EECS 564________________________PROBLEM SET #3________________________Winter 1999

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ASSIGNED: January 22, 1999
DUE DATE: January 29, 1999

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READ: Sections 5.5-5.6 (estimation) and 3.6 and 4.6 (UMP and GLRT) of Srinath.
We will skip Chapter 4 until after we have studied the Karhunen-Loeve expansion.
THIS WEEK: Multichannel estimators and their Cramer-Rao bounds and performance.

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1. We observe n independent identically distributed random variables (iidrvs) r₁...r_n,
   where each r_i is Gaussian with mean A₁ and variance A₂. A₁ and A₂ are unknown constants.
   Compute the joint maximum likelihood estimates of A₁ and A₂, each using R₁..R_n.
   Show that the estimate of A₁ is unbiased, but the estimate of A₂ is biased.
   

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2. We wish to transmit 2 constants A₁ and A₂ over an unsecure communications channel.
   The transmitted signals are s₁=x₁₁A₁+x₁₂A₂ and s₂=x₂₁A₁+x₂₂A₂ where x_ij are known.
   The received signals are r₁=s₁+n₁ and r₂=s₂+n₂ where n₁,n₂~N(0,s _n) are independent.
   
   1. Compute the maximum likelihood estimates of A₁ and A₂, each using R₁ and R₂.
   2. Show that each estimator is unbiased, and compute their error covariance matrix.
   3. Compute the vector Cramer-Rao bound, and show that the estimators are efficient.

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3. We observe K iidrvs r₁...r_K where each observation r is a 3-vector r=m+n.
   The noise vector n is a 0-mean Gaussian random vector with covariance matrix s²I. m is:
   H₀: [A,0,B]; H₁: [0,A,B]; H₂: [-A,0,B]; H₃: [0,-A,B].
   The criterion is MEP (min Pr[error]), and the 4 hypotheses are equally likely a priori.
   (1) Draw the decision regions in a 2-D plane. (2) Compute the attained Pr[error].

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   Some hints for this problem (which requires some thought (uh-oh!)):
   • Compute the log-likelihood log p_r|Hi(R|H_i) and discard all terms independent of i and R.
   • Show the dot products of r with each of 4 mean vectors are sufficient statistics.
   • Use this in turn to come up with two very simple sufficient statistics.
   • Compute Pr[error] by rotating the plane 45° (a common trick in communication).

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4. We observe the N-vector r=a+n where a~N(0,K _a) is independent of n~N(0,K _n).
   (1) Compute a_MAP(R). (2) Show a_MAP(R) is efficient. (3) Compute error covariance matrix.

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"A cynic is someone who, when he smells roses, looks around for a coffin"-Oscar Wilde