1.3 Homework 3 of elec 430

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

• Find the power spectral density of $Y_t$ . Find the autocorrelation of $Y_t$ ( i.e. , $R_Y()$ ).
• 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 $\frac{N_0}{2}=5$ . 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 $Y_t$ , $R_Y(t+, t)$
• 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
  $$Z_k=\int_{(k-1)T}^{kT} X_{}\,d$$
  Find the autocorrelation of the discrete-time processes $Z_k$ 's, that is, $R_Z(k+m, k)=E(Z_{\mathrm{k+m}}{Z}_k)$
• 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