Introduction to stochastic processes

Received signal at an antenna as in .

$X_t=\cos (2\pi f_{0}t+())$ where $f_0$ is the deterministic carrier frequency and $():(, \mathbb{R})$ is a random variable defined over $\left[-\pi , \pi \right]$ and is assumed to be a uniform random variable; i.e. , $f_{}()=\begin{cases}\frac{1}{2\pi } & \text{if $\in \left[-\pi , \pi \right]$}\\ 0 & \text{otherwise}\end{cases}$

The second order stationarity can be determined by first considering conditional densities and the joint density. Recall that

Every $T$ seconds, a fair coin is tossed. If heads, then $X_t=1$ for $nT\le t< (n+1)T$ .If tails, then $X_t=-1$ for $nT\le t< (n+1)T$ .

Second order probability mass function

The conditional pmf