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Homework 2 of Elec 430

Module by: Behnaam Aazhang. E-mail the author

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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 -ππ 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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