EENG 851: Advanced Signal Processing

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EENG 851 Advanced Signal Processing

• Chapter 2
• Stochastic (random) processes using MATLAB
  command window
• Mathematical Necessities (Matrices)
• The Adaptive Linear Combiner
• Examples

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Standard Normal Random Process
figure(1)clf plot(110,randn(10,1),'')gri
d
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Standard Normal Random Process
plot(1100,randn(100,1),'')grid
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Standard Normal Random Process
plot(1100,randn(100,1))grid
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Standard Normal Random Process
plot(11000,randn(1000,1))grid
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Standard Uniform Random Process
plot(11000,rand(1000,1))grid
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Standard Normal Random Process
plot(1100,randn(100,4))grid
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Zero Mean Uniform Random Process
plot(1100,rand(100,4)-.5)grid
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Mathematical Necessities
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Mathematical Necessities (Cont)
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Mathematical Necessities (Cont)
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Mathematical Necessities (Cont)
A1 2 34 5 6 A 1 2 3 4
5 6 B3 2 1 04 3 2 15 4 3 2 B
3 2 1 0 4 3 2 1
5 4 3 2 AB ans 26
20 14 8 62 47 32 17
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Mathematical Necessities (Cont)
A1 2 34 5 67 8 9 A 1 2 3
4 5 6 7 8 9 B 2 1
03 2 14 3 2 B 2 1 0 3
2 1 4 3 2
AB ans 20 14 8 47 32
17 74 50 26 BA ans 6 9
12 18 24 30 30 39 48
AB-BA ans 14 5 -4 29 8
-13 44 11 -22
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Mathematical Necessities (Cont)
A.B ??? Error using gt . Matrix dimensions
must agree. A.B(12,13) ans 3 4
3 16 15 12 A.B(2 3,4 1
2) ans 1 8 9 8 25 24
A1 2 34 5 6 A 1 2 3 4
5 6 B3 2 1 04 3 2 1 5 4 3 2 B
3 2 1 0 4 3 2
1 5 4 3 2
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Mathematical Necessities (Cont)
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Mathematical Necessities (Cont)
A1 2 j4 5 -j A 1.0000
2.0000 0 1.0000i 4.0000
5.0000 0 - 1.0000i
A' ans 1.0000 4.0000
2.0000 5.0000 0
- 1.0000i 0 1.0000i
A.' ans 1.0000 4.0000
2.0000 5.0000
0 1.0000i 0 - 1.0000i
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Mathematical Necessities (Cont)
A' ans 1 4 2 5 3
6 B' ans 3 4 5 2 3
4 1 2 3 0 1 2
(AB)' ans 26 62 20 47 14
32 8 17 B'A' ans 26 62
20 47 14 32 8 17
A1 2 34 5 6 A 1 2 3 4
5 6 B3 2 1 04 3 2 1 5 4 3 2 B
3 2 1 0 4 3 2
1 5 4 3 2
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Mathematical Necessities (Cont)
A1 2 00 1 -20 0 2 A 1 2
0 0 1 -2 0 0 2
inv(A) ans 1.0000 -2.0000 -2.0000
0 1.0000 1.0000 0 0
0.5000
B2 -1 01 1 -11 0 -1 B 2 -1
0 1 1 -1 1 0 -1
inv(B) ans 0.5000 0.5000 -0.5000
0.0000 1.0000 -1.0000 0.5000 0.5000
-1.5000
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Mathematical Necessities (Cont)
A1 2 00 1 -20 0 2 A 1 2
0 0 1 -2 0 0 2 B2
-1 01 1 -11 0 -1 B 2 -1 0
1 1 -1 1 0 -1
inv(AB) ans 0.5000 -0.5000
-0.7500 0 1.0000 0.5000 0.5000
-0.5000 -1.2500 inv(B)inv(A) ans
0.5000 -0.5000 -0.7500 0.0000 1.0000
0.5000 0.5000 -0.5000 -1.2500
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Mathematical Necessities (Cont)
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Mathematical Necessities (Cont)
A1 2 32 4 53 5 6 A 1 2 3
2 4 5 3 5 6
inv(A) ans 1.0000 -3.0000 2.0000
-3.0000 3.0000 -1.0000 2.0000 -1.0000
0
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Mathematical Necessities (Cont)
B1 1j 2j1-j 2 3j2-j 3-j 3 B
1.0000 1.0000 1.0000i 2.0000
1.0000i 1.0000 - 1.0000i 2.0000
3.0000 1.0000i 2.0000 - 1.0000i 3.0000 -
1.0000i 3.0000 inv(B) ans
1.0000 0.0000i -1.0000 0.5000i 0.5000 -
0.5000i -1.0000 - 0.5000i 0.5000 0.0000i
0.0000 0.5000i 0.5000 0.5000i 0.0000 -
0.5000i 0.0000 0.0000i
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The Adaptive Linear Combiner (ALC)
Weight Vector
Input Signal Vector
Output Signal
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The ALC (Cont)
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The ALC (Cont)
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ALC Notation
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ALC Notation (Cont)
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ALC Notation (Cont)
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Desired Response and Error
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Desired Response and Error (Cont)
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The Performance Function
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The Performance Function (Cont)
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The Performance Function (Cont)
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The Performance Function (Cont)
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The Performance Function (Cont)
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The Performance Function (Cont)
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Gradient and Minimum Mean-Square Error
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Gradient and Minimum MSE (Cont)
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Gradient and Minimum MSE (Cont)
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Gradient and Minimum MSE (Cont)
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Gradient and Minimum MSE (Cont)
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Gradient and Minimum MSE (Cont)
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Example of a Performance Surface
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Example of a Performance Surface (Cont)
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Example of a Performance Surface (Cont)
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Example of a Performance Surface (Cont)
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Example of a Performance Surface (Cont)
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Example of a Performance Surface (Cont)
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Example of a Performance Surface (Cont)
adsp performance surface 7/19/99 clear N5
P0 -sin(2pi/N)' R.51 cos(2pi/N)cos(2pi
/N) 1 Edk22 W-3.13-3.13 CEdk2ones
(61,61)W.'RW-2P.'W x-10.210y-10.210 C
m0 for w0x mm1 n0 for w1y
nn1 Ww0 w1'
C(m,n)Edk2W'RW-2P.'W end end
figure(1)clf mesh(x,y,C)grid grid title('Perform
ance Surface') ylabel('w0') xlabel('w1') zlabel('M
SE')
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Example of a Performance Surface (Cont)
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Example of a Performance Surface (Cont)
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Example of a Performance Surface (Cont)
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Example of a Performance Surface (Cont)
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Example of a Performance Surface (Cont)
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Alternative Expression for the Gradient
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Alternative Expression for the Gradient (Cont)
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Alternative Expression for the Gradient (Cont)
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Decorrelation of Error and Input Components
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Class Example 1
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Class Example (Cont)
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Class Example (Cont)
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Class Example (Cont)
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Class Example (Cont)
Class Example WS page 276
10/1/99 w-1.13 cw.2-2w4 figure(1)clf pl
ot(w,c)grid axis(-1 3 0 7) title('Example
1') xlabel('Weight, w1') ylabel('Mean Square
Error')
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Class Example (Cont)
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Exercise 3. To be handed in.
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Exercise 3. (Cont)
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ALC Example
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ALC Example (Cont)
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ALC Example (Cont)
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ALC Example (Cont)
EENG 851 ALC example WS 264 8/23/00
saved as adspxmpl26_4.m clear w0-4.058 w1-10
.052 v4 4 W0,W1meshgrid(w0,w1) N5 c5.
5W0.2.5W1.2cos(2pi/N)W0.W12sin(2pi/N)
W12 figure(1)clf contour(w0,w1,c5,v) clabel(c
) axis('square') grid title('ALC Example WS
26-4') ylabel('w1') xlabel('w0') hold
on N10 c10.5W0.2.5W1.2cos(2pi/N)W0.W1
2sin(2pi/N)W12 contour(w0,w1,c10,v,'r') hold
off