presentation-8 discrete fourier...
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MEH329DIGITAL SIGNAL PROCESSING
-8-Discrete Fourier Transform
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Discrete Fourier TransformIntroduction
MEH329 Digital Signal Processing 2
• Frequency analysis should be performed inreal applications.
• X(ejΩ) is not computationally convenientrepresentation (continuous function of freq.)!
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Discrete Fourier TransformDTFT of Finite Length Signal
MEH329 Digital Signal Processing 3
2
1
nj j n
n n
X e x n e
11 211 1 21 ...j nj n j njX e x n e x n e x n e
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Discrete Fourier TransformFrequency-Domain Sampling
MEH329 Digital Signal Processing 4
2
2
21
0
, 0,1,..., 1
j kj N
kN
N j knN
n
X k X e X e
x n e k N
for example:
2 4 6 8 10 12 140, , , , , , , for 8
8 8 8 8 8 8 8N
Discrete Fourier Transform (DFT)
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Discrete Fourier TransformFrequency-Domain Sampling
MEH329 Digital Signal Processing 5
2 4 6 8 10 12 140, , , , , , , for 8
8 8 8 8 8 8 8
3 30, , , , , , ,
4 2 4 4 2 4
N
jX e
......
4
2
3
4
3
4
2
4
0
2
8
j
kX e
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Discrete Fourier Transform
MEH329 Digital Signal Processing 6
• DTFT: 2
0
1
2j j nx n X e e d
2 2k d dk
N N
21
0
1 , 0,1,..., 1
N j knN
k
x n X k e n NN
Inverse Discrete Fourier Transform (IDFT)
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Discrete Fourier Transform
MEH329 Digital Signal Processing 7
• Alternative representation of DFT and IDFT:
1
0
, 0,1,..., 1N
knN
n
X k x n W k N
1
0
1 , 0,1,..., 1
Nkn
Nk
x n X k W n NN
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MEH329 Digital Signal Processing 8
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Discrete Fourier Transform
MEH329 Digital Signal Processing 9
• If we write n→n+N or k→k+N in DFT and IDFT,are the results change?
2 21 12
0 0
(periodic with !)
N Nj k n N j kn j kN N
n n
x n e x n e e
X k N
2 21 1
0 0
1 1
(periodic with !)
N Nj k n N j knN N
k k
X k e X k eN N
x n N
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Discrete Fourier Transform
MEH329 Digital Signal Processing 10
• Example: Find the DFT of [2 3 1]x n
223
0
, 0,1, 2j kn
n
X k x n e k
2
0
0
0 0 1 2 6j
n
X x n e x x x
2 2 423 3 3
0
1 0 1 2 1.732j n j j
n
X x n e x x e x e j
4 4 823 3 3
0
2 0 1 2 1.732j n j j
n
X x n e x x e x e j
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Discrete Fourier TransformPeriodic Sequences
MEH329 Digital Signal Processing 11
• Given a periodic sequence with period N
• The Fourier series representation:
• The Fourier series representation of continuous-time periodic signals require infinite manycomplex exponentials.
[ ] [ ]x n x n rN
2 /1[ ] j N kn
k
x n X k eN
[ ]x n
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Discrete Fourier TransformPeriodic Sequences
MEH329 Digital Signal Processing 12
• Not that for discrete-time periodic signals
• Due to the periodicity of the complexexponential we only need N exponentials fordiscrete time Fourier series
1
2 /
0
1[ ]
Nj N kn
k
x n X k eN
2 / 2 / 2 2 /j N k mN n j N kn j mn j N kne e e e
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Discrete Fourier TransformDiscrete Fourier Series
MEH329 Digital Signal Processing 13
• A periodic sequence in terms of Fourier seriescoefficients
• The Fourier series coefficients can be obtainedvia
1
2 /
0
[ ]N
j N kn
n
X k x n e
1
2 /
0
1[ ]
Nj N kn
k
x n X k eN
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Discrete Fourier TransformDiscrete Fourier Series
MEH329 Digital Signal Processing 14
• Analysis eq.:
• Synthesis eq.:
2 /j NNW e
1
0
1[ ]
Nkn
Nk
x n X k WN
1
0
[ ]N
knN
n
X k x n W
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Discrete Fourier TransformDiscrete Fourier Series
MEH329 Digital Signal Processing 15
• Example: DFS of periodic impulse train
1[ ]
0r
n rNx n n rN
else
1 1
2 / 2 / 2 / 0
0 0
[ ] [ ] 1N N
j N kn j N kn j N k
n n
X k x n e n e e
1
2 /
0
1[ ]
Nj N kn
r k
x n n rN eN
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Discrete Fourier TransformDiscrete Fourier Series
MEH329 Digital Signal Processing 16
• For N=4:
32 /4
0
30
0
32 /4 2 /4 4 /4 6 /4
0
32 /4 2 4 /4 8 /4 12 /4
0
32 /4 3 6 /4 12 /4 18 /4
0
2 /4 4
1[ ]
4
1[0] 1
4
1 1[1] 1 0
4 4
1 1[2] 1 0
4 4
1 1[3] 1 0
4 4
1[4]
4
j kn
k
j
k
j k j j j
k
j k j j j
k
j k j j j
k
j
x n e
x e
x e e e e
x e e e e
x e e e e
x e
3
8 /4 16 /4 24 /4
0
11 1
4k j j j
k
e e e
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MEH329 Digital Signal Processing 17
Discrete Fourier TransformDiscrete Fourier Series
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MEH329 Digital Signal Processing 18
Discrete Fourier TransformDiscrete Fourier Series
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Discrete Fourier TransformDiscrete Fourier Series
MEH329 Digital Signal Processing 19
• Example: DFS of periodic rectangular pulse train
• The DFS coefficients
2 /10 54
2 /10 4 /10
2 /100
sin / 21
sin /101
j kj kn j k
j kn
keX k e e
ke
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Discrete Fourier TransformDiscrete Fourier Series
MEH329 Digital Signal Processing 20
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Discrete Fourier TransformProperties of Discrete Fourier Series
MEH329 Digital Signal Processing 21
• Linearity:
• Shift:
• Duality:
1 1
2 2
1 2 1 2
DFS
DFS
DFS
x n X k
x n X k
ax n bx n aX k bX k
2 /
2 /
DFS
DFS j km N
DFSj nm N
x n X k
x n m e X k
e x n X k m
DFS
DFS
x n X k
X n Nx k
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Discrete Fourier TransformProperties of Discrete Fourier Series
MEH329 Digital Signal Processing 22
• Proof (Duality):
n↔k:
1
2 /
0
[ ]N
j N kn
n
X k x n e
1
2 /
0
1[ ]
Nj N kn
k
x n X k eN
12 /
0
12 /
0
[ ]
[ ]
Nj N kn
k
Nj N kn
n
Nx n X k e
Nx k X n e
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Discrete Fourier TransformProperties of Discrete Fourier Series
MEH329 Digital Signal Processing 23
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Discrete Fourier TransformProperties of Discrete Fourier Series
MEH329 Digital Signal Processing 24
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Discrete Fourier TransformProperties of Discrete Fourier Series
MEH329 Digital Signal Processing 25
• Periodic Convolution:
1 1
2 2
DFS
DFS
x n X k
x n X k
3 1 2X k X k X k
1
3 1 20
1
2 10
N
m
N
m
x n x m x n m
x m x n m
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MEH329 Digital Signal Processing 26
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MEH329 Digital Signal Processing 27
Discrete Fourier TransformDFS – DFT Relation
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Discrete Fourier TransformDFS – DFT Relation
MEH329 Digital Signal Processing 28
1Nn0 of outside 0nx
r
x n x n rN
• The DFS coefficients of the periodic sequence aresamples of the DTFT of x[n].
• Since x[n] is of length N there is no overlap betweenterms of x[n-rN] and we can write the periodicsequence as
mod NN
x n x n x n
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Discrete Fourier TransformDFS – DFT Relation
MEH329 Digital Signal Processing 29
• We choose one period of as the Fouriertransform of x[n]
kX~
mod NN
X k X k X k
0 1
0
X k k NX k
else
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Discrete Fourier TransformDFS – DFT Relation
MEH329 Digital Signal Processing 30
1
2 /
0
[ ]N
j N kn
n
X k x n e
1
2 /
0
1[ ]
Nj N kn
k
x n X k eN
,0 1[ ]
0 ,otherwise
x n n Nx n
,0 1[ ]
0 ,otherwise
X k k NX k
1
2 /
0
[ ]N
j N kn
n
X k x n e
1
2 /
0
1[ ]
Nj N kn
k
x n X k eN
DFT Pair
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Discrete Fourier TransformDiscrete Fourier Transform
MEH329 Digital Signal Processing 31
• To evaluate the relation between size of asignal (L) and the number of frequency sample(N).
• N must be equal or greater than L toreconstruct the signal without a loss.
• If L is bigger than N, aliasing occurs in the timedomain!
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MEH329 Digital Signal Processing 32
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MEH329 Digital Signal Processing 33
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Discrete Fourier TransformProperties of DFT
MEH329 Digital Signal Processing 34
• Linearity:
1 1
2 2
1 2 1 2
DFT
DFT
DFT
x n X k
x n X k
ax n bx n aX k bX k
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MEH329 Digital Signal Processing 35
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Discrete Fourier TransformProperties of DFT
MEH329 Digital Signal Processing 36
• Circular Shift:
2 / 0 n N-1
DFT
j k N mDFT
N
x n X k
x n m X k e
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The n Modulo N Operation
MEH329 Digital Signal Processing 37
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Circular Shift of a Sequence
MEH329 Digital Signal Processing 38
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Circular Shift of a Sequence
MEH329 Digital Signal Processing 39
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Circular Shift of a Sequence
MEH329 Digital Signal Processing 40
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Circular Shift of a Sequence
MEH329 Digital Signal Processing 41
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Circular folding (or reversal)
MEH329 Digital Signal Processing 42
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Circular folding (or reversal)
MEH329 Digital Signal Processing 43
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Circular Convolution:
MEH329 Digital Signal Processing 44
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Discrete Fourier TransformProperties of DFT
MEH329 Digital Signal Processing 45
• Circular Convolution:
1
3 1 20
N
Nm
x n x m x n m
1
3 2 10
N
Nm
x n x m x n m
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Discrete Fourier TransformProperties of DFT
MEH329 Digital Signal Processing 46
• Example: Circular convolution
• DFT of each sequence
• Multiplication of DFTs
• the inverse DFT
1 2
1 0 1
0
n Lx n x n
else
21
1 20
0
0
N j knN
n
N kX k X k e
else
2
3 1 2
0
0
N kX k X k X k
else
3
0 1
0
N n Nx n
else
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Circular Convolution-Graphical Interpretation
MEH329 Digital Signal Processing 47
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Linear Convolution (Review)
MEH329 Digital Signal Processing 48
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Linear Convolution Using DFT
MEH329 Digital Signal Processing 49
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Linear Convolution Using DFT
MEH329 Digital Signal Processing 50
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Linear Convolution Using DFT
MEH329 Digital Signal Processing 51
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Discrete Fourier TransformProperties of DFT
MEH329 Digital Signal Processing 52
• Example: if N=2L=12• DFT of each sequence
• Multiplication of DFTs
2
1 2 2
1
1
Lkj
N
kjN
eX k X k
e
22
3 2
1
1
Lkj
N
kjN
eX k
e
Result of the linearconvolution!
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Duality Property
MEH329 Digital Signal Processing 53
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Discrete Fourier TransformWindowing
54
• Periodicity causes unwanted spectral effects infrequency domain
• This issue is called as spectral leakage.
0 2 4 6 8 10 12 14 16-1
0
1x[n]
0 2 4 6 8 10 12 14 160
5
10DFT Coefficients
0 5 10 15 20 25 30 35 40 45-1
0
1x~[n]
0 5 10 15 20-1
0
1x[n]
0 5 10 15 200
5
10DFT Coefficients
0 10 20 30 40 50 60 70-1
0
1x~[n]
Spectral leakage
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Windowing Property
MEH329 Digital Signal Processing 55
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Discrete Fourier TransformWindowing
56
• Windowing is utilized to overcome this effect.• Well known window functions:
• Triangular• Trapezoid• Hamming• Hanning• Blackman• Parzen• Welch• Nuttall• Kaiser
0 5 10 15 20 250
0.2
0.4
0.6
0.8
1
Window Function (Hamming)
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Discrete Fourier TransformWindowing
57
• The effect of windowing:
0 5 10 15 20-1
0
1x[n]
0 5 10 15 200
5
10DFT Coefficients
0 10 20 30 40 50 60 70-1
0
1x~[n]
0 5 10 15 20-1
0
1Windowed Signal
0 5 10 15 200
5
10DFT Coefficients
0 10 20 30 40 50 60 70-1
0
1Periodic form of windowed function
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Discrete Fourier TransformLinear Transform Perspective
MEH329 Digital Signal Processing 58
• Computation of each point of the DFT can beaccomplished by N complex multiplications and(N-1) complex additions.
• The N point DFT coefficients can be computed in atotal of N^2 complex multiplications and (N-1)Ncomplex additions.
1
0
[ ] , 0,1,..., 1N
knN
n
X k x n W k N
1
0
1[ ] , n 0,1,..., 1
Nkn
Nk
x n X k W NN
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Discrete Fourier TransformLinear Transform Perspective
MEH329 Digital Signal Processing 59
N=4 point DFT calculation:
10 0 0 0 0
0
10 1 2 3
0
10 2 4 6
0
10 3 6 9
0
0 [ ] 0 1 2 3
1 [ ] 0 1 2 3
2 [ ] 0 1 2 3
3 [ ] 0 1 2 3
Nn
N N N N Nn
NnN N N N N
n
NnN N N N N
n
NnN N N N N
n
X x n W x W x W x W x W
X x n W x W x W x W x W
X x n W x W x W x W x W
X x n W x W x W x W x W
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Discrete Fourier TransformLinear Transform Perspective
MEH329 Digital Signal Processing 60
Alternative representation:
0
1
1
N
x
x
x N
x
0
1
1
N
X
X
X N
X
1 2 1
2 12 4
1 2 1 1 1
1 1 1 1
1
1
1
NN N N
NN N N N
N N N NN N N
W W W
W W W
W W W
W
1 1N N N N NN
x W X W X
N N NX W x
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Discrete Fourier TransformLinear Transform Perspective
MEH329 Digital Signal Processing 61
Example:0
1
2
3
N
x1 2 3 1 2 3 1 2 1
4 4 4 4 4 4 4 4 42 4 6 2 0 2 2 0 2
4 4 4 4 4 4 4 4 43 6 9 3 2 1 1 2 1
4 4 4 4 4 4 4 4 4
1 1 1 1 1 1 1 1 1 1 1 1
1 1 1
1 1 1
1 1 1
N
W W W W W W W W W
W W W W W W W W W
W W W W W W W W W
W
1 1 1 1 0 6
1 1 1 2 2
1 1 1 1 2 2
1 1 3 2 2
N N N
j j j
j j j
X W x