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现代数字信号处理 英文版课件

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2 = ( 21 π)
◮
nD Fourier: Fn g (ω ) = f (x)
f (x) exp −j ω ′ x
dx dω
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n = ( 21 π)
g (ω ) exp +j ω ′ x
A. Mohammad-Djafari, Advanced Signal and Image Processing
g (s) =
D
f (r ) h(r , s) dr
f (r ) −→ h(r , s) −→ g (s)
◮
1–D : g (t ) =
Dபைடு நூலகம்
f (t ′ ) h(t , t ′ ) dt ′ f (x ′ ) h(x , x ′ ) dx ′
D
g (x ) =
◮
2–D : g (x , y ) =
D
f (x ′ , y ′ ) h(x , y ; x ′ , y ′ ) dx ′ dy ′ f (x , y ) h(x , y ; r φ) dx dy
. Advanced Signal and Image processing
Ali Mohammad-Djafari
Groupe Probl` emes Inverses Laboratoire des signaux et syst` emes (L2S) UMR 8506 CNRS - SUPELEC - UNIV PARIS SUD 11 Sup´ elec, Plateau de Moulon, 91192 Gif-sur-Yvette, FRANCE. djafari@lss.supelec.fr http://djafari.free.fr http://www.lss.supelec.fr
Huazhong & Wuhan Universities, September 2012, 10/59
Examples:
A. Mohammad-Djafari, Advanced Signal and Image Processing
2D Fourier Transform: F2
g (ωx , ωy ) = f (x , y )
A. Mohammad-Djafari, Advanced Signal and Image Processing
Linear Transformations: Separable systems
g (s) =
D
f (r ) h(r , s) dr hj (rj , sj )
j
h(r , s) = Examples: ◮ 2D Fourier Transform g (ωx , ωy ) =
h(r , r ′ ) = h(r − r ′ ) f (r ) −→ h(r ) −→ g (r ) = h(r ) ∗ f (r )
◮
1–D : g (t ) =
D
f (t ′ ) h(t − t ′ ) dt ′ f (x ′ ) h(x − x ′ ) dx ′
g (x ) =
D ◮
2–D : g (x , y ) =
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A. Mohammad-Djafari, Advanced Signal and Image Processing
Huazhong & Wuhan Universities, September 2012,
Fourier Transform
[Joseph Fourier, French Mathematicien (1768-1830)] ◮ 1D Fourier: F1 g (ω ) = f (t ) exp {−j ω t } dt
f (x , y ) exp {−j (ωx x + ωy y )} dx dy
h(x , y , ωx , ωy ) = h1 (ωx x ) h2 (ωy y ) exp {−j (ωx x + ωy y )} = exp {−j (ωx x )} exp {−j (ωy y )}
◮
nD Fourier Transform g (ω ) = f (x) exp −j ω ′ x) dx
90
100
1D signal
2D signal=image
3D signal
A. Mohammad-Djafari, Advanced Signal and Image Processing
Huazhong & Wuhan Universities, September 2012,
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Linear Transformations
Wuhan University, September 2012
September 2012
A. Mohammad-Djafari, Advanced Signal and Image Processing Huazhong & Wuhan Universities, September 2012, 1/59
|g (ω )|2 is called the spectrum of the signal f (t ) For real valued signals f (t ), |g (ω )| is symetric f (t ) exp {−j ω0 t } sin(ω0 t ) cos(ω0 t ) exp −t 2 1 (t − m)2 /σ 2 exp − 2 exp {−t /τ } , t > 0 1 if |t | < T /2 g (ω ) δ(ω − ω0 ) ? ? ? ? ? ?
◮
2D Fourier: F2 g (ωx , ωy ) = f (x , y )
f (t )
=
1 2π
g (ω ) exp {+j ω t } dω
f (x , y ) exp {−j (ωx x + ωy y } dx dy g (ωx , ωy ) exp {+j (ωx x + ωy y } dωx dωy
Huazhong & Wuhan Universities, September 2012,
1D Fourier Transform F1
g (ω ) = f (t ) = f (t ) exp {−j ω t } dt
1 2π
g (ω ) exp {+j ω t } dω
◮ ◮
D
Huazhong & Wuhan Universities, September 2012, 6/59
g (r , φ) =
A. Mohammad-Djafari, Advanced Signal and Image Processing
Linear and Invariant systems: convolution
Content
1. Introduction: Signals and Images, Linear transformations (Convolution, Fourier, Laplace, Hilbert, Radon, ..., Discrete convolution, Z transform, DFT, FFT, ...) 2. Modeling: parametric and non-parametric, MA, AR and ARMA models 3. Parameter Estimation: Deterministic (LS, WLS) and Probabilistic methods (ML and Bayesian) 4. Bayesian estimation 5. Kalman Filtering and smoothing 6. Case study: Signal deconvolution 7. Case study: Image restoration 8. Case study: Image reconstruction and Computed Tomography
A. Mohammad-Djafari, Advanced Signal and Image Processing
Huazhong & Wuhan Universities, September 2012,
3/59
Representation of signals and images
◮
Signal: f (t ), f (x ), f (ν )
◮
f (x , y ) exp {−j (ωx x + ωy y } dx dy g (ωx , ωy ) exp {+j (ωx x + ωy y } dωx dωy
2 = ( 21 π)
|g (ωx , ωy )|2 is called the spectrum of the image f (x , y ) ◮ For real valued image f (x , y ), |g (ωx , ωy )| is symetric with respect of the two axis ωx and ωy . Examples: f (x , y ) exp {−j (ωx 0 x + ωy 0 y )} exp −(x 2 + y 2 ) 1 2 + (y − m )2 /σ 2 ] exp − 2 [(x − mx )2 /σx y y exp {−(|x | + |y |)} 1 if |x | < Tx /2 & |y | < Ty /2 1 if (x 2 + y 2 ) < a
◮
◮
◮
A. Mohammad-Djafari, Advanced Signal and Image Processing

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数字信号处理英文课件：chapter 3.3_3.4

数字信号处理英文课件：chapter 3.3_3.4

yC[n] g[n] L h[n] yL[n] g[n] * h[n]
ge[n] 0 N 1
he[n]
n L 1 0
n M 1 L 1
ge[n] L he[n]
0
n L 1
L N M 1
3.4 Linear Convolution Using the DFT
• the relationship between yL[n] and yC[n]:
X[k] X *[N k] X *[kN ]
• Using the above symmetry properties and symmetric relations, we can make the DFT computation more efficient.
• For example, we can get N-point DFTs of two real sequences using a single N-point DFT operation.
[n]
h[n],0 0, M
n
n
M 1 L 1
g[n]
h[n]
0 N 1n
0
n M 1
ge[n]
he[n] n
0 n
0 N 1 L 1 0 M 1 L 1
g[n] h[n]
n L 1 L N M 1
3
3.4 Linear Convolution Using the DFT
So that
X [k] Xcs[k] Xca[k]
X cs [k ]
1 2
X
[k ]
X
*[N
k]
X ca [k ]
1 2
X [k ]
X
数字信号处理 DSP 英文版课件4.0

数字信号处理 DSP 英文版课件4.0


w1(n), w2(n): contents of the two registers
At time n:
Example 4.1.5： An IIR filter h(n) 0.75 u (n) . Using convolution, derive
n
closed-form expressions(闭合形式表示) for the output signal y(n) when the input is: （a）A unit step(单位阶跃) x(n)=u(n) （b）An alternating step x(n) (1) u (n) (正负交替的阶跃)
Algorithmic form:
for each input sample x，do：
y：= w1
w1：＝x
The delay register is typically initialized (初始化) to zero： w1(0)=0.
double delay：
I/O equation: y(n)=x(n-2)
State-space representation：
y(n)=w1(n) (output equation) w1 (n+1)=x (n) (state updating equation) w1(n)： content of the register or internal state at time n
4.1.6 Flip-and-Slide Form
Flip-and-Slide Form of convolution:
h(n) is flipped(反转) around and slid (slide平移) over the input sequence x(n). y(n) is obtained by computing the dot product(点积) of flipped h with the input samples aligned(对准) below it.
ADSP(Chapter2)

ADSP(Chapter2)


⎡ Rsx (0 ) ⎤ ⎢ R (1) ⎥ sx ⎢ ⎥ P = ⎢ Rsx (2 ) ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ Rsx ( N − 1)⎥ ⎣ ⎦ = E [s(n )x(n )]
Rxx (1) Rxx (2 ) ⎡ Rxx (0 ) ⎢ R (1) Rxx (0 ) Rxx (1) xx ⎢ R = ⎢ Rxx (2 ) Rxx (1) Rxx (0 ) ⎢ ⎢ ⎢ Rxx ( N − 1) Rxx ( N − 2 ) Rxx (N − 3) ⎣ = E x(n )xT (n )
h = [h(0 ) h(1) x(n ) = [x(n ) x(n − 1)
h( N − 1)]
T T
x(n − N + 1)]
Rsx (m ) = ∑ h(i )Rxx (m − i ), m = 0, 1,
i =0
N −1
, N −1 Rxx (N − 1) ⎤ ⎡ h(0 ) ⎤ Rxx (N − 2 )⎥ ⎢ h(1) ⎥ ⎥⎢ ⎥ Rxx ( N − 3)⎥ ⎢ h(2 ) ⎥ ⎥⎢ ⎥ ⎥⎢ ⎥ Rxx (0 ) ⎥ ⎢h( N − 1)⎥ ⎦ ⎦⎣
Causality: Low diagonal matrix
(2) Orthogonal equations
E [e(n )x(n − j )] = 0 , ∀j
⎡ ∂ξ (n ) ∂e(n ) ⎤ = 2 E ⎢e(n ) ⎥ = −2 E [e(n )x(n − j )] = 0 , ∂h( j ) ∂h( j ) ⎦ ⎣
Chapter 2 Wiener Filtering and Kalman Filter
2.1 Normal equations of Wiener filter (1) Wiener Filtering Problem
数字信号处理双语版ppt第五章

数字信号处理双语版ppt第五章


Eq 5-8
for any integer n. Looking at the so-called twiddle factor in front of the second summation in Eq. (5-7), we can simplify it as
Eq 5-9
Understanding DSP, Second Edition
As we'll verify in later sections of this chapter, the number of complex multiplications, for an N-point FFT, is approximately:
Eq 5-2
Understanding DSP, Second Edition
Understanding DSP, Second Edition
7
Figure 5-1-1 Number of complex multiplications in the DFT and the radix-2 FFT as a function of N.
Understanding DSP, Second Edition
This capability to subdivide an N/2-point DFT into two N/4-point DFTs gives the FFT its capacity to greatly reduce the number of necessary multiplications to implement DFTs. Following the same steps we used to obtained A(m), we can show that Eq.(5-13)’s B(m) is