Transcript Document

The Discrete Fourier Series
Quote of the Day
Whoever despises the high wisdom of mathematics
nourishes himself on delusion.
Leonardo da Vinci
Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck, ©1999-2000 Prentice Hall
Inc.
Discrete Fourier Series
x[n] with period N so that
• Given a periodic sequence ~
~
x[n]  ~
x[n  rN]
• The Fourier series representation can be written as
1 ~
~
x[n]   Xk e j2  / Nkn
N k
• The 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kne 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 ~
~
x[n]   Xk e j2  / Nkn
N k 0
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Discrete Fourier Series Pair
• A periodic sequence in terms of Fourier series coefficients
1 N1 ~
~
x[n]   Xk e j2  / Nkn
N k 0
• The Fourier series coefficients can be obtained via
~
Xk  
N 1
 ~x[n]e
 j2  / N kn
n0
• For convenience we sometimes use
• Analysis equation
WN  e j2 / N
~
Xk  
N 1
~
kn
x
[
n
]
W

N
n0
• Synthesis equation
Copyright (C) 2005 Güner Arslan
1 N1 ~
~
x[n]   Xk WNkn
N k 0
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Example 1
• DFS of a periodic impulse train
~
x[n] 
1 n  rN
n  rN 

r  
0 else

• Since the period of the signal is N
~
Xk  
N 1
N 1
n0
n0
 ~x[n]e j2 / Nkn 
 [n]e
 j2  / N kn
 e  j2  / Nk 0  1
• We can represent the signal with the DFS coefficients as
~
x[n] 
Copyright (C) 2005 Güner Arslan
1 N1 j2  / Nkn
n  rN   e

N k 0
r  

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Example 2
• DFS of an periodic rectangular pulse train
• The DFS coefficients
4
~
1  e j2 / 10k5
 j2  / 10kn
 j4 k / 10 sink / 2
Xk    e

e
 j2  / 10k
sink / 10
1e
n0
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Properties of DFS
• Linearity
~
x1 n
~
x n
~
 

X1 k 
~
DFS


X2 k 
~
~
DFS
 
 aX1 k   bX2 k 
DFS
2
a~
x1 n  b~
x2 n
• Shift of a Sequence
~
x n
~
x n  m
e j2 nm / N~
x n
~
Xk 
~
DFS

 e j2 km / NXk 
~
DFS


Xk  m
DFS


• Duality
~
~
x n DFS


Xk 
~
Xn DFS

 N~
x  k 
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Symmetry Properties
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Symmetry Properties Cont’d
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Periodic Convolution
• Take two periodic sequences
~
~
x1 n DFS

 X1 k 
~
~
x n DFS

 X k 
2
2
• Let’s form the product
~
~ ~
X3 k   X1 k X2 k 
• The periodic sequence with given DFS can be written as
N 1
~
~
x n 
x m~
x n  m
3

1
2
m0
• Periodic convolution is commutative
~
x3 n 
N 1
~
~


x
m
 2 x1 n  m
m0
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Periodic Convolution Cont’d
~
x3 n 
N 1
 ~x m~x n  m
1
2
m0
• Substitute periodic convolution into the DFS equation
N 1 N 1
~


X3 k      ~
x1[m]~
x2[n  m]WNkn
n0  m0

• Interchange summations
~
X3 k  
 N1 ~
~
kn 
x
[
m
]

x
[
n

m
]
W

1
N 
 2

m0
 n0

• The inner sum is the DFS of shifted sequence
N 1
~
kn
km~
x
[
n

m
]
W

W
 2
N
N X2 k 
N 1
• Substituting
~
X3 k  
n0
 N1 ~
~
kn 
x
[
m
]

x
[
n

m
]
W

1
N 
 2

m0
 n0

N 1
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~ ~
~
km~


x
[
m
]
W
X
k

X1 k X2 k 
 1
N
2
N 1
m0
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Graphical Periodic Convolution
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The Fourier Transform of Periodic Signals
• Periodic sequences are not absolute or square summable
– Hence they don’t have a Fourier Transform
• We can represent them 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
Xe 
   2N ~Xk    2Nk 



k  
– This is periodic with 2 since DFS is periodic
• The inverse transform can be written as
1 2    ~ j jn
1 2     2 ~ 
2k  jn


X
e
e
d


X
k




e d



0


0


2
2
N 

k   N
 
2 k
j
n
1  ~ 2   
2k  jn
1 N1 ~
N
 Xk 0      N e d  N k0 Xk e
N k  
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Example
• Consider the periodic impulse train
~
p[n] 

 n  rN
r  
• The DFS was calculated previously to be
~
P k   1 for all k
• Therefore the Fourier transform is
~ j
Pe 
   2N    2Nk 



k  
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Relation between Finite-length and Periodic Signals
• Consider finite length signal x[n] spanning from 0 to N-1
• Convolve with periodic impulse train
~
x[n]  x[n]  ~
p[n]  x[n] 


r  
r  
 n  rN   xn  rN
• The Fourier transform of the periodic sequence is
~ j
j ~ j
X e  X e P e  X e j
 
  
   2N    2Nk 



k  
~ j
Xe 
 
2 k
2  j N  
2k 
X e
  



N 
k   N

 

• This implies that
 j 2Nk 
~
  X e j
Xk   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
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Example
• Consider the following
sequence
1 0  n  4
x[n]  
else
0
• The Fourier transform
 
X e j  e  j2
sin5 / 2
sin / 2
• The DFS coefficients
~
sink / 2
Xk   e  j4 k / 1 0
sink / 10 
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