现代数字信号处理英文版课件

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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

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)

⎡ 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

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

数字信号处理英文课件Chapter7

■
Four Types of Linear Phase FIR Filters
h(n)
Type I
h(n)
Type II
0
M/2
M
n
0
M/2
M
n
h(n)
H(z) =
z –M ⁄ 2
h(M ⁄ 2) +
M⁄2–1 n=0
Type III
2
3
4
and
Φ ( ω ) = –Φ ( –ω )
y ( n ) = 0.5 y ( n – 1 ) + x ( n ) – x ( n – 2 )
1 – cos 2 ω + j sin 2 ω 1 – e –j 2 ω = ------------------------------------------------H ( j ω ) = -------------------------ω – j 1 – 0.5 cos ω + j sin ω 1 – 0.5 e
Page 8 of 25
Page 7 of 25
C h a p t e r
7 :
A
F r a m e w o r k
f o r
D i g i t a l
F i l t e r
D e s i g n
C h a p t e r
7 :
A
F r a m e w o r k
f o r
D i g i t a l
2.3 Finite Impulse Response (FIR) Filters
• Duration of h ( n ) : N = M + 1 < ∞ , i.e., h ( n ) = 0 as • FIR filters are nonrecursive; i.e., all coefficients a k = 0 . • Causal FIR filters can be represented as follows: n → ∞.

数字信号处理(英文版)1-连续时间信号系统

Unit impulse function δ(t)
With a gate signal pτ(t), short the duration τ and keep the unit area
4/τ 2/τ 1/τ
1/τ
-τ/2
τ/2
-τ/4
τ/4
-τ/8 τ/8
When τ0, the amplitude tends to , which means it is impossible to define δ(t) by a regular function.
Typical signals and their representation
Gate signal
p (t ) 0
1
|t |

2
1 -τ/2 τ/2
|t |

2
The gate signal can be represented by unit step signals:

(t )dt (t )dt (t )dt u (t )
0
0

Properties of δ(t)
δ(t) is a even function, that is
δ(t) = δ(-t) We got δ(t) from a gate signal, and gate signal is an even function. It is also easy to give the math show of the even property.
Typical signals and their representation
Sinusoidal Asin(ωt+υ)

数字信号处理DigitalSignalProcessingppt课件

处理系统中集成了几十万甚至更多的晶体管，而模拟信号处理系统中大量使用的是电阻、电容、电感等无源器件，随着系统的复杂性增加这一矛盾会更加突出。
17
5. 数字信号处理的应用领域
▪ 语音处理
▪ 语音信号分析 ▪ 语音合成 ▪ 语音识别 ▪ 语音增强 ▪ 语音编码
▪ 图像处理：恢复，增强，去噪，压缩 ▪ 通信：信源编码，信道编码，多路复用，数据压缩 ▪ 电视：高清晰度电视，可视电话，视频会议 ▪ 雷达：对目标探测，定位，成像
统，其性能取决于运算程序和乘法器的各系数，这些均存储在数字系统中，只要改变运算程序或系数，即可改变系统的特性参数，比改变模拟系统方便得多。
15
▪ 可以实现模拟系统很难达到的指标或特性：例如：
有限长单位脉冲响应数字滤波器可以实现严格的线性相位；在数字信号处理中可以将信号存储起来，用延迟的方法实现非因果系统，从而提高了系统的性能指标；数据压缩方法可以大大地减少信息传输中的信道容量。
▪ 由一维走向多维，像高分辨率彩色电视、雷达、
石油勘探等多维信号处理的应用领域已与数字信号处理结下了不解之缘。
22
各种数字信号处理系统均几经更新换代：在
图像处理方面，图像数据压缩是多媒体通信、影碟机(VCD或DVD)和高清晰度电视(HDTV)的关键技术。国际上先后制定的标准H.261、JPEG、 MPEG—1和MPEG—2中均使用了离散余弦变换 (DCT)算法。近年来发展起来的小波(Wavelet)变换也是一种具有高压缩比和快速运算特点的崭新压缩技术，应用前景十分广阔，可望成为新一代压缩技术的标准。
5
▪ 信息科学
▪ 信息科学是研究信息的获取、传输、处理和利用的一门科学。
▪ 信号

现代信号处理ModernSignalProcessing40页PPT

凡不是广义平稳的信号
遍历性
若 N li m E 2N 11tN Nx(tt1)Lx(ttk)(t1,L,tk)2 0
则 {x(t)}称为均方遍历信号。
2.两个随机信号的二阶统计量
互相关函数
Rxy()@E{x(t)y*(t)}
相同部分相乘（相同符号）不同（随机）部分相乘（平均意义上，相互抵消）。
考核方式习题（11%）计算机仿真（实验3次，24%）考试（65%）
第一章随机信号
本章主要介绍随机信号的基本概念：相关函数、功率谱密度、两个信号的正交、统计不相关和统计独立、相干信号以及它们的几个典型应用。
1.信号分类
信号——信息的载体
连续时间信号s(t) t 离散时间信号s(k) k为整数
▪ 时分多址(TDMA: time-division multiple access): 各个用户的信号波形在时域上无重叠正交(时域正交)
用户1和用户2之间有一个保护时隙
b
a si
(t)s*j (t)dt
0,
i j
共享：整个频带
正交的两个典型应用(续)
▪ 频分多址(FDMA: frequency-division multiple access): 各个用户的信号波形在频域上无重叠频域正交
E wi 2 qiHqi
im1
im1
由wi qiHx得：E wi 2 E qiHxxHqi qiHE xxH qi qiHRxqi
正交的两个典型应用(续)
M
最优化： min Em min
q
H i
R
x
q
i
im 1
约
束

2019年-现代数字信号处理Advanced Digital Signal Processing_ch5 wavelet-精品文档-PPT精选文档

waves Wave transform
wavelets
(wide-sense) wavelet transform
Orthogonal transform Orthogonal basis function
g (i,t),g (j,t) g (i,t)g * (j,t)d t 0 c i,,ii jj
• Main Tools of Time-Frequency Analysis
Short time Fourier transform (STFT) Wavelet transform Wigner distribution (WD)
Quadric transform (non-linear transform) Time-frequency distribution Wiger-Ville distribution (non-stationary
Signals are different, but spectrums are similar
Deficiency of wave transform (e.g. FT)
Wave transforms are not suitable for timevariant signal since they don’t include position (time) information in the transform results (e.g. FT analyzes the global frequency distribution of a signal, but it can not characterize the local behavior of the signal).

数字信号处理英文影印版课件4-1

数字信号处理Digital Signal Processing 电子信息工程系韩建峰KeywordsSections⏹Sampling sinusoids⏹Sampling theorem⏹Discrete-to-Continuous Conversion SummaryLECTURE 1Reading assignments This lecture •Chapter 4•Section 4-1KeywordsPart AContinuous-to-Discrete Conversion Sampling & ReconstructionAliasing & FoldingLECTURE OBJECTIVES•SAMPLING can cause ALIASING •Spectrum for digital signals,x [n]•Normalized Frequencyππωω22ˆ+==ss f f T ALIASINGReviewSignalsSampling Reconstruction•Continuous-time Signal•But the key point is that any computer represent ation is discrete.•So, do sampling!•And, how?()cos()x t A tωϕ=+•Sample a continuous-time signal at equally spaced time instants.Take a “snapshot” every Ts.Speech, audio andso on.•Or, compute the values of a discrete-time signal directly from a formula.2=-+[]53x n n nSAMPLING x(t)•SAMPLING PROCESS•Convert x(t) to numbers x[n]•“n” is an integer; x[n] is a sequence ofvalues•Think of “n” as the storage address inmemory•UNIFORM SAMPLING at t = nTs•IDEAL: x[n] = x(nT)sSAMPLING RATE, f s •SAMPLING RATE (f)s–f=1/T ss•NUMBER of SAMPLES PER SECOND –T= 125 microsec f s= 8000 samples/secs–UNITS ARE HERTZ: 8000 Hz •UNIFORM SAMPLING at t = nT= n/f ss–IDEAL: x[n] = x(nT)=x(n/f s)s•Examples of continuous-time signals exist in the “real-world” outside the computer.•Simple mathematical formula.•More general continuous-time signals can be represented as sum of sinusoids.•So, we will use sinusoidal signal as the basis for our study of sampling.sf s T n A n x ωωωϕω==+=ˆ)ˆcos(][)cos()(][)cos()(ϕωϕω+==+=s s nT A nT x n x t A t x •Change x(t) into x[n] DERIV ATION))cos((][ϕω+=n T A n x s DEFINE DIGITAL FREQUENCYDigital Frequency ωˆ•V ARIES from 0to 2π, as f varies from 0 to the sampling frequency•UNITS are radians, not rad/sec–DIGITAL FREQUENCY is NORMALIZEDss f f T πωω2ˆ==Sample RateHow to select theT sSample TheoremA interesting phenomenon•Exercise 4.1•Is this the only possible answer?Hz 1000at sampled )2400cos()(2==s f t t x π21000[]cos(2400)cos(2.4)cos(0.42)cos(0.4)nx n n n n n πππππ===+=()cos(400)x t t π⇒=Aliasing[]cos(0.4)x n n π=Illustration of aliasingDifferent frequency, but same values at n=0,1,2,3…•2.4πis an alias of 0.4π•Exercise 4.2Aliasing•How does aliasing arise in a mathematical treat ment of discrete-time signal?•The last example:12[]cos(0.4)[]cos(2.4)x n n x n n ππ==2[]cos(0.42)cos(0.4)x n n n n πππ=+=Periodic function with period 2πAliasing Derivation-1and we substitute: t ←n f sIf x (t )=A cos(2π(f + f s )t +ϕ)then: x [n ]=A cos(2π(f + f s )n f s +ϕ)or, x [n ]=A cos(2πf f s n +2π n +ϕ)Aliasing Derivation-22ˆs sfT f πωω==+2π 2()22ˆthen: s s s s sf f f f f f f πππω+==+ˆand we want: []cos()x n A n ωϕ=+If x (t )=A cos(2π(f + f s )t +ϕ)t ←nf sFolded Aliasx (t )=A cos(2π(-f + f s )t -ϕ)SAME DIGITAL SIGNALˆ[]cos()x n A n ωϕ=+x [n ]=A cos((2πf T s )n -2π n +ϕ) x [n ]=A cos((-2πfT s )n +(2π f s T s )n -ϕ)x [n ]=x (nT s )=A cos(2π(-f + f s )nT s -ϕ)Aliasing2ˆs sfT f πωω==+2π 2ˆ2s sfT f πωωπ==-+Folded AliasAlisingPrincipal Aliasingˆˆˆ, 2, 2 integer l l l ωωππω+-=General FormulaSpectrum of a Discrete-Time Signal•PLOT versus NORMALIZED FREQUENCY •INCLUDE ALL SPECTRUM LINES –ALIASES•ADD MULTIPLES of 2π•SUBTRACT MULTIPLES of 2π–FOLDED ALIASES•ALIASES of NEGATIVE FREQS12X*–0.5π12X–1.5π12X0.5π2.5π–2.5πˆω12X12X*12X*1.5π))80/)(100(2cos(][ϕπ+=n A n x 80s f Hz=sf fπω2ˆ=ˆ2sff ωπ=f s =125Hz12X*0.4π12X–0.4π1.6π–1.6πˆω12X12X*))125/)(100(2cos(][ϕπ+=n A n x•DEMO: Strobe Movies 12•What is the meaning of this DEMO?•Can you give us more examples in the real world?f Camera: 30 Frames/s Human Eyessf'fSummary2ˆs sfT f πωω==+2π 2ˆ2s sfT f πωωπ==-+Folded AliasAlisingPrincipal Aliasingˆˆˆ, 2, 2 integer l l l ωωππω+-=General FormulaHomeworkP-4.1Review: Chapter 4, Section 4-1 Preview:Chapter 4, Section 4-2,4-4。

DSP数字信号处理2

1, n = 0 d [ n] = 0, n 0
Shift in time: d[n - k ] Can express any sequence with d: {a0,a1,a2..}= a0d[n] + a1d[n-1] + a2d[n-2]..
x[n] =
k = -
x[k ]d [n - k ]
Example: There is no integer N such that the signal x[n] = cos[n] satisfies the condition x[n + N] = x[n] for all N.
d [n] = [n] - [n - 1] [n] = k =- d [k ] = k =0 d [n - k ]
n
Exponential sequences… Exponential sequences= eigenfunctions General form: x[n] = A· an If A and a are real:
数字信号处理课程ppt全英文版本
Discrete Time Signal &System
Contents of this lecture • What is a signal ? • What is signal processing ? • Basic signals (sequences) • Basic operation • Discrete time system
+
x[n]
x[n] = a-3d[n+3] + a1d[n-1] - a2d[n-2] – a7d[n-7]
More basic sequences… Unit step sequence:

数字信号处理DigitalSignalProcessing课件

re j eTs e jTs
得到：
r eTs
Ts
s与z
z re j |r1 e j
Ts 2 f fs
X (e j ) x(n)e jn n
离散时间序列旳傅里叶变换，
DTFT
Im[ z ]
z 平面
0 Re[z]
z 平面 Im[z]
r 1
0 Re[z]
Ts 2 f fs
n0
if az1 1, that is z a
ROC
then X (z) 1 1 az1
X (z) z za
a1
例2：x(n) anu(n 1)
{ u(n 1)
1 n 1,,
0 其他
1
X (z) an zn 1 (a1z)n
n
n0
1
1
1 a
1 z
z
z a
ROC : a1z 1, z a
极零分析旳应用
1. 稳定性: 鉴别条件1：
h(n)
n0
h(n) l1
稳定性: 鉴别条件2 ：
| pk | 1, k 1,, N
全部极点都必需在单位
圆内！
证明： H (z) N ck z k 1 z pk
p N
n
h(n) ck k
k 1
p
N
n
h(n)
ck k
n0
n0 k 1
x(n)zn zm1dz
c
c
n0
x(n) zmn1dz c n0
z re j
x(n) rmn1e j(mn1)dz n
dz rje jd x(n)rmn j e j(mn)d n
X (z)zm1dz x(n)r mn j e j(mn) d

现代数字信号处理Advanced Digital Signal Processing_ch3 adaptive filter

Quadratic function with single global optimum
One weight: parabola
wnw0n Rxxrxx0, Rxd rxd0 nEd2nrxx0w02n2rxd0w0n
Two weights: paraboloid
Forarbitraryconstantsc1 andc2,
c1 and c2
are concentric ellipses.
wnw0n w1nT
Rxx rrxxxx10
rxx1 rxx0
,
Rxd rrxxdd10
Ed2nw0n w1nrrxxxx10 rrxxxx10w w10nn2rxd0 rxd1w w10nn
Ed2nrxx0w02nw12n2rxx1w0nw1n2rxd0w0n2rxd1w1n
• The Classes of Adaptive Linear Filter
By the length of linear filter
FIR: always stable; good convergence properties; possibly linear-phased
IIR: probably less estimation error (residual) than FIR
vnTRxxvn0
The performance function in v(n) coordinate system
m invnTRxxvn
The v(n) coordinate system is a shifting of the w(n) coordinate system.
• Principle Axes Coordinate System

现代数字信号处理Advanced Digital Signal Processing_ch5 wavelet

Inner product
F f( t)g * (,t)d tf( t) ,g (,t)
g ( ,t):b a sisfu n c tio n
• Wave & Wavelet Transform
Waves Waves are non-compact (infinite) support functions
5.2 Time-Frequency Analysis
• Basic Idea
In FT, the local behavior of a signal is not represented in the signal’s frequency spectrum
The FT is not the most proper representation for the time-variant signals or the signals containing transient or localization components
• The Conflicting Requirements between the Frequency Resolution & the Time Resolution in STFT
Frequency resolution requirement
The window width T should be wide enough to give the desired frequency resolution.
w h e n 0 .
lim ejit 0, lim ejit 0,
t
t
i.e. the FT is an orthonormal wave transform.

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2 = ( 21 π)
◮
nD Fourier: Fn g (ω ) = f (x)
f (x) exp −j ω ′ x
dx dω
9/59
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 ) =
8/59
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 )}
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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 ) ? ? ? ? ? ?
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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
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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
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Signal: f (t ), f (x ), f (ν )
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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
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A. Mohammad-Djafari, Advanced Signal and Image Processing