Consider an LTI system subjected to a wide-sense stationary input {x(n)}, which is a white noise sequence. The cross-correlation ϕ_x[m] between the input x(n), and output y(n) is:

Where  and h[⋅] is the impulse response.

Derivation:

The cross-correlation function between the input and output processes is given by:

R_xy (t₁, t₂) = E{X(t₁) Y*(t₂)}

= R_xx (t₁, t₂) * (h* (t²))

∴ The output cross-correlation between the input and the output is the convolution of the autocorrelation function of the input with the conjugate of the impulse response of the system.

Analysis:

Given the autocorrelation of the input wide-sense stationary signal as:

The cross-correlation will be:

ϕ_x[m] = ϕ_xx [m] * h[⋅]

Since x(n) * δ(n) = x(n)