Homework 3

ECE 630: Statistical Communication Theory
Prof. B.-P. Paris
Homework 3
Due Feb. 11, 2003

Reading

Wozencraft & Jacobs: Chapter 3, pages 129-199.

Problems

1. Wozencraft & Jacobs: Problem 3.10
2. Let $X_t(\omega)$ be a stochastic process defined on $\Omega = \{\omega_1,\ldots,\omega_4\}$ having probabibility assignments $Pr\{\omega_i\}=\frac{1}{4}$ for $i=1,2,3,4$. The sample functions are
   
   $$\begin{array}{ll} X_t(\omega_1) = t & X_t(\omega_2) = -t \\... ...a_3) = \cos 2\pi t & X_t(\omega_4) = -\cos 2 \pi t \end{array}$$
   
   
   1. Compute the joint probability $Pr\{X_0(\omega)=1,X_1(\omega)=1\}$.
   2. Compute the conditional probability $Pr\{X_1(\omega)=1\vert X_0(\omega)=0\}$.
   3. Compute the mean and correlation function of $X_t(\omega)$.
   4. Is this process stationary? wide-sense stationary?
3. 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 $p_{X_t}(x)$.

4. Let $X_t$ be a wide-sense stationary stochastic process. Let $\dot{X}_t$ denote the derivative of $X_t$.
   1. Compute the expected value and correlation function of $\dot{X}_t$ in terms of the expected value and correlation function of $X_t$.
   2. Under what conditions are $\dot{X}_t$ and $X_t$ orthogonal, i.e., $\mbox{\bf E}[\dot{X}_t X_t] = 0$?
   3. Compute the mean and correlation function of $Y_t = X_t - \dot{X}_t$.
   4. The bandwidth of the process $X_t$ can be defined by
      
      $$B_X^2 = \frac{\int_{-\infty}^{\infty} f^2 S_X(f) df} {\int_{-\infty}^{\infty} S_X(f) df}$$
      
      
      Express this definition in terms of the mean and correlation functions of $X_t$ and $\dot{X}_t$.