Page 71 - Algorithm Collections for Digital Signal Processing Applications using MATLAB

2. Probability and Random Process 59

If the columns of the B matrix is orthonormal to each other

-1
( i.e.) [B] = [B] T

-1
-1
-1
-1 T
-1 T
-1 T
[B] [U] [B] [B ] [U ] [B ]
-1 T
-1
-1 T
-1
=[B] [U] [U ] [B ]
T
-1 T
-1
-1
= [B] [B ] (Because [U] =[U] )
-1
=[B] [B]
= [I] (Identity Matrix )

Computing Covariance Matrix of the LHS of the equation

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

T
E (([A][Z]) ([A][Z]) )

T
T
=AE[ZZ ]A
= [A][I][A] T

-1
-1 T
= [B] [B ]
= [I]

-1
T
(Because [B] =[B] is assumed )
Thus Covariance matrix of the LHS and RHS are Identity matrix if

T
• E[XX ] is the Identity matrix (Assumed)
T
• E[ZZ ] is the Identity matrix (Property)
-1
T
• [B] = [B] (Assumed)
T
-1
• [U] =[U] (Property)