Autocorrelation of Uniformly Distributed Random Variable in the Interval (0,T)

[wildman]

Homework Statement

The random variable C is uniform in the interval (0,T). Find the autocorrelation
R_x(t_1,t_2) if X(t) = U(t-C) where U is a unit step function.

Homework Equations

The Attempt at a Solution

R_x (t_1,t_2) = \int_{-\infty}^{\infty} U(t_1-c) U(t_2-c) f(c) dc

R_x (t_1,t_2) = \frac{1}{T}\int_0^T U(t_1-c) U(t_2-c) dc

I get stuck here. How do you integrate two shifted unit step functions?

 

[CompuChip]

By first thinking about them
Divide the interval into three parts (assuming t1 < t2)

1. c < t1
2. t1 < c < t2
3. t2 < c

On each of these, what are the values of the step functions? What is their product? Now split the integral and do each part separately.

 

[wildman]

Oh yeah, THANKS!