Let \(\mathbf{V}(t)\) be a linearly filtered complex-valued, wide-sense stationary random process with sample functions given by

\[ \mathbf{v}(t)=\int_{-\infty}^{\infty} \mathbf{h}(t-\tau) \mathbf{u}(\tau) d \tau \]

where \(\mathbf{U}(t)\) is a complex-valued input process and \(\mathbf{h}(t)\) is the impulse response of a time-invariant linear filter.

(a) Show that

\[ \boldsymbol{\Gamma}_{V}(\tau)=\mathbf{H}(\tau) \otimes \boldsymbol{\Gamma}_{U}(\tau) \]

where

\[ \mathbf{H}(\tau)=\int_{-\infty}^{\infty} \mathbf{h}(\xi+\tau) \mathbf{h}^{*}(\xi) d \xi \]

and \(\otimes\) represents a convolution operation.

(b) Show that the mean-square value \(\overline{|\mathbf{v}|^{2}}\) of the output is given by

\[ \overline{|\mathbf{v}|^{2}}=\int_{-\infty}^{\infty} \mathbf{H}(-\eta) \boldsymbol{\Gamma}_{U}(\eta) d \eta \]