite-Length Sequence
with an Infinite-Length Sequence
– Overlap-Add Method
– Overlap-Save Method
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Discrete-Time Signals in the Transform-Domain 24
The z-Transform
• Definition
For a given sequence g[n], its z-transform G(z) is
defined as

G  z   Z  g  n  

g  n z
n
n  
where z  R e  z   j Im  z  is a complex variable.
If we let z  re
j
, then the right-hand side of the
above expression reduces to
G  re
j

 

g n r
n
e
 j n
n  
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Discrete-Time Signals in the Transform-Domain 25
The z-Transform
For a given sequence, the set R of values of z
for which its z-transform converges is called
the region of convergence (ROC).
If


g n r
n

n  
In general, the region of convergence R of a ztransform of a sequence g[n] is an annular region
of the z-plane:
Rg  z  Rg
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Discrete-Time Signals in the Transform-Domain 26
The z-Transform
• Rational z-Transforms
– An alternate representation as a ration of two
polynomials in z:
– An alternate representation in factored form as
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Discrete-Time Signals in the Transform-Domain 27
Region of Convergence of a Rational zTransform
• The ROC of a rational z-transform is bounded by
the locations of its poles.
– A finite-length sequence ROC:
0 z 
– A right-sided sequence ROC:
Rg  z
– A left-sided sequence ROC:
z  Rg
– A two-sided sequence ROC:
Rg  z  Rg
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Discrete-Time Signals in the Transform-Domain 28
Inverse z-Transform
• General Expression
– By the inverse Fourier transform relation. We have
g n r
n

1
2



G  re
j
e
j n
d
j
– By making the change of variable z  re , the above
equation can be converted into a contour integral given
by
1
n 1
g n 
G
z
z
dz


'

2 j C
'
Where C is a counterclockwise contour of integration
defined by
z r
云南大学滇池学院课程：数字信号处理
Discrete-Time Signals in the Transform-Domain 29
Inverse z-Transform
• Inverse Transform by Partial-Fraction Expansion
G z
can be expressed as
G z 
P z
D z
• We can divide P(Z) by D(Z) and re-express G(Z) as
G z 
M N

0
云南大学滇池学院课程：数字信号处理
z


P z
D z
Discrete-Time Signals in the Transform-Domain 30
Inverse z-Transform
• Simple Poles
p168
• Multiple Poles
p169
云南大学滇池学院课程：数字信号处理
Discrete-Time Signals in the Transform-Domain 31
z-Transform Properties
• P174 Table 3.9
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Discrete-Time Signals in the Transform-Domain 32
Summary
• Three different frequency-domain representations of an
aperiodic discrete-time sequence have been introduced and
their properties reviewed .Two of these representations, the
discrete-time Fourier transform (DTFT) and the z-transform,
are applicable to any arbitrary sequence, whereas the third
one , the discrete Fourier transform (DFT), can be applied
only to finite-length sequences.
• Relation between these three transforms have been
established. The chapter ends with a discussion on the
transform-domain representation of a random discrete-time
sequence.
• For future convenience we summarize below these three
frequency-domain representations.
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Discrete-Time Signals in the Transform-Domain 33
Assignment and Experiment
• Assignment
– A03: 3.2, 3.12, 3.20, See p180~182
– A04:
– A05:
• Experiment
– E03: Q3.3 See p32
– E04:
– E05
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Discrete-Time Signals in the Transform-Domain 34