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Homework3

Module by: Roy Ha, Dinesh Rajan, Mohammad Borran, Behnaam Aazhang. E-mail the authors

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

Problem 1

Suppose that a white Gaussian noise X t X t is input to a linear system with transfer function given by

H⁢f={1  if  |f|≤20  if  |f|>2 H f 1 f 2 0 f 2
(1)
Suppose further that the input process is zero mean and has spectral height N 0 2=5 N 0 2 5 . Let Y t Y t denote the resulting output process.

a) Find the power spectral density of Y t Y t . Find the autocorrelation of Y t Y t (i.e., R Y τ R Y τ).

b) Form a discrete-time process (that is a sequence of random variables) by sampling Y t Y t at time instants T seconds apart. Find a value for T such that these samples are uncorrelated. Are these samples also independent?

Figure 1
Figure 1 (HW3Fig1.png)

What is the variance of each sample of the output process?

Problem 2

Suppose that X t X t is a zero mean white Gaussian process with spectral height N 0 2=5 N 0 2 5 . Denote Y t Y t as the output of an integrator when the input is Y t Y t .

Figure 2
Figure 2 (HW3Fig2.png)
  • a) Find the mean function of Y t Y t . Find the autocorrelation function of Y t Y t , R Y ⁢t+τt R Y t τ t
  • b) Let Z k Z k be a sequence of random variables that have been obtained by sampling Y t Y t at every T seconds and dumping the samples, that is
    Z k =∫(k−1)⁢Tk⁢T X τ dτ Z k τ k 1 T k T X τ
    (2)
    Find the autocorrelation of the discrete-time processes Z k Z k 's, that is, R Z ⁢k+mk=E⁢ Z k+m ⁢ Z k R Z k m k E Z k+m Z k
  • d) Is Z k Z k a wide sense stationary process?

Problem 4

Proakis and Salehi problem 3.63 parts 1, 3, and 4

Problem 5

Proakis and Salehi problem 3.54

Problem 6

Proakis and Salehi problem 3.62

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