(a) Let W be a normalized IID Gaussian n-rv and let Y be a Gaussian m-rv. Suppose we would like the joint covariance E rWYTl to be some arbitrary real-valued n × m matrix [K]. Find the matrix [A] such that Y = [A]W achieves the desired joint covariance. Note: This shows that any real-valued n × m matrix is the joint covariance matrix for some choice of random rv s.
(b) Let Z be a zero-mean Gaussian n-rv with non-singular covariance [KZ], and let Y be a Gaussian m-rv. Suppose we would like the joint covariance E rZYT l to be some arbitrary n×m matrix [K∗]. Find the matrix [B] such that Y = [B]Z achieves the desired joint covariance. Note: This shows that any real-valued n × m matrix is the joint covariance matrix for some choice of rv s Z and Y where [KZ] is given (and non-singular).
(c) Now assume that Z has a singular covariance matrix in (b). Explain the constraints this places on possible choices for the joint covariance E rZYT l. Hint: Your solution should involve the eigenvectors of [KZ].
Text Book: Stochastic Processes: Theory for Applications By Robert G. Gallager.