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# Homework Assignments

## Homework 9

1. Suppose is a random process with power spectral density

Find the autocorrelation function of .
2. Suppose that is a random variable with p.d.f. and is a random variable independent of uniformly distributed in . Define a random process by

where is a constant. Find the power spectral density of .
3. Suppose events occur randomly in in the following way:
1. The numbers of events in nonoverlapping intervals are independent of one another.
2. exactly one event in  , where is a continuous nonnegative function on .
3. more than one event in an interval of length  .
Define a random process by and is the number of events occurring in .
1. For , show that is a Poisson random variable with parameter .
2. Find the mean and autocorrelation functions of .

4. Suppose that is a w.s.s., zero-mean, Gaussian random process with autocorrelation function and power spectral density . Define the random process by . Find the mean, autocorrelation, and power spectral density of .
5. Suppose and are independent random variables with and . Define random processes by

Find the autocorrelation and cross-correlation function s of and . Are and jointly wide sense stationary? Are they individually wide sense stationary?