EECS 564________________________PROBLEM SET #3________________________Winter 1999

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

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.

1. We observe n independent identically distributed random variables (iidrvs) r1...rn,
where each ri 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..Rn.
Show that the estimate of A1 is unbiased, but the estimate of A2 is biased.

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 xij 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.

3. We observe K iidrvs r1...rK 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].
Some hints for this problem (which requires some thought (uh-oh!)):
• Compute the log-likelihood log pr|Hi(R|Hi) 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).

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 aMAP(R). (2) Show aMAP(R) is efficient. (3) Compute error covariance matrix.

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