1.1 Homework 2 of elec 430

Elec 430 homework set 2. Rice University Department of Electrical and Computer Engineering.

Problem 1

Suppose A and B are two Gaussian random variables each zero mean with $\langle A^2\rangle$∞ and $\langle B^2\rangle$∞ . The correlation between them is denoted by $\langle AB\rangle$ . Define the random process $X_t=A+Bt$ and $Y_t=B+At$ .

• a) Find the mean, autocorrelation, and crosscorrelation functions of $X_t$ and $Y_t$ .
• b) Find the 1st order density of $X_t$ , $f_{X_t}(x)$
• c) Find the conditional density of $X_{t_2}$ given $X_{t_1}$ , $f_{X_{t_2}{X}_{t_1}}(x_2{x}_1)$ . Assume $t_2> t_1$
  see Proakis and Salehi problem 3.28
• d) Is $X_t$ wide sense stationary?

Problem 2

Show that if $X_t$ is second-order stationary, then it is also first-order stationary.

Problem 3

Let a stochastic process $X_t$ be defined by $X_t=\cos (t+)$ where $$ and $$ are statistically independent random variables. $$ is uniformaly distributed over $\left[-\pi , \pi \right]$ and $$ has an unknown density $f_{}()$ .

• a) Compute the expected value of $X_t$ .
• b) Find an expression for the correlation function of $X_t$ .
• c) Is $X_t$ wide sense stationary? Show your reasoning.
• d) Find the first-order density function $f_{X_t}(x)$ .