ϕ(t) = R(t) = autocorrelation
ϕ(f) = S(f) = power spectral density
Inputs v[t] and w[t] are zero mean, jointly WSS processes with auto-correlation se-
quences ϕvv(τ ) and ϕww(τ ), cross-correlation sequence ϕvw(τ ), power spectrums ϕvv(ω)
and ϕww(ω), and cross power spectrum ϕvw(ω). Let x(t) be the output of a LTI system
h1(t) with input v(t), i.e. x(t) = h1(t)v(t), and let y(t) be the output of a LTI system
h2(t) with input w(t), i.e. y(t) = h2(t) w(t). Assume all quantities to be complex.
(a) Show that x(t) and y(t) are jointly WSS. Determine an expression for ϕxy(τ ) and
ϕxy(ω) in terms of the input statistics and the system parameters.
(b) Is the cross power spectrum always non-negative, i.e. is ϕxy(ω) ≥ 0, for all ω? Justify
(c) Specialize this formula to obtain an expression for ϕxx(ω).
ϕxy(τ) = E[x(t)y(t+τ)]
ϕx(f) = ϕv(f)|H|²
ϕx(f) = Fourier Transform( ϕx(t) )
The Attempt at a Solution
I'm at a complete loss. For jointly WSS, I need to prove x(t) and y(t) are WSS which they are because of v(t) and w(t) going through an LTI system. But I also need to show ϕxy(τ) only depends on τ.
I feel that I'm missing a crucial point and don't know how to start.