CHAPTER 3 Discrete-Time Signals in the
Transform-Domain
CHAPTER 3 Discrete-Time Signals in the
Transform-Domain
Wang Weilian
School of Information Science and Technology
Yunnan University
云南大学滇池学院课程:数字信号处理 Discrete-Time Signals in the Transform-Domain 2
OutlineOutline
• 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
云南大学滇池学院课程:数字信号处理 Discrete-Time Signals in the Transform-Domain 3
OutlineOutline
• Region of Convergence of a Rational z-Transform
• Inverse z-Transform
• z-Transform Properties
云南大学滇池学院课程:数字信号处理 Discrete-Time Signals in the Transform-Domain 4
The Discrete-Time Fourier TransformThe 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 where is the real frequency variable.
• The discrete-time Fourier transform of a sequence x[n] is defined by
j xe
jX e
[ ]j j n
n
X e x n e
云南大学滇池学院课程:数字信号处理 Discrete-Time Signals in the Transform-Domain 5
The Discrete-Time Fourier TransformThe Discrete-Time Fourier Transform
• In general is a complex function of the real variable and can be written in rectangular form as
where and are, respectively, the real and imaginary parts of , and are real functions of .
• Polar form
jX e
j
imX e jreX e
j j jre imX e X e jX e
jX e
where arg
jj j
j
e X e e
X e
X
云南大学滇池学院课程:数字信号处理 Discrete-Time Signals in the Transform-Domain 6
The Discrete-Time Fourier TransformThe Discrete-Time Fourier Transform
• Convergence Condition:
If x[n] is an absolutely summable sequence, i.e.,
Thus the equation is a sufficient condition for the existence of the DTFT.
n
j j n
n n
if x n
then X e x n e x n
云南大学滇池学院课程:数字信号处理 Discrete-Time Signals in the Transform-Domain 7
The Discrete-Time Fourier TransformThe Discrete-Time Fourier Transform
• Bandlimited Signals:
– A full-band discrete-time signal has a spectrum occupying the whole frequency rang .
– If the spectrum is limited to a portion of the frequency range , it is called a bandlimited signal.
– A lowpass discrete-time signal has a spectrum occupying the frequency range , where is called the bandwidth of the signal.
– A bandpass discrete-time signal has a spectrum occupying the frequency range , where is its bandwidth.
0
0
0 p p
0 L H H L
云南大学滇池学院课程:数字信号处理 Discrete-Time Signals in the Transform-Domain 8
The Discrete-Time Fourier TransformThe 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.
云南大学滇池学院课程:数字信号处理 Discrete-Time Signals in the Transform-Domain 9
The Discrete-Time Fourier TransformThe Discrete-Time Fourier Transform
• Energy Density Spectrum
* *
-
2
Parseval's relation:
1 [ ] [ ] ( ) ( )
2
Total energy of a finite-energy sequence [ ] :
If [ ] [ ],
j j
n
gn
g n h n G e H e d
g n
g n
h n g n
2 2
-
2
then from Parseval's relation we observe
1 | [ ] | | ( ) |
2
The quantity:
is called the energy density spectrum of the sequence [ ].
jg
n
j jgg
g n G e d
S e G e
g n
云南大学滇池学院课程:数字信号处理 Discrete-Time Signals in the Transform-Domain 10
The Discrete Fourier TransformThe 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: Program 3_1
云南大学滇池学院课程:数字信号处理 Discrete-Time Signals in the Transform-Domain 11
The Discrete Fourier TransformThe Discrete Fourier Transform
• Definition
The simplest relation between a finite-length
sequence x[n], defined for , and its
DTFT is obtained by uniformly sampling
on the -axis between at
, .
0 1n N
0 2
2 /k k N 0 1k N
1
2 /
2 /0
From [ ]
[ ] , 0 1
j j n
n
Nj j kn N
k Nn
X e x n e
X k X e x n e k N
jX e
jX e
云南大学滇池学院课程:数字信号处理 Discrete-Time Signals in the Transform-Domain 12
The Discrete Fourier TransformThe Discrete Fourier Transform
• The sequence X[k] is called the discrete Fourier transform (DFT) of the sequence x[n].
• Using the commonly used notation
• We can rewrite as
• Inverse discrete Fourier transform (IDFT)
2 /j NNW e
1
0
[ ] [ ] , 0 1N
knN
n
X k x n W k N
1 12 /
0 0
[ ] [ ] [ ] , 0 1N N
j kn N knN
n n
x n X k e X k W n N
云南大学滇池学院课程:数字信号处理 Discrete-Time Signals in the Transform-Domain 13
The Discrete Fourier TransformThe Discrete Fourier Transform
• Matrix Relations
The DFT samples defined in can
be expressed in matrix form as
where X is the vector composed of the N DFT samples,
x is the vector of N input samples,
1
0
[ ] [ ]N
knN
n
X k x n W
X xND
X 0 1 1T
X X X N
x [0] [1] [ 1]T
x x x N
云南大学滇池学院课程:数字信号处理 Discrete-Time Signals in the Transform-Domain 14
The Discrete Fourier TransformThe Discrete Fourier Transform
• is the DFT matrix given by
• IDFT relations
ND N N
1 1 1
1
1
1
ND
1 2 N-1N N N
2 4 2(N-1)N N N
N-1 2(N-1) (N-1) (N-1)N N N
1
W W W
W W W
W W W
1 *1x X XN ND D
N
云南大学滇池学院课程:数字信号处理 Discrete-Time Signals in the Transform-Domain 15
The Discrete Fourier TransformThe 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.
云南大学滇池学院课程:数字信号处理 Discrete-Time Signals in the Transform-Domain 16
Relation between the DTFT and the DFT, and their Inverses
Relation between the DTFT and the DFT, and their Inverses
• DTFT from DFT by Interpolation
We could express in terms of X[k]:
1 1 1
0 0 0
1 12 /
0 0
12 / 1 / 2
0
1 1[ ] [ ]
1 [ ]
2sin
1 2
2sin
2
N N Nj j n j nkn
Nn n k
N Nj kn N j n
k n
Nj k N N
k
X e x n e eX k WN N
X k e eN
N k
X k eN kNN
jX e
云南大学滇池学院课程:数字信号处理 Discrete-Time Signals in the Transform-Domain 17
Relation between the DTFT and the DFT, and their Inverses
Relation between the DTFT and the DFT, and their Inverses
• Sampling the DTFT
– Consider the following question
– We obtain the relation
– Example 3.14
[ ] ( )
2 / , 0 -1
[ ], 0 -1 [ ] ( ), 0
?
-1 k
DTFT j
k
DFT j
x n X e
k N k N
y n n N Y k X e k N
[ ] [ ], 0 -1m
y n x n mN n N
云南大学滇池学院课程:数字信号处理 Discrete-Time Signals in the Transform-Domain 18
Relation between the DTFT and the DFT, and their Inverses
Relation between the DTFT and the DFT, and their Inverses
• Numerical Computation of the DTFT Using the DFT
– Let be the DTFT of length-N sequence x[n]. We wish to evaluate at a dense grid of frequencies:( )jX e
( )jX e
1 12 /
0 0
12 /
0
2 / , 0 1, where
[ ] [ ]
[ ] 0 1Define a new sequence [ ]
0 -1
then { [ ]} [ ]
k k
k
k
k
N Nj j nj j kn M
n n
e
Mj j kn M
e en
k M k M M N
X e X e x n e x n e
x n n Nx n
N n M
X e DFT x n x n e
云南大学滇池学院课程:数字信号处理 Discrete-Time Signals in the Transform-Domain 19
Discrete Fourier Transform PropertiesDiscrete 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.
云南大学滇池学院课程:数字信号处理 Discrete-Time Signals in the Transform-Domain 20
Discrete Fourier Transform PropertiesDiscrete Fourier Transform Properties
• Circular Shift of a Sequence
– Time-shifting property of the DTFT
– Circular shifting property of the DFT
0
0 0
0
[ ], 0 -1 [ ], 0 -1
[ ], 0 -1 [ ] [ ], 0 -1
We obtain [ ] [ ] [( )% ]
For >0, [
?
]
DFT
DFT knc c N
c N
c
x n n N X k k N
x n n N X k W X k k N
x n x n n x n n N
n x n
0 0
0 0
[ ] 1
[ ] 0 n
x n n n n N
x n n N n
01 0 1[ ] [ ] ( ) ( )
DTFT j nj jx n x n n X e e X e
云南大学滇池学院课程:数字信号处理 Discrete-Time Signals in the Transform-Domain 21
Computation of the DFT of Real Sequences
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
where, and
x n g n jh n
Re{ [ ]}g n x n [ ] Im{ [ ]}h n x n
云南大学滇池学院课程:数字信号处理 Discrete-Time Signals in the Transform-Domain 22
Computation of the DFT of Real Sequences
Computation of the DFT of Real Sequences
we arrive at:
Note that
*
*
1{ [ ] },
21
2
N
N
G k X k X k
H k X k X kj
* *
N NX k X N k
云南大学滇池学院课程:数字信号处理 Discrete-Time Signals in the Transform-Domain 23
Linear Convolution Using the DFTLinear 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,
, 1
0. 1
, 1
0. 1
e
e
g n n Ng n
n L
h n n Mh n
n L
0
N
0
M
云南大学滇池学院课程:数字信号处理 Discrete-Time Signals in the Transform-Domain 24
Linear Convolution Using the DFTLinear Convolution Using the DFT
obtained by appending g[n] and h[n] with
zero-valued samples. Then
• Linear Convolution of a Finite-Length Sequence with an Infinite-Length Sequence
– Overlap-Add Method
– Overlap-Save Method
L c
e e
y n g n h n y n
g n linear convolution h n
云南大学滇池学院课程:数字信号处理 Discrete-Time Signals in the Transform-Domain 25
The z-TransformThe z-Transform
• Definition
For a given sequence g[n], its z-transform G(z) is defined as
where is a complex variable.
If we let , then the right-hand side of the above expression reduces to
n
n
G z Z g n g n z
Re Imz z j z jz re
j n j n
n
G re g n r e
云南大学滇池学院课程:数字信号处理 Discrete-Time Signals in the Transform-Domain 26
The z-TransformThe 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
In general, the region of convergence R of a z-transform of a sequence g[n] is an annular region of the z-plane:
n
n
g n r
g gR 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 27
The z-TransformThe z-Transform• Rational z-Transforms
– An alternate representation as a ration of two polynomials in z:
– An alternate representation in factored form as
云南大学滇池学院课程:数字信号处理 Discrete-Time Signals in the Transform-Domain 28
Region of Convergence of a Rational z-Transform
Region of Convergence of a Rational z-Transform
• The ROC of a rational z-transform is bounded by the locations of its poles.
– A finite-length sequence ROC:
– A right-sided sequence ROC:
– A left-sided sequence ROC:
– A two-sided sequence ROC:
0 z
gR z
gz R
g gR 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-TransformInverse z-Transform
• General Expression
– By the inverse Fourier transform relation. We have
– By making the change of variable , the above equation can be converted into a contour integral given by
Where is a counterclockwise contour of integration
defined by
1
2n j j ng n r G re e d
jz re
'
11
2n
Cg n G z z dz
j
'C
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 30
Inverse z-TransformInverse z-Transform
• Inverse Transform by Partial-Fraction Expansion
can be expressed as
• We can divide P(Z) by D(Z) and re-express G(Z) as
G z
P zG z
D z
0
M N P zG z z
D z
云南大学滇池学院课程:数字信号处理 Discrete-Time Signals in the Transform-Domain 31
Inverse z-TransformInverse 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 32
z-Transform Propertiesz-Transform Properties
• P174 Table 3.9
云南大学滇池学院课程:数字信号处理 Discrete-Time Signals in the Transform-Domain 33
SummarySummary
• 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.
云南大学滇池学院课程:数字信号处理 Discrete-Time Signals in the Transform-Domain 34
Assignment and ExperimentAssignment 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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