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.
  1. 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 Kx(t).
    1. An old Exam #3 problem involving power spectral densities. All RPs are 0-mean.
    2. WSS RP x(t) with PSD Sx(w)=6/(w2+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 Sy(w) and variance Var[y(t)] of the output y(t).
    3. We observe y(t)=x(t)+v(t) where E[x(t)v(s)]=0. v(t) is white with PSD Sv(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).
    4. A Gauss-Markov RP generated from system dx/dt+4x(t)=u(t) where Su(w)=1 is
      input into a system with transfer function 1/(jw+3). Compute PSD Sy(w) of y(t).
    5. 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.

  2. See problem set handout.
  3. See problem set handout.