### Chapter 8 The Discrete Fourier Transform

```Biomedical Signal processing
Chapter 8 The Discrete Fourier
Transform
Zhongguo Liu
Biomedical Engineering
School of Control Science and Engineering, Shandong
University
2015/4/13
1
Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Chapter 8 The Discrete Fourier Transform
8.0 Introduction
8.1 Representation of Periodic Sequence: the
Discrete Fourier Series
8.2 Properties of the Discrete Fourier Series
8.3 The Fourier Transform of Periodic Signal
8.4 Sampling the Fourier Transform
8.5 Fourier Representation of Finite-Duration
Sequence: the Discrete Fourier Transform
8.6 Properties of the Discrete Fourier Transform
8.7 Linear Convolution using the Discrete
Fourier Transform
2
Filter Design Techniques
8.0 Introduction
3
8.0 Introduction
Discrete Fourier Transform (DFT) for
finite duration sequence
DFT is a sequence rather than a function
of a continuous variable
DFT corresponds to sample, equally
spaced in frequency, of the Fourier
transform of the signal.
4
8.0 Introduction
The relationship between periodic
sequence and finite-length sequences：
The Fourier series representation of the
periodic sequence corresponds to the
DFT of the finite-length sequence.
5
8.1 Representation of Periodic Sequence:
the Discrete Fourier Series
Given a periodic sequence ~
x[n] with period N so
that
~
~
x[n]  x[n  rN]
The Fourier series representation can be written
as
j  2 / N kn
1
x[n]   X  k  e
N k
Fourier series representation of continuous-time periodic
signals require infinite many complex exponentials
Not that for discrete-time periodic signals we have
e
6
j2 / Nk  mNn
e
j2 / Nkn
e
j2mn
e
j2 / Nkn
8.1 Representation of Periodic Sequence:
the Discrete Fourier Series
e
j2 / Nk  mNn
e
j2 / Nkn
e
j2mn
e
j2 / Nkn
Due to the periodicity of the complex exponential
we only need N exponentials for discrete time
Fourier series
1 N 1
j  2 / N  kn
x[n]   X  k  e
N k 0
No need
7
j  2 / N kn
1
x[n]   X  k  e
N k
Discrete Fourier Series Pair
A periodic sequence in terms of Fourier series
coefficients
j  2 / N kn
1 N 1
x[n]   X  k  e
N k 0
To obtain the Fourier series coefficients we multiply
both sides by
e
 j (2 / N ) rn for 0nN-1 and then
sum both the sides , we obtain
N 1
 x(n)e
j
2
rn
N
n 0
N 1
 x(n)e
n 0
8
j
2
rn
N
N 1
1

n 0 N
N 1
 X (k )e
j
2
( k r ) n
N
k 0
1 j 2N ( k r ) n
  X ( k ) e
k 0
n 0 N
N 1
N 1
Discrete Fourier Series Pair
j  2 / N kn
1 N 1
x[n]   X  k  e
N k 0
N 1
 x(n)e
j
2
rn
N
n 0
1
N
N 1
e
j
2
( k r ) n
N
n 0
N 1
1 j 2N ( k r ) n
  X ( k ) e
k 0
n 0 N
N 1
N 1
1, k - r  mN , m an integer

0, otherwise
 x(n)e
j
2
rn
N
Problem 8.51, HW
 X (r )
n 0
N 1
9
X ( k )   x ( n) e
n 0
j
2
kn
N
8.1 Representation of Periodic
Sequence: the Discrete Fourier Series
x n with period N,
a periodic sequence ~
~
x n  ~
x n  rN  for any integer r
The Fourier series coefficients of ~x n is
N 1
X  k    x  n e
 j  2 N kn
n 0
j  2 N kn
1 N 1
x  n   X  k  e
N k 0
10
8.1 Representation of Periodic
Sequence: the Discrete Fourier Series
N 1
X  k    x  n e
 j  2 N kn
n 0
~
The sequence X k  is periodic with period N
~
~
~
~
X 0  X N , X 1  X N 1
N 1
X  k  N    x  n e
 j  2 N  k  N n
n 0
 j  2 N kn   j 2 n

   x  n e
 X k 
e
 n 0

N 1
11
Discrete Fourier Series (DFS)
N 1
X  k    x  n e
 j  2 N kn
n 0
Analysis equation:
Let WN  e j 2 N 
N 1
~
kn
~
X k    x nWN
n 0
Synthesis equation:
DFS
~
~
x n  X k 
N 1
1
~
~
x n   X k WNkn
N k 0
F
discrete  periodic
F
periodic discrete
12
Ex. 8.1 DFS of a impulse train
Consider the periodic impulse train
~
x n 
1, n  rN , r is any integer
 n  rN   

r  
0, therwise

~
x n
N points
-N -N+1…… -2
-1
0
1
2
……
N-1 N N+1 N+2
……
n
N 1
~
X k     nWNkn  WN0  1
13
n 0
Ex. 8.1 DFS of a impulse train
N 1
~
X k     nWNkn  WN0  1
n 0
~
X k 
N points
-N -N+1…… -2
-1
0
1
2
……
N-1 N N+1 N+2
……
k
 1, n  rN , r is any integer
x  n     n  rN   
r 
0, therwise

N 1
1 N 1
1
j  2
 kn
  X  k WN   e
N k 0
N k 0
14
N  kn
N points
1
-N
-N+1……
-2
-1
0
-N
15
……
N points
1
-N+1……
1
2
-2
-1
0
1
2
……
~
x n
N-1 N N+1 N+2
……k
~
X k 
N-1 N N+1 N+2
…… n
Example 8.2 Duality in the Discrete
Fourier Series
The discrete Fourier series coefficients
is the periodic impulse train N points
N
Y k 

~
Y k  
 N k  rN 
r  
N 1
1
~
~
x n   X k WNkn
N k 0
-N
…
… -2
-1 0 1
…
2…
N
N 1
~
X k   ~
x nWNkn
n 0
N 1
N 1
1
1
~
~
y n   Y k WNkn   N k WNkn  WN0  1
N k 0
N k 0
16
…
…
Y k 
N points
N
k
-N -N+1…… -2
-1
0
2
1
……
N-1 N N+1 N+2
N points
……
y  n
1
-N -N+1…… -2
17
-1
0
1
2
……
N-1 N N+1 N+2
n
……
N points
~
x n
1
-N+1……
-N
-2
-1
0
1
……
N-1 N N+1 N+2
-N+1……
-2
-1
2
0
……
1
N points
N-1 N N+1 N+2
-1
0
2
1
……
N-1 N N+1 N+2
N points
1
18
-N -N+1…… -2
-1
0
1
2
…… n
Y k 
N
-N -N+1…… -2
……k
~
X k 
N points
1
-N
2
……
……
y  n
N-1 N N+1 N+2
……
Example 8.3 The Discrete Fourier Series
of a Periodic Rectangular Pulse Train
Periodic sequence with period N=10
1
4
X  k   W
n 0
kn
10
4
 e
 j  2 10 kn
n 0
 4 k 10 sin  k 2 
1  W105k
e

k
sin  k 10
1  W10
19
magnitude
phase
20
X k   e
 4 k 10
sin  k 2 
sin  k 10 
magnitude
phase
21
X k   e
 4 k 10
sin  k 2 
sin  k 10 
8.2 Properties of the Discrete
Fourier Series
Linearity: two periodic sequence, both
with period N
DFS
~
~
x1n  X1k 
DFS
~
~
x2 n  X 2 k 
DFS
~
~
~
~
ax1n  bx2 n  aX1k   bX 2 k 
22
8.2 Properties of the Discrete
Fourier Series
Shift of a sequence
DFS
~
~
x n  X k 
DFS
km ~
~
x n  m  WN X k 
DFS
WN nl x  n  X  k  l 
Problem 8.52, HW
23
8.2 Properties of the Discrete
Fourier Series
Duality
DFS
~
~
x n  X k 
~
x n
1
~
X k 
1
0
1
2
0
……
n
N-1
0
X  n
1
24
DFS
~
X n  N~
x  k 
1
2
……
N-1
2
1
……
0
k
Nx k 
N
n
N-1
1
2
……
k
N-1
8.2.4 Symmetry
Problem 8.53, HW
25
8.2.5 Periodic Convolution
x2 n are two periodic sequences,
x1 n and ~
~
each with period N and with discrete Fourier
~
~
X
series X 1k  and 2 k 
~
~
~
X 3 k   X1k X 2 k 
N 1
N 1
m 0
m 0
x3  n   x1  m x2  n  m   x2  m x1  n  m
N 1
X 3  k    x3  nW
N 1
n 0
N 1
kn
N
N 1 N 1
  x1  m x2  n  mW
n 0 m 0
kn
N
N 1

  x1  m x2  n  mWNkn   x1  m WNkm X 2  k 
m0
N 1
26
n 0
m0

km 
   x1  mWN  X 2  k   X1  k  X 2  k 
 m0


8.2.5 Periodic Convolution
N 1
 x  m x  n  m 
m0
1
2
DFS 

X1 k  X 2 k 
The sum is over the finite interval 0  m  N  1
The value of ~x2 n  m in the interval 0  repeat
m  N 1
periodically for m outside of that interval
N 1
1
~
~ ~
~
~
~
DFS
x3 n  x1nx2 n  X 3 k    X 1l X 2 k  l 
N l 0
27
Example 8.4 Periodic Convolution
x2 m
x1 m
x2 m
x2 1 m
N 1
x3 1   x1  m x2 1  m
m0
x2 2  m
N 1
28
x3  2   x1  m x2  2  m
m0
8.2.5 Periodic Convolution
~
x3 n  ~
x1n~
x2 n
1 N 1 ~ ~
~
X 3 k    X 1 l X 2 k  l 
N l 0
29
8.1 Representation of Periodic
Sequence: the Discrete Fourier Series
x n with period N,
a periodic sequence ~
~
x n  ~
x n  rN  for any integer r
The Fourier series coefficients of ~x n is
N 1
X  k    x  n e
 j  2 N kn
n 0
j  2 N kn
1 N 1
x  n   X  k  e
N k 0
30
Review
Discrete Fourier Series (DFS)
N 1
X  k    x  n e
 j  2 N kn
n 0
Let WN  e j 2 N 
N 1
~
X k    ~
x nWNkn
Analysis equation:
n 0
Synthesis equation:
DFS
N 1
1
~
 kn
~
x n   X k WN
N k 0
~
~
x n  X k 
31
8.2 Properties of the Discrete
Fourier Series
Shift of a sequence
DFS
~
~
x n  X k 
DFS
km ~
~
x n  m  WN X k 
DFS
WN nl x  n  X  k  l 
32
WN  e
 j 2 N 
8.2 Properties of the Discrete
Fourier Series
Duality
DFS
~
~
x n  X k 
~
x n
1
~
X k 
1
0
1
2
0
……
n
N-1
0
X  n
1
33
DFS
~
X n  N~
x  k 
1
2
……
N-1
2
1
……
0
k
Nx k 
N
n
N-1
1
2
……
k
N-1
8.2.5 Periodic Convolution
~
~
~
X 3 k   X1k X 2 k 
N 1
~
~
~
x3 n   x1 mx2 n  m
m 0
~
x3 n  ~
x1n~
x2 n
1 N 1 ~ ~
~
X 3 k    X 1 l X 2 k  l 
N l 0
34
Example 8.4 Periodic Convolution
x2 m
x1 m
x2 m
x2 1 m
N 1
x3 1   x1  m x2 1  m
m0
x2 2  m
N 1
35
x3  2   x1  m x2  2  m
m0
8.3 The Fourier Transform of
Periodic Signal
Periodic sequences are neither absolutely summable
nor square summable, hence they don’t have a
strict Fourier Transform
xn  1 for all n
xn  e
jw0 n
   2  w  2 r 
X e jw 
F

F


r  
    2  w  w  2 r 

Xe
jw
0
r  
x  n    ak e
k
36
jwk n
    2 a  w  w  2 r 

F

Xe
jw
r   k
k
k
8.3 The Fourier Transform of
Periodic Signal
We can represent Periodic sequences as sums of
complex exponentials: DFS
We can combine DFS and Fourier transform
Fourier transform of periodic sequences
Periodic impulse train with values proportional
to DFS coefficients
j  2 N kn
1 N 1
x  n   X  k  e
N k 0

2
j
X e
37
 
k - 
2 k 

X  k    
N
N 

8.3 The Fourier Transform of
Periodic Signal
2
2 k 

X e   
X  k    
N
N


k - 
This is periodic with 2 since DFS is periodic
j

The inverse transform can be written as
1
2
2 -

0-
1
j
j n
X  e e d 
2
2 -

0-
2
2 k  jn

X  k    
e d
N 

k -  N

2 k
n
N
1
2 k  jn
1

  X  k     e d   X  k  e

0-
N k - 
N 
N k 0

 x  n

38
2 -
N -1
j
Ex. 8.5 Fourier Transform of a
periodic impulse train
Consider the periodic impulse train
p[n] 
N points
p  n
1

   n  rN 
r 
-N
…
… -2
-1 0 1
…
2…
N
The DFS was calculated previously to be
P k   1 for all k
-N
…
…
-2 -1 0 1
Therefore the Fourier transform is
2 
2 k 
P e   
  

N 

k  N
j
39

P k 
N points
1
…
2 … N-1
…
…
N
…
… n
Relation between Finite-length and
Periodic Signals
Consider finite length signal x[n]
spanning from 0
p  n
to N-1 1 …
…
-N
N
- -1 0 1 2
…periodic
2
with
Convolve
impulse train

…
x[n]  x[n]  p[n]  x[n]     n  rN  
r 

 x n  rN 
r 
The Fourier transform of the periodic sequence is

2 
2 k 
j
j
j
j
X e   X e  P e   X e  
  

N
N


k 
2 k
j

 
2
2 k 
j
N
X e   
X e
   

N 
k  N

 

40
Relation between Finite-length and
Periodic Signals
2
2 k 

X e   
X  k    
N 

k -  N

j
2 k
j

 
2

2 k 
j
N
X e   
X e
   

N 
k  N

 

This implies that
 j 2N k 
j
X k   X  e

X
e
    2 k

N


DFS coefficients of a periodic signal can be thought
as equally spaced samples of the Fourier transform
of one period
41
Relation between Finite-length and
Periodic Signals
~
If x n is periodic with period N, the DFS are
N 1
~
X k    ~
x ne  j 2
N kn
n 0
~


x
n

x n for 0  n  N  1 and xn  0 otherwise
If
then
    xne
N 1
X e
jw
n 0
X  k   X  e jw 
42
 jwn
N 1
 jwn
~
  x ne
n 0
w2 k N
Ex. 8.5 Relation between FS coefficients and FT
Consider the sequence
1 0  n  4
x[n]  
else
0
The Fourier transform
X e
jw
X e
j
43

   x  n e
 jwn
n 
e
 j 2
sin  5 / 2
sin  / 2
Ex. 8.5 Relation between FS coefficients and FT
 Consider the sequence
1 0  n  4
x[n]  
else
0
The DFS coefficients
N 1
X  k    x  n e
 j  2 N kn
n 0
~
sink / 2
Xk   e j4k / 10
sink / 10
The Fourier transform
N 1
X  e jw    x  n e jwn
  e
Xe
44
j
n 0
 j2
sin5 / 2
sin / 2
Ex. 8.5 Relation between FS coefficients and FT
 Consider the sequence
1 0  n  4
x[n]  
else
0
The DFS coefficients
N 1
X  k    x  n e
 j  2 N kn
n 0
~
sink / 2
Xk   e j4k / 10
sink / 10
The Fourier transform
N 1
X  e jw    x  n e jwn
  e
Xe
45
j
n 0
 j2
sin5 / 2
sin / 2
8.4 Sampling the Fourier Transform
Consider an aperiodic sequence xn with
Fourier transform X e jw  ,and assume that a
sequence X~ k  is obtained by sampling
at frequency
wk  2 k N
j  2 N k 

X k   X e 
 X e

w 2 N  k


j  2 N k
X  k   X  z  z e j2 N k  X e
jw


~
 X k  is Fourier series coefficients of periodic sequence
x  n
46
Sampling the Fourier Transform

j  2 / N k 

 jm
j
X k   X  e
X
e

x
m
e


  



j  2 / N kn
1 N 1
x[n]   X  k  e
N k 0
m 
 j  2 / N km  j  2 / N kn
1 N 1  
x[n]     x  m e
e

N k 0  m

 1 N 1 j 2 / N k  nm  
  x  m   e
  x  m p  n  m 

m
 N k 0
 m


1 N 1 j  2 / N k  nm
p  n  m   e
    n  m  rN 
N k 0
r 
47
Sampling the Fourier Transform
X  k   X  e

N 1
j  2 / N k
1
x[n]   X  k  e
N k 0




j 2 / N kn


N points
-N
…
… -2
-1 0 1
…
2…
N  12

 x  m p  n  m
m 

 x  n     n  rN  
r 

 x n  rN 
r 

1 N 1 j  2 / N k  nm
p  n  m   e
    n  m  rN 
N k 0
r 
48
p  n
1
 x  n 0  n  N  1
x  n  
else
 0
N
…
…
Sampling the Fourier Transform
j  2 / N k 

X k   X  e



N 7
j  2 / N kn
1 N 1
x[n]   X  k  e
N k 0
x[n] 

 x n  rN 
r 
49
2
N
Sampling the Fourier Transform
Samples of the DTFT of an aperiodic sequence
can be thought of as DFS coefficients
of a periodic sequence
obtained through summing periodic
replicas of original sequence
If the original sequence is of finite length,
and we take sufficient number of samples of its
DTFT,
then the original sequence can be recovered by
 x  n 0  n  N  1
x  n  
else
 0
50
Sampling the Fourier Transform
It is not necessary to know the DTFT at all
frequencies
To recover the discrete-time sequence in time
domain
Discrete Fourier Transform is used in
Representing a finite length sequence by
samples of DTFT
51
8.5 Fourier Representation of Finite-Duration
Sequence: Discrete Fourier Transform
Consider a finite-length sequence xn of
length N samples such that xn  0 outside
N 9
the range 0  n  N  1
To each finite-length sequence of length N,
we can associate a period sequence
x  n 

 x n  rN 
r 
x n, 0  n  N  1
~
xn  
 0, otherwise x  n  x  n mod N   x   n   


N

52
Discrete Fourier Transform
~
X k  with period N
The Discrete Fourier Transform of xn is
For ~x n , the DFS is
~
 X k , 0  k  N  1
X k   
 0, otherwise
X  k   X  k mod N    X   k   N 
53
Discrete Fourier Transform
N 1
~
kn
~
X k    x nWN
n 0
N 1
1
~
 kn
~
x n   X k WN
N k 0
 N 1
kn
 xnWN , 0  k  N  1
X k    n 0

otherwise
 0,
 1 N 1
  X k WN kn , 0  n  N  1
xn   N k 0

otherwise
 0,
54
Discrete Fourier Transform pairs
Analysis equation
N 1
X k    xnW
Synthesis equation
kn
N
n 0
1 N 1
xn   X k WNkn
N k 0
xn
55
DFT

X k 
Discrete Fourier Transform
Time
Fourier transform
(FT)
Fourier series (FS)
Frequency
continuous
continuous
continuous
periodic
discrete
Continuous impulse
train
Discrete Fourier
series (DFS)
discrete
periodic
continuous impulse
train, periodic
Discrete Fourier
transform (DFT)
discrete
discrete
Discrete-time
Fourier transform
(DTFT)
56
continuous
periodic

57
Ex. 8.7 The DFT of a Rectangular Pulse
x[n] is of length 5
We can consider x[n] of
any length greater than 5
Let’s pick N=5
Calculate the DFS of the
periodic form of x[n]
4
X k    e
 j  2 k /5n
n 0
1  e j 2 k
5 k  0, 5, 10,...


 j  2 k /5
else
0
1 e
58
2
k
5
Ex. 8.7 The DFT of a Rectangular Pulse
If we consider x[n] of
length 10
We get a different set of
DFT coefficients
Still samples of the
DTFT but in different
places
59
Review
Relation of DTFT,DFS, DFT
X e



m 
x  m e jm
DFS
j  2 / N k 

X k   X  e



j
j  2 / N kn
1 N 1
x[n]   X  k  e
N k 0
N  12


 x  n  rN 
r 
N 1
~
X k    ~
x nWNkn
n 0
Let
WN  e
DFS
 j 2 N 
N 1
1
~
~
x n   X k WNkn
N k 0
 x  n  , 0  n  N  1 DFT
 X k , 0  k  N 1
x  n  
X k   
else
else
 0,
 0,
60
Discrete Fourier Transform
N 1
~
kn
~
X k    x nWN
n 0
N 1
1
~
 kn
~
x n   X k WN
N k 0
 N 1
 xnWNkn , 0  k  N  1
X k    n 0

otherwise
 0,
 1 N 1
  X k WN kn , 0  n  N  1
xn   N k 0

otherwise
 0,
61
Review
X e
j



m 
x  m e jm
DFS
j  2 / N k 

X k   X  e



Relation of DTFT,DFS, DFT
j  2 / N kn
1 N 1
x[n]   X  k  e
N 7
N k 0


 x  n  rN 
r 
N 1
~
X k    ~
x nWNkn
n 0
 x  n  , 0  n  N  1 DFT
 X k , 0  k  N 1
x  n  
X k   
else
else
 0,
 0,
62
Sampling of DTFT of
Linear Convolution
Consider x1 n of
Linear
length L and x2 n
Convolution
of length P
x3 n 

 x1mx2 n  m
m  

X3 k   X3 e
j  2 k N 
  X e
1
X 3 e
L  P 1
jw
j  2 k N 
  X e X e 
jw
1
 X e
2
jw
2
j  2 k N 
N ?
  X k  X k 
1
2
0  k  N 1
1
x3 p  n   X 3  k WN kn , 0  n  N  1
N k 0
 
x3 n  rN , 0  n  N  1

The inverse DFT x n  
r  
3p
of X 3 k  is ：

otherwise
 0,
63
N 1
8.6 Properties of the Discrete Fourier Transform
8.6.1 Linearity
DFT
ax1n  bx2 n  aX1k   bX2 k 
If x1 n has length N1 and x2 nhas length N 2 ,
N3  maxN1, N2 
X1  k  
X 2 k  
64
N3 1
kn
x
n
W
 1   N3 , 0  k  N 3  1
n 0
N3 1
kn
x
n
W
 2   N3 , 0  k  N 3  1
n 0
8.6.2 Circular Shift of a Sequence
DFS
~
~
x n  X k 
DFS
 j  2 k N m
x  n  m  e
xn, 0  n  N  1
x   n  m   N  , 0  n  N  1
65
DFT

DFT

 j  2 k N m
e
X k 
X k 
X k 
Ex. 8.8 Circular Shift of a Sequence
circular
shift
66
Figure 8.12
8.6.2 Circular Shift of a Sequence
 
xn  X e jw
DFS
~
~
x n  X k 
x  n  m  e jwm X  e jw 
DFS
x  n  m  e
DFT
xn, 0  n  N  1
x1  n , 0  n  N  1
 j  2 k N m
DFT


X k 
X k 
 j  2 k N m
X1  k   e
x  n  x  n  N   X k   X  k  N 
DFS
x1  n  x1  n  N   X1 k   X1  k  N 
DFS
67
X k 
8.6.2 Circular Shift of a Sequence
x  n  x  n  N   X k   X  k  N 
DFS
x1  n  x1  n  N   X1 k   X1  k  N 
DFS
X1  k   e
 j  2 k N m

X1  k   e
 j  2   k  


N
 j  2 k N m
e
x1  n 
68

N m

X k 
X   k   N 
X   k  N   e
x   n  m   N   x  n  m
DFS
 j  2 k N m
X k 
 j  2 k N m
e
X k 
8.6.2 Circular Shift of a Sequence
X1  k   e
x1  n 
 j  2 k N m
X k 
x   n  m   N   x  n  m
DFS
 j  2 k N m
e
X k 

 x1  n  x   n  m   N  , 0  n  N  1
x1  n  
otherwise

 0,
xn, 0  n  N  1

xn  mN , 0  n  N  1
69
DFT
DFT

X k 
e j 2 k
N m
X k 
8.6.3 Duality
DFS
~
~
x n  X k 
x  n  x  n  N   X k   X  k  N 
DFS
x1 n  X n
DFS
~
X n  N~
x  k 
x1 n  X n
X1 k   Nx k 

 Nx  k   Nx   k   N  , 0  k  N  1
X1  k   
0,
otherwise


xn
X  n
70
DFT

DFT

X k 
Nx  k  N  , 0  k  N 1
Ex.8.9 The Duality Relationship for the DFT
71
8.6.4 Symmetry Properties
x  n  x  n  N   X k   X  k  N 
DFS
DFT
x* n  X *  k N , 0  n  N  1
DFT
x*  n N   X * k , 0  n  N  1
1
xe  n    x  n   x*  n 
2
1
*
xo  n    x  n   x  n 
2
1
xep n  xe  n   x   n     x*   n    ,0  n  N 1
N
N

2 
1
*


  n    ,0  n  N 1
x
n

x
n

x
xop n  o  




N
N


2
72




8.6.4 Symmetry Properties

73

1
xep n  xn N   x*  n N  , 0  n  N  1
2
1
xop n  xn N   x*  n N , 0  n  N  1
2
for 0  n  N  1,   n   N  N  n,   n   N  n
1
xep  n    x  n   x*  N  n  , 0  n  N  1
2
1
1
*
xep  0   x  0  x  N    x  0  x*  0  Re  x 0
2
2
1
*
xop  n    x  n   x  N  n  , 0  n  N  1
2
1
x0 p  0   x  0  x*  0  j Im  x  0
2
8.6.4 Symmetry Properties
1
xep  n    x  n   x*  N  n  , 0  n  N  1
2
1
*
xop  n    x  n   x  N  n  , 0  n  N  1
2
1
*
xe  n    x  n   x  n 
2
1
xo  n    x  n   x*  n 
2
xep  n    xe  n   xe  n  N  , 0  n  N  1
xop  n    xo  n   xo  n  N  , 0  n  N  1
74
8.6.4 Symmetry Properties
x n  xe n  xo n
x n  x n  xe n  xo n , 0  n  N 1
xep n  xe n , 0  n  N 1
xop n  xo n , 0  n  N 1
x n  xep n  xop n
DFT
Rexn  X ep k 
DFT
xep n  ReX k 
75
DFT
j Imxn  X op k 
DFT
xop n  j ImX k 
8.6.4 Symmetry Properties
76
8.6.4 习题答案修正
Problem 8.56的证明上式中应该没有

0≦n ≦N-1，所以在下式中，

77
8.6.5 Circular Convolution
N 1
DFS  X k X k
x
m
x
n

m





1 2
1  2  
m0
For two finite-duration sequences x1  n and x2 n ,
both of length N, with DFTs X1 k  and X 2  k 
X 3 k   X1 k  X 2 k 
N 1
x3  n   x1  m x2  n  m, 0  k  N  1
m 0
N 1
x3  n   x1   m   N  x2   n  m   N , 0  k  N  1
m0
N 1
  x1  m x2   n  m   N , 0  k  N  1
m0
78
since
  m 
N
 m,
for 0  k  N  1
8.6.5 Circular Convolution
N 1
x3  n   x1  m x2   n  m   N 
m0
 x1  n N x2  n
 x2 n N x1 n
N 1
  x2  m x1   n  m   N 
m0
DFT
x3 n  X 3 k   X 1 k X 2 k , 0  k  N  1
79
8.6.5 Circular Convolution
DFT
x1 n N x2 n  X 1 k X 2 k 
if
x3 n  x1nx2 n
1 N 1
X 3 k    X 1 l X 2 k  l N 
N l 0
DFT
x1 nx2 n 
80
1
X 1 k  N X 2 k 
N
Ex. 8.10 Circular
Convolution with a
Delayed Impulse
Sequence
x1 n   n  n0 
0, 0  n  n0

  1, n  n0
 0, n  n  N  1
0

N 1
x3  n   x1  m x2   n  m   N 
m0
x2 [n] N  [n  1]
 x2 [((n  1)) N ], n0  n  N  1
81
Ex. 8.10 Circular Convolution with a Delayed
Impulse Sequence
0, 0  n  n0

x1 n   n  n0    1, n  n0
 0, n  n  N  1
0

X1k   WNkn0
X 3 k   X 1 k X 2 k 
 WNkn0 X 2 k 
x[n] N  [n  1]
 x[((n  1)) N ], n0  n  N  1
82
Example 8.11 Circular Convolution
of Two Rectangular Pulses
 1, 0  n  L  1
x1n  x2 n  
0, otherwise
N L6
N 1
k 0
 N,
kn
X 1k   X 2 k   WN  
n 0
0, otherwise
 N 2,
k 0
X 3 k   X 1 k X 2 k   
0, otherwise

N 1
x3  n   x1  m x2   n  m   N 
m0
83
 N , 0  n  L 1

0, otherwise
1 N 1
 kn
  X 3  k WN
N k 0
Ex. 8.11 Circular
Convolution of Two
Rectangular Pulses
N  2 L  12
L 1
X1  k   X 2  k   WNkn
n 0
1  WNLk

1  WNk
1W
X 3 k   X 1 k X 2 k   
 1W
Lk
N
k
N
N 1



2
x3  n   x1  m x2   n  m   N 
m0
84
8.6.6 Summary of Properties of the
Discrete Fourier Transform
85
8.6.6 Summary of Properties of the
Discrete Fourier Transform
86
8.7 Linear Convolution using the
Discrete Fourier Transform
Implement a convolution of two sequences
by the following procedure:
1. Compute the N-point DFT X 1 k  and X 2 k 
of the two sequence x1 n and x2 n
2. Compute X 3 k   X1k X 2 k  for 0  k  N  1
3. Compute x3 n  x1n N x2 n as the inverse
DFT of X 3 k 
87
8.7 Linear Convolution using the
Discrete Fourier Transform
In most applications, we are
interested in implementing a linear
convolution of two sequence.
To obtain a linear convolution, we will
discuss the relationship between linear
convolution and circular convolution.
88
8.7.1 Linear Convolution of Two
Finite-Length Sequences
x1 n x2 n
length L
x3 n 
P

 x mx n  m
m  
1
x2  1  m
L
x2 n  m
2
for x3  n  0, 0  n  L  P  2
x2  L  P 1 m
L  p 1 is maximum length of x3 n
89
L
L
L  P 1
8.7.2 Circular Convolution as Linear
Convolution with Aliasing
circular convolution x1 n N x2 n corresponding
to DFTs: X1 k  X 2 k  and linear convolution
x1  n * x2  n , Whether they are same?
depends on the length of the DFT in relation
to the length of x1  n and x2 n
90
8.7 Linear Convolution using the
Discrete Fourier Transform
Implement a convolution of two sequences
by the following procedure:
1. Compute the N-point DFT X 1 k  and X 2 k 
of the two sequence x1 n and x2 n
2. Compute X 3 k   X1k X 2 k  for 0  k  N  1
3. Compute x3 n  x1n N x2 n as the inverse
DFT of X 3 k 
Review
91
8.7.2 Circular Convolution as Linear
Convolution with Aliasing
DTFT  X  e jw 
For finite sequence x  n 
x  n 


r 

DFS  X k  X e j  2 k
x  n  rN  
 

j  2 k

X e
X k   

 0,
N
,
N

0  k  N 1
otherwise
The inverse DFT of X  k  is one period of x  n :
 x  n , 0  n  N  1
DFT X k
xp n 
x p  n  
 
otherwise
0,
If N≧length of x[n], then xp[n]= x[n]
92
8.7.2 Circular Convolution as Linear
Convolution with Aliasing
Linear convolution:
x3 n 

 x mx n  m
m  
1
2
The Fourier transform of x3 n is
X 3 e jw  X1 e jw X 2 e jw
Define a DFT
 

X3 k   X3 e
j  2 k N 
  X e
1
   
j  2 k N 
 X e
2
j  2 k N 

0  k  N 1
 X1  k  X 2  k 
 
x3 n  rN , 0  n  N  1

The inverse DFT x n  
r  
3p
of X 3 k  is ：

otherwise
 0,
x3 p n  x1n N x2 n
93
8.7.2 Circular Convolution as Linear
Convolution with Aliasing
From X 3 k   X1 k X 2 k  0  k  N 1
And
x3 p n  x1n N x2 n
Linear convolution:
x3 n 

 x mx n  m
m  
1
2
 
  x3 n  rN , 0  n  N  1
x3 p n  r  

otherwise
 0,
The circular convolution of two-finite sequences
is equivalent to linear convolution of the two
sequences, followed by time aliasing as above.
94
8.7.2 Circular Convolution as Linear
Convolution with Aliasing
If x1 n has length L and x2 n has length P,
then x3 n has maximum length L  P  1
if N, the length of the DFTs, satisfies
N  L  P 1
DFT
The circular convolution corresponding to
X1 k  X 2 k  is identical to the linear convolution
jw
jw
X
e
X
e
corresponding to 1   2  
x3 p n  x1n N x2 n
95

x3 n 

 x mx n  m
m  
1
2
x1 n  x2 n
x3  n
Ex. 8.12 Circular Convolution
as Linear Convolution with
Aliasing.
linear convolution
x3 p  n 
x3 n  N 

 x3 n  rN 
r 
N=6
6 points shift right of
the linear convolution
x3 n  N  6 points shift left of
the linear convolution
6 points circular convolution=
linear convolution with aliasing
N=12
96
12 points circular convolution
= linear convolution
Which points of Circular
Convolution equal that of
Linear Convolution when
Aliasing?
Fig.8.19
Consider x1 n of
length L and x2 n
of length P, where N  L
Linear
Convolution
L  P 1
P<L
Fig.8.20
Circular Convolution
x3 p  n 

 x n  rN 
r 
3
0  n  N 1
97
P 1 L  P 1
L  8, P  4
N  L8
N  L  P  1  11
98
8.7.3 Implementing Linear TimeInvariant Systems Using the DFT
Linear time-invariant systems can be
implemented by linear convolution.
Linear convolution can be obtained from
the circular convolution.
So, circular convolution can be used to
implement linear time-invariant systems.
99
Consider an L-point input sequence xn
and a P-point impulse response hn
The linear convolution of these two
sequence yn has finite duration with
length (L+P-1)
For the circular convolution and linear
convolution to be identical, the circular
convolution must have a length of at least
(L+P-1) points.
100
The circular convolution can be achieved by
multiplying the DFTs of xn and hn .
Since the length of the linear convolution is
(L+P-1) points, the DFTs that we compute
must also be of at least that length, i.e.,
both xn and hn must augmented with
sequence values of zero.
101
Block Convolution
If the input signal is of indefinite duration,
the input signal to be processed is
segmented into sections of length L.
Each section can be convolved with the
finite-length impulse response and the
output sections fitted together in an
appropriate way.
The processing of each section can then be
implemented using the DFT.
102
Block Convolution

x  n   xr  n  rL
r 0
 x  n  rL , 0  n  L  1
xr  n  
otherwise
 0,
(1) segmentx(n) into sections of length L;
(2) fill 0 into h(n) and some section
of x(n) , then do L+P-1 points FFT ;
(3) calculate y (n)
y(n)  IFFT {H (k ) X (k )}，n  0,...L  P  2
103

x  n   xr  n  rL
r 0
L=16
(1) segment x(n) into sections of length L;
(2) fill 0 into h(n) and some section
of x(n), then do L+P-1 points FFT
(3) calculate y (n)
y(n)  IFFT {H (k ) X (k )}
n  0,..., L  P  2

y  n  x  n  h  n   yr  n  rL 
r 0
where yr n  xr n  h n
(4)add the points n=0…P-2 in y[n]
to the last P-1 points in the former
section y[n]，the output for this
section
is the points n=0…L-1
104
P-1 points
8.7.2 Circular Convolution as Linear
Convolution with Aliasing
L  8, P  4
N  L8
105
overlap-save method
(1) segment x(n) into sections of length L,
overlap P-1 points;
(2) fill 0 into h(n) and some section
of x(n), then do L points FFT
L=25
(3) calculate y (n)
y(n)  IFFT {H (k ) X (k )}
n  0,..., L  1
(4) the output for this section is
L-P+1 points of y[n]
n=P-1,…L-1
106
P-1 points
Chapter 8 HW
8.3, 8.4, 8.7, 8.10,
8.51, 8.52, 8.53,
107

Zhongguo Liu_Biomedical Engineering_Shandong Univ.

```