I'm a bit rusty with probability and stochastic processes so I'm stuck on a certain part of a problem.

So I have the following equation:

z = Ax + By

x and y are zero-mean independent random vectors. B is a matrix and A is a random matrix that is independent of both x and y. Assume variables are complex and the * represents the conjugate transpose.

I need to find the variance of z.

E[zz*] = E[ (Ax+By)(Ax+By)* ] = E[Axx*A*] + E[Byy*B*] + E[Axy*B*] + E[Byx*A*]
= E[Axx*A*] + BE[yy*]B* + E[Axy*]B* + BE[yx*A*] (factor constants)
= E[Axx*A*] + BVar(y)B* (independence reduction)

I'm not sure what to do with the first term. I know A and x are independent, but I don't know if I can factor anything out due to the order of the matrices/vectors.

Thoughts?

This post was edited by thundercock on Feb 8 2014 01:27am

= Avar(x)A* + Bvar(y)B* (using linearity of expectation, zero-mean of x)

But A is also a random variable (matrix) so how does that play into the expectation?