EECS 501_________________________PROBLEM SET #10__________________________Fall 2001

**ASSIGNED:** November 30, 2001. Read Stark and Woods pp. 468-492 on power spectral density.

**DUE DATE:** December 07, 2001. Also read pp. 442-449 on ergodicity (not required for below).

THIS WEEK: Applications of power spectral density.

- Stark and Woods #10.4. Write the RP as the sum of a constant mean and a 0-mean RP.

Then treat them separately. Note that the autocorrelation for the 0-mean RP is K_{x}(t).

An old Exam #3 problem involving power spectral densities. All RPs are 0-mean.

- WSS RP x(t) with PSD S
_{x}(w)=6/(w^{2}+16) is input into a LTI system

with impulse response h(t)=d(t)+e^{-3t}, t > 0, where d(t)=an impulse.

Compute the PSD S_{y}(w) and variance Var[y(t)] of the output y(t).

- We observe y(t)=x(t)+v(t) where E[x(t)v(s)]=0. v(t) is white with PSD S
_{v}(w)=5.

x(t) is a random telegraph wave from a Poisson process with average arrival rate 5.

Compute transfer function of infinite smoothing filter for estimating x(t) from y(s).

- A Gauss-Markov RP generated from system dx/dt+4x(t)=u(t) where S
_{u}(w)=1 is

input into a system with transfer function 1/(jw+3). Compute PSD S_{y}(w) of y(t).

- On problem set #8 you showed a WSS RP with periodic autocorrelation is itself periodic

with probability one. Interpret this result using the spectral interpretation of a WSS RP.

- See problem set handout.
- See problem set handout.