EENG 851: Advanced Signal Processing

1
EENG 851 Advanced Signal Processing

• Chapter 3
• Properties of the Quadratic
• Performance Surface
• Examples

2
Properties of the Quadratic Performance Surface
3
Properties of the Quadratic Performance Surface
4
Normal Form of the Input Correlation Matrix
5
Normal Form of the Input Correlation Matrix(Cont)
6
Eigenvalues and Eigenvectors of the Input
Correlation Matrix
7
Example with Two Weights
Matlab xroots(1 -.5 -.1875) x
0.7500 -0.2500
8
Example with Two Weights (Cont)
9
Example with Two Weights (Cont)
10
Example with Two Weights (Cont)
MATLAB Direct Execution from the Command Window.
R.5 .25.25 .5 R 0.5000 0.2500
0.2500 0.5000 Q,Leig(R) Q 0.7071
0.7071 -0.7071 0.7071 L 0.2500
0 0 0.7500
11
Geometrical Significance of Eigenvectors and
Eigenvalues
12
Geometrical Significance of Eigenvectors and
Eigenvalues (Cont)
13
Geometrical Significance of Eigenvectors and
Eigenvalues (Cont)
14
Geometrical Significance of Eigenvectors and
Eigenvalues (Cont)
15
Geometrical Significance of Eigenvectors and
Eigenvalues (Cont)
16
Geometrical Significance of Eigenvectors and
Eigenvalues (Cont)
17
Geometrical Significance of Eigenvectors and
Eigenvalues(Cont)
The eigenvectors of the input correlation matrix
define the principal axes of the error surface.
18
Geometrical Significance of Eigenvectors and
Eigenvalues(Cont)
19
Geometrical Significance of Eigenvectors and
Eigenvalues(Cont)
The eigenvalues of the input correlation matrix,
R, give the second derivatives of the error
surface with respect to the principal axes of
.
20
Geometrical Significance of Eigenvectors and
Eigenvalues(Cont)
21
Geometrical Significance of Eigenvectors and
Eigenvalues (Cont)
22
Geometrical Significance of Eigenvectors and
Eigenvalues (Cont)
adsp Principal axes 8/4/99 Figure
3.1 clear N5 R.5 .5cos(2pi/N).5cos(2pi/N
) .5 P0 -sin(2pi/N)' Edk22 v.01 1 2 3
4 5 6 w0-5.055 w1-7.5.052.5 w0star2/tan
(2pi/N) w1star-2/sin(2pi/N) W0,W1meshgrid(
w0,w1) c.5W0.2.5W1.2cos(2pi/N)W0.W12s
in(2pi/N)W12Figure 3.1
23
Geometrical Significance of Eigenvectors and
Eigenvalues (Cont)
adsp Principal axes 8/4/99 Figure 3.1
Continued figure(1)clf contour(w0,w1,c,v) clabe
l(c) axis('square') grid title('Mean-Square-Error
') ylabel('w1') xlabel('w0') hold on plot(-5
5,w1star w1star,'k') plot(w0star
w0star,-7.5 2.5,'k') plot(w0star,w1star,'rx
') plot(-3 4,3w1starw0star
-4w1starw0star,'r') plot(-3
4,-3w1star-w0star 4w1star-w0star,'r') hold
off
24
A Second Example
25
A Second Example (Cont)
26
A Second Example (Cont)
MATLAB Solution
R2 11 2 R 2 1 1 2
P7 8' P 7 8 WstarR\P Wstar
2.0000 3.0000 Cmin42Wstar'RWstar-2P'
Wstar Cmin 4
27
A Second Example (Cont)
28
A Second Example (Cont)
29
Two Parameter Principal Axes
30
Two Parameter Principal Axes (Cont)
31
Two Parameter Principal Axes (Cont)
32
Exercise (To be handed in)
33
Properties of the Quadratic Performance Surface
34
Properties of the Quadratic Performance Surface
35
Properties of the Quadratic Performance Surface
eig(3 22 3) ans 1 5
MATLAB
36
Properties of the Quadratic Performance Surface
gtgt eig(3 11 3) ans 2 4
MATLAB
37
Properties of the Quadratic Performance Surface
38
Properties of the Quadratic Performance Surface
39
Properties of the Quadratic Performance Surface
40
Properties of the Quadratic Performance Surface
41
Properties of the Quadratic Performance Surface
42
Properties of the Quadratic Performance Surface
43
Properties of the Quadratic Performance Surface
44
Properties of the Quadratic Performance Surface
roots(1 -5 5) ans 3.6180 1.3820
format long roots(1 -5 5) ans
3.61803398874990 1.38196601125011
eig(2 11 3) ans 1.38196601125011
3.61803398874990
45
Properties of the Quadratic Performance Surface
eig(4 3 03 6 20 2 4) ans
1.25834261322606 4.00000000000000
8.74165738677394
46
Properties of the Quadratic Performance Surface
47
Properties of the Quadratic Performance Surface
V,Deig(2 11 3) V 0.85065080835204
0.52573111211913 -0.52573111211913
0.85065080835204 D 1.38196601125011
0 0
3.61803398874990
48
Properties of the Quadratic Performance Surface
V,Deig(4 3 03 6 20 2 4) V
0.66231967581349 -0.55470019622523
0.50362718288235 -0.60528454386659
0.00000000000000 0.79600918396474
0.44154645054233 0.83205029433784
0.33575145525490 D 1.25834261322606
0 0
0 4.00000000000000 0
0 0 8.74165738677394
49
Properties of the Quadratic Performance Surface
50
Properties of the Quadratic Performance Surface
V,Deig(4 3 03 6 20 2 4) V
0.66231967581349 -0.55470019622523
0.50362718288235 -0.60528454386659
0.00000000000000 0.79600918396474
0.44154645054233 0.83205029433784
0.33575145525490 D 1.25834261322606
0 0
0 4.00000000000000
0 0
0
8.74165738677394 V'V ans
1.00000000000000 0.00000000000000
0 0.00000000000000 1.00000000000000
0.00000000000000 0
0.00000000000000 1.00000000000000