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## Utah State University ECE 6010 Stochastic Processes Homework # 8 Solutions

1. Suppose is a Wiener process. Define a process by for a fixed positive number .
1. Find the mean and autocorrelation functions of .
Mean :

Autocorrelation :

So .

2. Show that is a stationary and find its spectrum.
is a constant and W.S.S.

Since Gaussian too Strictly Stationary.

So,

2. Suppose and are zero mean and individually and jointly W.S.S. Show that the mean-square error associated with the noncausal Wiener filter for estimation of from is

3. Suppose for , where and are zero-mean, W.S.S., and orthogonal. Suppose that we wish to estimate , with an estimate of the form , where and are impulse responses of linear time-invariant systems. show that

where and are the transfer functions of and , respectively, and and are the power spectral densities of and . (Note the case that for some fixed .)

4. Consider the situation of the previous problem with ,

1. Find the noncausal Wiener filter for estimating from . Find the corresponding mean-square error.

When ,

and

From the previous problem

2. Find the causal Wiener filter for estimating from . Consider and .

We have

where

so

Now

Taking the inverse Laplace transform,

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Let

(the thing whose we need to compute. Then

where . Then

For we have delay, so that the filter performs smoothing .

Transforming

Then