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Homework 3 of Elec 430

Module by: Behnaam Aazhang. E-mail the author

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Exercise 1

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

Hf=1if|f|≤20if|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.

  1. 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 ).
  2. Form a discrete-time process (that is a sequence of random variables) by sampling Y t Y t at time instants T T seconds apart. Find a value for T T such that these samples are uncorrelated. Are these samples also independent?
  3. What is the variance of each sample of the output process?
Figure 1
Figure 1 (HW3Fig1.png)

Exercise 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)
  1. 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
  2. Let Z k Z k be a sequence of random variables that have been obtained by sampling Y t Y t at every T T seconds and dumping the samples, that is
    Z k =∫k−1TkT 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
  3. Is Z k Z k a wide sense stationary process?

Exercise 3

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

Exercise 4

Proakis and Salehi, problem 3.54

Exercise 5

Proakis and Salehi, problem 3.62

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