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] Xk e j2 / Nkn
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 j2 / Nk mNn e j2 / Nkne j2 mn e j2 / Nkn
• Due to the periodicity of the complex exponential we only
need N exponentials for discrete time Fourier series
1 N1 ~
~
x[n] Xk e j2 / Nkn
N k 0
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Discrete Fourier Series Pair
• A periodic sequence in terms of Fourier series coefficients
1 N1 ~
~
x[n] Xk e j2 / Nkn
N k 0
• The Fourier series coefficients can be obtained via
~
Xk
N 1
~x[n]e
j2 / N kn
n0
• For convenience we sometimes use
• Analysis equation
WN e j2 / N
~
Xk
N 1
~
kn
x
[
n
]
W
N
n0
• Synthesis equation
Copyright (C) 2005 Güner Arslan
1 N1 ~
~
x[n] Xk WNkn
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
~
Xk
N 1
N 1
n0
n0
~x[n]e j2 / Nkn
[n]e
j2 / N kn
e j2 / Nk 0 1
• We can represent the signal with the DFS coefficients as
~
x[n]
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1 N1 j2 / Nkn
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 j2 / 10k5
j2 / 10kn
j4 k / 10 sink / 2
Xk e
e
j2 / 10k
sink / 10
1e
n0
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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
~
Xk
~
DFS
e j2 km / NXk
~
DFS
Xk m
DFS
• Duality
~
~
x n DFS
Xk
~
Xn 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
m0
• Periodic convolution is commutative
~
x3 n
N 1
~
~
x
m
2 x1 n m
m0
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Periodic Convolution Cont’d
~
x3 n
N 1
~x m~x n m
1
2
m0
• Substitute periodic convolution into the DFS equation
N 1 N 1
~
X3 k ~
x1[m]~
x2[n m]WNkn
n0 m0
• Interchange summations
~
X3 k
N1 ~
~
kn
x
[
m
]
x
[
n
m
]
W
1
N
2
m0
n0
• 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
n0
N1 ~
~
kn
x
[
m
]
x
[
n
m
]
W
1
N
2
m0
n0
N 1
Copyright (C) 2005 Güner Arslan
~ ~
~
km~
x
[
m
]
W
X
k
X1 k X2 k
1
N
2
N 1
m0
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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 ~Xk 2Nk
k
– This is periodic with 2 since DFS is periodic
• The inverse transform can be written as
1 2 ~ j jn
1 2 2 ~
2k jn
X
e
e
d
X
k
e d
0
0
2
2
N
k N
2 k
j
n
1 ~ 2
2k jn
1 N1 ~
N
Xk 0 N e d N k0 Xk 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 2Nk
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 xn rN
• The Fourier transform of the periodic sequence is
~ j
j ~ j
X e X e P e X e j
2N 2Nk
k
~ j
Xe
2 k
2 j N
2k
X e
N
k N
• This implies that
j 2Nk
~
X e j
Xk 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
sin5 / 2
sin / 2
• The DFS coefficients
~
sink / 2
Xk e j4 k / 1 0
sink / 10
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