Chapter 8 The Discrete Fourier Transformcmliu/Courses/dsp/chap8.pdf · Chapter 8 The Discrete...
Transcript of Chapter 8 The Discrete Fourier Transformcmliu/Courses/dsp/chap8.pdf · Chapter 8 The Discrete...
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Chapter 8 The Discrete Fourier Transform
IntroductionDefinition of the Discrete Fourier TransformProperties of the DFTLinear Convolution Using the DFTDCT Concluding Remarks
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1. Introduction
Fourier Transform Pair
Required TransformFinite Signal Length Discrete Frequency Samples
DFT ApplicationsSignal AnalysisComputing benefits from the convolution
X e x nj jn
n
( ) ( ) e x pω ω= −
= − ∞
∞
∑
x n X e e dj jn( ) ( )=−∫
1
2πωω ω
π
π
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2. Discrete Fourier Transform
ConsiderationsThe Discrete-time Fourier transform is defined for sequences with finite or infinite length.The DFT is defined only for sequences with finite length.
Formulation-- Forward TransformConsider the sequence of length Nx[n] for n=0,1, 2, ..., N-1
The discrete-time Fourier transform is
It is periodic with period 2π.
X x n e x n ej n
k
j n
k
N
( ) [ ] [ ]ω ω ω= =−
= − ∞
∞−
=
−
∑ ∑0
1
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2. Discrete Fourier Transform (c.1)
Formulation-- Forward Transform (c.1)The usually considered frequency interval is (-π , π ]. There are infinitely many points in the interval. If x[n] has N points, we compute N equally spaced w in (-π , π ]. These N points are
So we define
k and n are in the interval from 0 to N-1.
Formulation-- Inverse TransformCompute x[n] from N frequency components
So
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~( ) ( ) [ ] /X k X k
Nx n e jnk N
n
N
= = −
=
−
∑2 2
0
1π π
x n X e d X eN
jkk
jk n N
k
N
[ ] ( ) ( ) ( )/= ≈ ⋅= =
−
∫ ∑1
2
1
2
2
0
2
2
0
1
πω ω
πω
πω
ω
ππ
x nN
X k e jk n N
m
N
[ ] ( ) /≈=
−
∑1 2
0
1π
ω π= = −k
Nk N
20 1 2 1, , , , . . . ,
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2. Discrete Fourier Transform (c.2)
The Discrete Fourier Transform
Notes
Periodic property of X[k]– WknN =1; X[k] = X[k+-N]= X[k+-2N] = ...
DFT compute the samples of the discrete-time Fourier transform of x[n].
The Inverse yields a periodic x[n] with period N
The relation with the Discrete Fourier series– X[k] = Nck
1,,2,1,0,;
)(1)(1][
][][)(
/2
1
0
1
0
/2
1
0
1
0
/2
−==
==
==
−
−
=
−−
=
−
=
−
=
−
∑∑
∑∑
NnkeWwhere
WkXN
ekXN
nx
WnxenxkX
Nj
N
k
nkN
n
Njkn
N
n
knN
n
Nknj
Kπ
π
π
X kX k fo r k N
period ic ex ten sion o f X k[ ]
[ ]
[ ]=
≤ ≤ −⎧⎨⎩
0 1
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2. Properties of the DFT
Let x'[n] is the periodic extension of the x[n]
Linearity PropertiesD[a1x1[n] + a2x2[n]] = a1 D[x1[n]] + a2 D[x2[n]]
Symmetric PropertiesIf x'[n] is real then X[k] is conjugate symmetric, or X[-k] = X*[k].
If x'[n] is real and even, then X[k] is real and even.
If x'[n] is real and odd, then X[k] is imaginary and odd.
Duality PropertiesLet X[k] and x'[n] be the DFT pair. That isX[k] = D[x'[n]] and x'[n] = D-1 [X[k]].
then
x'[n] = {D[X [k]/N]}*
The DFT can be used to compute the inverse DFT.
Time-Shifting & Frequency-Shifting PropertiesD[x'[n-n0]] = e-jkn0 X[k] & D[x'[n]ejnk0 ] = X[k-k0]
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2. Properties of the DFT-- Periodic Convolution
TopicCompute convolution of two finite sequence using the DFT
Consideration
Convolution– Convolution of two sequences h[n] & u[n] for n=0, 1, 2, ..., N-1.
– The nonzero elements are at most in the interval [0, 2N-2].
Periodic or Cyclic Convolution– If u1[n] and u2[n] are two periodic sequence with period N. The cyclic convolution of the two sequence
is defined as
where n=0, 1, 2, ..., N-1.
Cyclic Convolution & DFT
The DFT of y[n] is equal to the multiplication of the DFT of u1[n] and u2[n].Y[k] = U1 [k]U2 [k]
y n h n i u i h i u n ii
n
i
n
[ ] [ ] [ ] [ ] [ ]= − = −= =
∑ ∑0 0
y k u k i u i u i u k ii
N
i
N
[ ] [ ] [ ] [ ] [ ]= − = −=
−
=
−
∑ ∑1 20
1
1 20
1
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2. Properties of the DFT-- Periodic Convolution (c.1)
Periodic Convolution
u[0] h[0]
h[1]
h[2]
u[2]
u[1]
u[1] h[0]
h[1]
h[2]
u[2]
u[0]
u[2] h[0]
h[1]
h[2]
u[0]
u[1]y[0]
y[1]
y[2]
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Periodic Convolution9
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2. Properties of the DFT-- Periodic Convolution (c.2)
Definition
1,,2,1,0,;
)(1)(1][
][][)(
/2
1
0
1
0
/2
1
0
1
0
/2
−==
==
==
−
−
=
−−
=
−
=
−
=
−
∑∑
∑∑
NnkeWwhere
WkXN
ekXN
nx
WnxenxkX
Nj
N
k
nkN
n
Njkn
N
n
knN
n
Nknj
Kπ
π
π
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3. Implementing LTI Systems Using the DFT
Linear ConvolutionFinite length h[n] and indefinite-length x[n]
Method 1Nonoverlapping-and-add
wherex n x n r Lr
r
[ ] [ ]= −=
∞
∑0
x nx n rL n L
otherwiser [ ][ ],
,=
+ ≤ ≤ −⎧⎨⎩
0 1
0
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3. Implementing LTI Systems Using the DFT (c.1)
Method 1 (c.1)
Method 2Overlapping-and-skipping
The output
where
y n x n h n y n rLr
r
[ ] [ ] * [ ] [ ]= = −=
∞
∑0
y n x n h nr r[ ] [ ] * [ ]=
x n x n r L P P n Lr [ ] [ ( ) ],= + − + − + ≤ ≤ −1 1 0 1
y n y n r L P Pr
r
[ ] [ ( ) ]= − − + + −=
∞
∑ 1 10
y ny n P n L
otherwiserrp[ ][ ],
,=
− ≤ ≤ −⎧⎨⎩
1 1
0
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The Discrete Cosine Transform (DCT)
General Transform
Four types of DCT is defined for different periodicity
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Type I Type II
Type III Type IV
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The Discrete Cosine Transform (DCT)
Type I DCTExtend x(n) by
We have
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The Discrete Cosine Transform (DCT)
Type II DCTExtend x(n) by
We have
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The Discrete Cosine Transform (DCT)
Other Variant with different normalization factor
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The Discrete Cosine Transform (DCT)
Energy Compaction
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DCT18
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DCT
Error Comparison
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Homeworks
Deadline= May, 11, Class time8.31, 8.33, 8.40, 8.41, 8.42
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