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Next: Problem 12 Up: Collected Problems Previous: M-ary Signal Sets

Problem 11

A stochastic process is defined by

X_t = \cos 2 \pi F t

where the frequency $F$ is uniformly distributed over the interval $[0,f_0]$.
  1. Find the mean and correlation function of $X_t$.
  2. Show that this process is non-stationary.
Now suppose we redefine the process $X_t$ to be

X_t = \cos(2 \pi F t + \Theta)

where $F$ and $\Theta$ are statistically independent random variables. $\Theta$ is uniformly disributed over $[-\pi,\pi]$ and $F$ is distributed as before.
  1. Compute the mean and correlation function of $X_t$.
  2. Is $X_t$ wide-sense stationary? Show your reasoning.
  3. Find the first order density $p_{X_t}(x)$.

Dr. Bernd-Peter Paris