Power spectral density of convolution of stochastic processes

Question

I am wondering what the result of convolving two WSS processes in terms of power spectral densities is.

I know that the output $Y(t)$ of a generic linear time invariant (LTI) system with impulse response $h(t)$, and whose input is a stochastic process $X(t)$, is given by the convolution

$$ Y(t) = h(t) \ast X(t) $$

which, in terms of power spectral densities, is translated to

$$ S_{Y}(f) = |H(f)|^2 S_X(f) $$

where $S_{X}(f)$ is the power spectral density of the input process. This result is obtained by direct computation of the autocorrelation of $Y(t)$ and successive application of the Wiener–Khinchin theorem. This result however holds if $h(t)$ is deterministic.

What if the impulse response $h(t)$ is WSS stochastic process as well? Is it true that

$$ R_y(\tau) = R_h(\tau) \ast R_x(\tau) $$

and thus that

$$ S_y(\tau) = S_h(\tau) \cdot R_x(\tau)? $$

Answer 1

It holds for weakly stationary stochastic process. For MIMO linear systems, $y(t) = h(t) * x(t)$, we still have $$R_y(t) = h(t) * R_x(t) * h^T(-t)$$ or $$R_y(f) = h(f) R_x(f) h^T(-f).$$

For details, refer to Power spectral density of the system output.