EECS 564________________________PROBLEM SET #5________________________Winter 1999

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ASSIGNED: February 05, 1999
DUE DATE: February 12, 1999

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READ: Section 2.5 (Karhunen-Loeve expansions) and Sections 4.3-4.6 (colored noise) of Srinath.
Read Section 4.5 on integral equations (pertinent now); skip the references to spectral factorization.
THIS WEEK: Karhunen-Loeve expansions. We will use these for detection/estimation of signals.

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1. A simple but illustrative example of asymptotic efficiency of ML estimation:
   We observe r=A²+n where A > 0 is an unknown constant and n~N(0,s).
   1. Compute Â_MLE(R). Be sure to consider R < 0 as a separate case.
   2. Compute the Cramer-Rao lower bound on unbiased estimates Â(R).
   3. Using MATLAB, and letting the actual value of A=5, do the following:
      • Use MATLAB function "randn" to generate realizations of n;
      • Compute the square error (A-Â(R))² for 100 realizations of n;
      • Plot EACH square error as a dot, and their MEAN as an X, for each of the following
      • values of noise variance s: s=10⁶,10⁵,10⁴,10³,10²,10,1. Plot on a log-log plot;
      • Plot the Cramer-Rao bound from (b) as a line on the same plot.
   4. Means should equal the C-R bound for smaller s, and lie BELOW the bound for larger s.
      Explain BOTH of these results.

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2. Prove that the largest eigenvalue of an integral equation is bounded below by
   &int &int f(t)K(t,u)f(u)dtdu, where f(t) is any function such that ||f(t)||=1.
   HINT: What projection &int f(t)x(t)dt of x(t) has the biggest variance (energy)?

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3. Let x(t) have total energy ||x(t)||²=&int |x(t)|²dt=E. Let X(f) be the Fourier transform of x(t).
   • Time-limit x(t) to |t| < T/2; call this x_T(t)=x(t) for |t| < T/2; 0 for |t| > T/2.
   • Band-limit x_T(t) to W Hertz; call this x_DB(t). X_DB(f)=X_T(f) for |f| < W; 0 for |f| > W.
   • Let E_DB=||x_DB(t)||² < E be the energy in x_DB(t). What f(t) maximizes the ratio E_DB/E?
   HINT: Use EITHER of the following:
   1. Parseval's theorem, convolution with a sinc, and the Cauchy-Schwarz inequality;
   2. The results of problem #2 above and a little thought (your choice!)

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4. Find the eigenvalues and eigenfunctions over the interval 0 < t < T < 1 of the covariance function
   K(t,s)=1-|t-s| for 0 < |t-s| < 1; K(t,s)=0 for |t-s| > 1. That is, solve the integral equation (ugh).

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5. Prove that the eigenfunctions of a covariance function are orthogonal if their eigenvalues differ.
   (This is required to show solving the integral equation is sufficient to find the K-L expansion.)
   HINT: Modify the proof of the analogous result for matrices.

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6. We observe random process r(t)=s(t)+n(t) under H₁; r(t)=n(t) under H₀, 0 < t < T,
   where s(t) is a known signal, and n(t) is a zero-mean white Gaussian noise random process.
   
   We now formally derive the matched filter for additive white Gaussian noise:
   1. Expand r(t) in a Karhunen-Loeve expansion whose first basis function is s(t).
   2. Show that coordinate random variables r_i, i > 1, are irrelevant to this detection problem.
   3. Derive the LRT for r₁, and re-derive the matched filter from Problem Set #4.

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"Diplomacy is the art of saying 'nice doggie' until you can find a stick."