Homework 3 of Elec 430 2.8 2001/10/05 2004/01/07 10:15:05.746 US/Central Behnaam Aazhang aaz@ece.rice.edu Dinesh Rajan dinesh@ece.rice.edu Mohammad Jaber Borran mohammad@ece.rice.edu Behnaam Aazhang aaz@ece.rice.edu Shawn Stewart mrshawn@alumni.rice.edu Roy Ha rha@rice.edu homework (Blank Abstract) Suppose that a white Gaussian noise X t is input to a linear system with transfer function given by H f 1 f 2 0 f 2 Suppose further that the input process is zero mean and has spectral height N 0 2 5 . Let Y t denote the resulting output process. Find the power spectral density of Y t . Find the autocorrelation of Y t (i.e., R Y τ ). Form a discrete-time process (that is a sequence of random variables) by sampling Y t at time instants T seconds apart. Find a value for T such that these samples are uncorrelated. Are these samples also independent? What is the variance of each sample of the output process?

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

Find the mean function of Y t . Find the autocorrelation function of Y t , R Y t τ t Let Z k be a sequence of random variables that have been obtained by sampling Y t at every T seconds and dumping the samples, that is Z k τ k 1 T k T X τ Find the autocorrelation of the discrete-time processes Z k 's, that is, R Z k m k E Z k+m Z k Is Z k a wide sense stationary process? Proakis and Salehi, problem 3.63, parts 1, 3, and 4 Proakis and Salehi, problem 3.54 Proakis and Salehi, problem 3.62