Wide-sense stationary processes and LTI systems

Let $x(\dot)$ be a wide-sense stationary process with:

• Mean $\mu_x$
• Autocorrelation $R_{xx}(\tau)$
• Autocovariance $C_{xx}(\tau)$
• $E[x^2(t)] \lt \infty$

Let $y(t) = h \ast x(t)$. Then, the following relations are true:

$$E[y(t)] = H(j0) \mu_x$$

$$R_{yx}{\tau} = h \ast R_{xx}(\tau)$$

$$C_{yx}(\tau) = h \ast C_{xx} (\tau)$$

$$R_{xy}(\tau) = \overleftarrow{h} \ast R_{xx}(\tau)$$

$$C_{xy}(\tau) = \overleftarrow{h} \ast C_{xx}(\tau)$$

$$R_{yy}(\tau) = h \ast \overleftarrow{h} \ast R_{xx}(\tau)$$

$$R_{yy}(\tau) = h \ast \overleftarrow{h} \ast C_{xx}(\tau)$$

• $y(t)$ is also wide-sense stationary.
• $y(t)$ is jointly wide-sense stationary with its input.

Given $y = h \ast x$ and $z = g \ast w$:

$$R_{yz}(\tau) = h \ast \overleftarrow{g} \ast R_{xw}(\tau)$$

Main article: Power spectral density

CT case:

$$R_{xx}(\tau) \leftrightarrow S_{xx}(j\omega)$$ $$C_{xx}(\tau) \leftrightarrow D_{xx}(j\omega)$$

DT case:

$$R_{xx}[m] \leftrightarrow S_{xx}(e^{j\Omega})$$ $$C_{xx}[m] \leftrightarrow D_{xx}(e^{j\Omega})$$