Here we consider some basic properties of covariances. In what follows, assume for generality that all RVs and constants are complex-valued (the definition of covariance for complex-valued RVs is...

Here we consider some basic properties of covariances. In what follows, assume for generality that all RVs and constants are complex-valued (the definition of covariance for complex-valued RVs is given in Equation (25a)). All RVs are denoted by Z (or subscripted versions thereof), while c with or without a subscript denotes a constant. Note that, as usual, Z ∗ denotes the complex conjugate of Z and that E{Z ∗ } = (E{Z}) ∗ .

(a) Show that cov

(b) Show that cov

(c) Show that co

(d) Suppose that at least one of the RVs Z₀
and Z₁
has a zero mean. Show that cov

(e) Show that

where j and k range over finite sets of integers.

Since real-valued RVs and constants are special cases of complex-valued entities, the results above continue to hold when some or all of the RVs and constants in question are real-valued. In particular, when Z0 and Z1 are both real-valued, part (b) simplifies to cov
  Also, when Z₁
and c₁,k in parts (d) and (e) are real-valued, we can simplify