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
1
Filter Design Techniques
8.0 Introduction
2
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.
3
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.
4
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
j2  / N k  mN n
e
j2  / N kn
e
j2 mn 
e
j2  / N kn
5
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
x[n] 
N
No need
N 1
 X k  e
j  2 / N  kn
k 0
j  2 / N kn
1
x[n]   X  k  e
N k
6
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
j
2
rn
N
N 1
1

n 0 N
N 1
N 1
 X (k )e
j
2
( k r ) n
N
j
2
( k r ) n
N
k 0
N 1
1
  X ( k ) e
k 0
n 0 N
7
N 1
Discrete Fourier1Series
Pairj  2 / N kn
x[n]   X  k  e
N
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
k 0
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
2
j
rn
N
Problem 8.51, HW
 X (r )
n 0
N 1
X (k )   x(n)e
n 0
j
2
kn
N
8
8.1 Representation of Periodic Sequence: the Discrete
Fourier Series
~
• a periodic sequence
n period N,
xwith
~
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
1
x  n 
N
N 1
 X k  e
j  2 N  kn
k 0
9
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 ,
N 1
~
~
X 1  X 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
10
Discrete
Fourier
Series
(DFS)
N 1
 j  2 N  kn
•Let
X k  x n e
    
WN  e
n 0
Analysis equation:
 j  2 N 
N 1
~
kn
~
X k    x nWN
n 0
Synthesis equation:
DFS
~
~
x n  X k 
1
~
x n 
N
N 1
~
 kn
 X k WN
k 0
F
discrete  periodic
F
periodic
 discrete
11
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
n 0
12
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
x  n 
-1
0
1
   n  rN 
r 
N 1
 X  k W
k 0
……
N-1 N N+1 N+2
……
 1, n  rN , r is any integer

0, therwise

1

N
2
k
 kn
N
1

N
N 1
e
k 0
j  2 N  kn
13
N points
1
-N
-N+1……
-2
-1
0
-N
……
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
14
Example 8.2 Duality in the Discrete Fourier Series
• The discrete Fourier series coefficients is the periodic impulse train
N points
~
Y k  
 N k  rN 
r  
1
~
x n 
N
1
~
y n 
N
N 1
Y k 
N

-N
-1 0 1
…
2…
…
…
N
N 1
N 1
~   ~  kn
X k  x n WN
~
 kn


X
k
W

N
k 0
1
~
 kn
Y k WN 

N
k 0
…
… -2
n 0
N 1
 kn
0


N

k
W

W

N
N 1
k 0
15
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
-1
0
1
2
……
N-1 N N+1 N+2
n
……
16
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
-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
……
17
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
18
magnitude
phase
X k   e
 4 k 10
sin  k 2 
sin  k 10 
19
magnitude
phase
X k   e
 4 k 10
sin  k 2 
sin  k 10 
20
8.2 Properties of the Discrete Fourier Series
• Linearity: two periodic sequence, both with period N
DFS
~
~
x1 n  X 1 k 
DFS
~
~
x2 n  X 2 k 
DFS
~
~
~
~
ax1 n  bx2 n  aX 1 k   bX 2 k 
21
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
22
8.2 Properties of the Discrete Fourier Series
• Duality
DFS
~
~
x n  X k 
DFS
~
X n  N~
x  k 
~
x n
1
~
X k 
1
0
1
2
n
N-1
0
X  n
1
0
……
1
2
……
N-1
2
1
……
0
k
Nx  k 
N
n
N-1
1
2
……
k
N-1
23
8.2.4 Symmetry
Problem 8.53, HW
24
~
 are two periodic sequences,
x2 nConvolution
x1n Periodic
8.2.5
and ~
each with period N and with discrete Fourier
~
~
series X 1 k  and X 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
n 0
m 0

km 
   x1  mWN  X 2  k   X 1  k  X 2  k 
 m0

25

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
1
~
~
~
~
DFS
x3 n  x1 nx2 n  X 3 k  
N
N 1
~ ~
 X 1l X 2 k  l 
l 0
26
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
x3  2   x1  m  x2  2  m 
27
m 0
8.2.5 Periodic Convolution
~
x3 n  ~
x1 n~
x2 n
1
~
X 3 k  
N
N 1
~ ~
 X 1l X 2 k  l 
l 0
28
8.1 Representation of Periodic Sequence: the Discrete
Fourier Series
~
• a periodic sequence
n period N,
xwith
~
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
1
x  n 
N
N 1
 X k  e
k 0
j  2 N  kn
Review
29
Discrete
Fourier
Series
(DFS)
N 1
 j  2 N  kn
•Let
X k  x n e
    
WN  e
n 0
 j  2 N 
N 1
~
kn
~
X k    x nWN
Analysis equation:
n 0
Synthesis equation:
1
~
x n 
N
N 1
~
 kn


X
k
W

N
k 0
DFS
~
~
x n  X k 
30
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 
WN  e
 j  2 N 
DFS
WN nl x  n  X  k  l 
31
8.2 Properties of the Discrete Fourier Series
• Duality
DFS
~
~
x n  X k 
DFS
~
X n  N~
x  k 
~
x n
1
~
X k 
1
0
1
2
n
N-1
0
X  n
1
0
……
1
2
……
N-1
2
1
……
0
k
Nx  k 
N
n
N-1
1
2
……
k
N-1
32
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
~
X 3 k  
N
N 1
~ ~
 X 1l X 2 k  l 
l 0
33
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
x3  2   x1  m  x2  2  m 
34
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
jwk n
    2 a  w  w  2 r 

F

Xe
jw
r   k
k
k
35
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
1
x  n 
N
N 1
 X k  e
j  2 N  kn
k 0
2
2 k 

X e   
X  k    
N 

k - N
j

36
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
1
 2 k  jn
  X  k     e d   X  k  e

0-
N k - 
N 
N k 0

 x  n

2 -
N -1
j
37
Ex. 8.5 Fourier Transform of a periodic impulse
train
• Consider the periodic impulse train
N points
p[n] 
1

   n  rN 
r 
-N
…
… -2
-1 0 1
…
2…
P  k   1 for all k
-N
P k 
N points
…
…
-2 -1 0 1
…
2 … N-1
…
…
N
The DFS was calculated previously to be
1
p  n
N
…
… n
Therefore the Fourier transform is
2 
2 k 
P e   
  

N 

k  N
j

38
Relation between Finite-length and Periodic
Signals
• Consider finite length signal x[n] pspanning
from 0 to N-1
 n
1
…
- -1 0 1 2
…periodic
2
Convolve with
-N
N
…
…
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

 

39
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
40
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
 
N 1
N 1
n 0
n 0
X e jw   xne  jwn   ~
x ne  jwn
X  k   X  e jw 
w2 k N
41
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

   x  n e
 jwn
n 
e
 j 2
sin  5 / 2 
sin  / 2 
42
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 / 1 0
sink / 10
The Fourier transform
N 1
X  e jw    x  n  e jwn
  e
Xe
j
n 0
 j2 
sin5 / 2
sin / 2
43
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 / 1 0
sink / 10
The Fourier transform
N 1
X  e jw    x  n  e jwn
  e
Xe
j
n 0
 j2 
sin5 / 2
sin / 2
44
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
X k   X e
jw
 w2
j  2 N k 

 X e

N k



X  k   X  z  z  e j  2  N k  X e
j  2 N  k

~
 X k  is Fourier series coefficients of periodic sequence
x  n
45
Sampling the Fourier Transform
j  2 / N k 

X k   X  e



X  e j  
j  2 / N kn
1 N 1
x[n]   X  k  e
N k 0
1
x[n] 
N


m 
x  m  e j m
 j  2 / N km  j  2 / N kn
 

  x  m e
e
k  0  m 

N 1
 1 N 1 j  2 / N k  nm  
  x  m   e
  x  m p  n  m

m 
 N k 0
 m

1
p  n  m 
N
N 1
e
k 0
j  2 / N k  n m 


   n  m  rN 
r 
46
j  2 /Fourier
N k 
Sampling
the
Transform

1
X k   X  e


N 1
1
x[n]   X  k  e
N k 0


j 2 / N kn


…
… -2 -1
N  12
-N
N points
01
…
2…
p  n
…
…
N

 x  m p  n  m
m 

 x  n      n  rN  
r 
1
p  n  m 
N
N 1
e
 x  n
x  n 
k 0

 0

 x  n  rN 
r 
j  2 / N k  n m 


   n  m  rN 
r 
0  n  N 1
else
47
Sampling the Fourier Transform
j  2 / N k 

X k   X  e



2
N
N 7
j  2 / N kn
1 N 1
x[n]   X  k  e
N k 0
x[n] 

 x  n  rN 
r 
48
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
49
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
50
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 
51
Discrete Fourier Transform
For ~x n , the DFS is
~
X k  with period N
The Discrete Fourier Transform of xn is
~
 X k , 0  k  N  1
X k   
 0, otherwise
X  k   X  k mod N    X   k   N 
52
Discrete Fourier Transform
N 1
~
X k    ~
x nWNkn
n 0
1
~
x n 
N
N 1
~
 kn
 X k WN
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,
53
Discrete Fourier Transform pairs
• Analysis equation
N 1
X k    xnW
Synthesis equation
n 0
1
xn 
N
xn
kn
N
N 1
 kn


X
k
W

N
k 0
DFT

X k 
54
Discrete Fourier Transform
Time
Fourier transform
(FT)
Fourier series (FS)
Discrete-time
Fourier transform
(DTFT)
Discrete Fourier
series (DFS)
Discrete Fourier
transform (DFT)
Frequency
continuous
continuous
continuous
periodic
discrete
Continuous impulse
train
discrete
periodic
continuous impulse
train, periodic
discrete
discrete
continuous
periodic
55
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
2
k
5
n 0
1  e j 2 k
5 k  0, 5, 10,...


 j  2 k /5
else
0
1 e
56
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
57
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 
1
~
x n 
N
N 1
~
 kn


X
k
W

N
k 0
 x  n , 0  n  N  1 DFT
 X k , 0  k  N 1
x  n  
X k   
else
else
 0,
 0,
58
DiscreteN 1Fourier Transform
~
kn
~
X k    x nWN
n 0
1
~
x n 
N
N 1
~
 kn


X
k
W

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,
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 7


 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,
60
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 3 e jw  X 1 e jw
  X e
1
j  2 k N 
L  P 1
X 2 e jw
 X e
2
j  2 k N 
  X k  X k 
1
2
N ?
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,
N 1
61
8.6 Properties of the Discrete Fourier Transform
8.6.1 Linearity
DFT
ax1 n  bx2 n  aX 1 k   bX 2 k 
If x1n has length N1 and x2 nhas length N 2 ,
N3  max N1 , N 2 
X1  k  
X 2 k  
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
62
8.6.2 Circular Shift of a Sequence DFS
~
~
x n  X k 
DFS
x  n  m  e
 j  2 k N m
xn, 0  n  N  1
x   n  m   N  , 0  n  N  1
DFT

e
DFT

 j  2 k N m
X k 
X k 
X k 
63
Ex. 8.8 Circular Shift of a Sequence
circular
shift
Figure 8.12
64
 
8.6.2 Circular
jw Shift of a Sequence  jwm
xn  X e
x  n  m  e
X  e jw 
DFS
~
~
x n  X k 
DFS
x  n  m  e
DFT
xn, 0  n  N  1
x1  n , 0  n  N  1
DFT


 j  2 k N m
X k 
X k 
X1  k   e
 j  2 k N m
X k 
x  n  x   n   N   X  k   X   k   N 
DFS
x1  n  x1   n   N   X1  k   X1   k   N 
DFS
65
DFS
8.6.2 Circular
Shift
of
a
Sequence



x  n  x   n   N   X  k   X   k   N 
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  
e
x1  n 


N

N m
 j  2 k N m

X k 
X   k   N 
X   k   N   e
x   n  m   N   x  n  m
DFS
 j  2 k N  m
e
X k 
 j  2 k N m
X k 
66
8.6.2 Circular Shift
 j of
2 ak Sequence
N m
X1  k   e
X k 
x1  n  x   n  m   N   x  n  m
DFS
e
 j  2 k N m
X k 

 x1  n  x   n  m   N  , 0  n  N  1
x1  n  
otherwise

 0,
xn, 0  n  N  1
DFT

xn  mN , 0  n  N  1
DFT

X k 
e  j 2 k
N m
X k 
67
DFS Duality
8.6.3
~
~
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
DFT

DFT

X k 
Nx   k   N  , 0  k  N  1
68
Ex.8.9 The Duality Relationship for the DFT
69
DFS
8.6.4x Symmetry
Properties
n  x   n   N   X  k   X   k   N 
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
xop  n  xo  n   x   n   N   x*   n   N  ,0  n  N 1
2




70

xn    x  n  ,
2
1
n  xn    x  n  ,
2
8.6.4 Symmetry
Properties
1
*
xep n 
xop
N
N
0  n  N 1
N
0  n  N 1
*
N
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
for
71
1
8.6.4
Symmetry
Properties
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
72
8.6.4 Symmetry
Properties
x  n  x  n  x  n
e
o
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 
DFT
j Imxn  X op k 
DFT
xop n  j ImX k 
73
8.6.4 Symmetry Properties
74
N 1
8.6.5
Circular Convolution
DFS
 x  m x  n  m
m 0
1
2

X1  k  X 2  k 
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
since
 m
N
 m,
for 0  k  N  1
75
8.6.5 CircularN Convolution
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
76
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
X 3 k  
N
N 1
 X l X k  l  
l 0
DFT
x1 nx2 n 
1
2
N
1
X 1 k  N X 2 k 
N
77
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
78
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

X 1 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
79
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
X 1 k   X 2 k   W
n 0
kn
N
k 0
 N,

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
 N , 0  n  L 1

0, otherwise
1

N
N 1
 X  k W
k 0
3
 kn
N
80
Ex. 8.11 Circular
Convolution of Two
Rectangular Pulses
N  2L  12
L 1
X 1  k   X 2  k   WNkn
n 0
1  WNLk

k
1  WN
1W
X 3 k   X 1 k X 2 k   
 1W
Lk
N
k
N



2
N 1
x3  n    x1  m  x2   n  m   N 
m 0
81
8.6.6 Summary of Properties of the Discrete
Fourier Transform
82
8.6.6 Summary of Properties of the Discrete
Fourier Transform
83
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   X 1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 
84
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.
85
8.7.1 Linear Convolution of Two FiniteLength Sequences
x1n x2 n
length
x3 n 
L
P

 x mx n  m
m  
1
L
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
L
86
L  P 1