Page 70 - Algorithm Collections for Digital Signal Processing Applications using MATLAB

58 Chapter 2

[X]=[B][U][Z]

[B][U][Z]=[X]

-1
[B][U][B][B] [Z]=[X]

-1
[A][Z]={[B][U][B]} [X]

Note that A=[B] -1

Column vectors of the transformation matrix [U] are orthonormal to
T
each other. (ie) [U] = [U] -1
T
If E [XX ] = [I] (Identity Matrix)
((i.e.) Covariance matrix of the Independent signals with mean zero and
variance one is Identity matrix )

Computing Covariance Matrix of the RHS of the equation

-1
[A][Z]={[B][U][B]} [X]

-1
T
-1
E [({[B][U][B]} [X]) ({[B][U][B]} [X]) ]
-1
T
-1 T
= E [({[B][U][B]} ) [X] [X] ({[B][U][B]} ) ]
-1
T
-1 T
= ({[B][U][B]} ) E [XX ] ({[B][U][B]} )
-1 T
-1
= ({[B][U][B]} ) ({[B][U][B]} )

-1
-1 T
-1
-1
-1
-1
= ([B] [U] [B] ) ( [B] [U] [B] )
-1
-1
-1 T
-1 T
-1 T
-1
=[B] [U] [B] [B ] [U ] [B ]