EECS 564________________________PROBLEM SET #4________________________Winter 1999

ASSIGNED: January 29, 1999
DUE DATE: February 5, 1999

READ: Sections 4.1-4.2 and 9.1-9.2 (through p. 384) of Srinath (note the similarity).
For now, assume the basis functions simply sample the continuous waveforms.
Next, we will do the Karhunen-Loeve expansion (not in the text) and use other basis functions.
THIS WEEK: Uniformly most powerful (UMP) tests and antipodal signaling.

  1. Random variables m and n have distributions n~N(0,s) and Pr[m=± 1]=½. a is a constant.
      Under H0, we observe r=n. Under H1, we observe r=ma+n.
    1. Show there is a UMP test. Specify it and compute its power function PD.
    2. Now let random variable m have an arbitrary pdf pm(M).
      Specify a necessary and sufficient condition for pm(M) for a UMP test to exist.

  2. We observe random variable r, which is known to have a Poisson distribution.
    Under H0, the average arrival rate is 3.
    Under H1, the average arrival rate is unknown, except it is known to be greater than 3.
    Show that a UMP test exists, and specify it. You need not specify the threshold.

  3. Let x and y be 2-vectors, each having energy ||x||2=||y||2=E.
    n is a zero-mean Gaussian random 2-vector with variances s2 and correlation coefficient p.
      Under H1, we observe r=x+n. Under H0, we observe r=y+n.
    1. Compute the sufficient statistic L in terms of s2,p, signals x,y and observation R.
    2. Define the Fisher discriminant as d2=(E[L|H1]-E[L|H0])2 /Var[L|H0].
      Choose x,y to maximize the Fisher discriminant d2, subject to x and y having energy E.
    3. Compute and sketch an expression for the optimal value of d2 as a function of p.
      Compute this expression for p=0 and ± 1, and interpret these results.

  4. We observe r(t)=s(t)+n(t) for 0< t< T, where s(t) is a known signal and n(t) is white noise.
    We process r(t) by filtering it with an impulse response h(t) and sampling the result at t=T.
    We wish to choose the filter h(t) in order to maximize the signal-to-noise ratio
    SNR=E[((h(t)*s(t))|t=T)2]/ E[((h(t)*n(t))|t=T)2].
    Use the Cauchy-Schwarz inequality to determine the optimal h(t) in terms of s(t).
    Note both this "matched filter" and the Cramer-Rao bound come from the same inequality.

  5. Let K be an NXN symmetric positive semi-definite matrix with (i,j)th element Kij.
    K has eigenvalues e1...eN and associated eigenvectors v1...vN.
    Prove the finite-dimensional version of Mercer's theorem (to be used shortly):
    Kij=e1v1(i)v1(j)+e2v2(i)v2(j) +e3v3(i)v3(j)+...+eNvN(i)vN(j).

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