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## Utah State University ECE 6010 Stochastic Processes Homework # 9 Solutions

1. Suppose is a ramdom process with power spectral density

Find the autocorrelation function of .

Similarly,

Therefore,

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 .

Therefore,

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 i s the number of events occurring in .
1. For , show that is a Poisson random variable with parameter .

k events in for and .

If satisfies the above equation then we can say that is a Poisson random variable with parameter .

Now, let and

Therefore,

So that does satisfy. And is Poisson.

2. Find the mean and autocorrelation functions of .

We have, . So,

Now,

So,

and

4. Suppose that is a w.s.s., zero-mean, Gaussian random process with auto-correlation function and power spectral density . Define the random process by . find the mean, autocorrelation, and powerspectral density of .

Mean :

Autocorrelation :

PSD :

5. Suppose and are independent random variables with and . Define random processes by

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

Similarly,

So, and are individually WSS, but not jointly WSS.