Homework 3 (Due: February 19)

ECE 630: Statistical Communication Theory
Prof. B.-P. Paris
Homework 3
Due: February 19, 2019

Reading

Madhow: Section 3.3.

Problems

Let X_t(ω) be a random process defined on Ω = {ω₁,…,ω₄} having probabibility assignments Pr{ω_i} = for i = 1, 2, 3, 4. The sample functions are $Xt (ω1) = t Xt (ω2) = - t X (ω ) = cos2πt X (ω ) = - cos 2πt t 3 t 4$
1. Compute the joint probability Pr{X₀(ω) = 1,X₁(ω) = 1}.
2. Compute the conditional probability Pr{X₁(ω) = 1|X₀(ω) = 0}.
3. Compute the mean and correlation function of X_t(ω).
Prove the following properties of a random process:
1. R_X(t,t) ≥ 0
2. R_X(t,u) = R_X(u,t) (symmetry)
3. |R_X(t,u)|≤(R_X(t,t) + R_X(u,u))
4. |R_X(t,u)|² ≤ R_X(t,t) ⋅ R_X(u,u)
A random process is defined by $Xt = cos2 πF t$
where the frequency F is uniformly distributed over the interval [0,f₀].
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
$X = cos(2πF t + Θ ) t$
where F and Θ are statistically independent random variables. Θ is uniformly disributed over [-π,π) 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).