For the zero-mean complex random vector z = xc ixs, with cov(z) = = C iQ, with = ,

For the zero-mean complex random vector z = xc − ixs, with cov(z) = Σ =

C − iQ, with Σ = Σ

∗

, define w = 2Re(a

∗

z), where a = ac − ias is an arbitrary non-zero complex vector. Prove cov(w) = 2a

∗Σa.

Recall ∗ denotes the complex conjugate transpose.