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# Homework 2 of Elec 430

Module by: Behnaam Aazhang. E-mail the author

Summary: (Blank Abstract)

Elec 430 homework set 2. Rice University Department of Electrical and Computer Engineering.

## Problem 1

Suppose A and B are two Gaussian random variables each zero mean with A2-< A 2 and B2-< B 2 . The correlation between them is denoted by AB- A B . Define the random process X t =A+Bt X t A B t and Y t =B+At Y t B A t .

• a) Find the mean, autocorrelation, and crosscorrelation functions of X t X t and Y t Y t .
• b) Find the 1st order density of X t X t , f X t x f X t x
• c) Find the conditional density of X t 2 X t 2 given X t 1 X t 1 , f X t 2 | X t 1 x 2 | x 1 f X t 2 | X t 1 x 2 | x 1 . Assume t 2 > t 1 t 2 t 1

### Hint:

see Proakis and Salehi problem 3.28
• d) Is X t X t wide sense stationary?

## Problem 2

Show that if X t X t is second-order stationary, then it is also first-order stationary.

## Problem 3

Let a stochastic process X t X t be defined by X t =cosΩt+Θ X t Ω t Θ where ΩΩ and ΘΘ are statistically independent random variables. ΘΘ is uniformaly distributed over π π and ΩΩ has an unknown density f Ω ω f Ω ω .

• a) Compute the expected value of X t X t .
• b) Find an expression for the correlation function of X t X t .
• c) Is X t X t wide sense stationary? Show your reasoning.
• d) Find the first-order density function f X t x f X t x .

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#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

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