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.

- We observe n independent identically distributed random variables (iidrvs) r
_{1}...r_{n},

where each r_{i}is Gaussian with mean A_{1}and variance A_{2}. A_{1}and A_{2}are unknown constants.

Compute the joint maximum likelihood estimates of A_{1}and A_{2}, each using R_{1}..R_{n}.

Show that the estimate of A_{1}is unbiased, but the estimate of A_{2}is biased.

- We wish to transmit 2 constants A
_{1}and A_{2}over an unsecure communications channel.

The transmitted signals are s_{1}=x_{11}A_{1}+x_{12}A_{2}and s_{2}=x_{21}A_{1}+x_{22}A_{2}where x_{ij}are known.

The received signals are r_{1}=s_{1}+n_{1}and r_{2}=s_{2}+n_{2}where n_{1},n_{2}~N(0,s_{n}) are independent.- Compute the
maximum likelihood estimates of A
_{1}and A_{2}, each using R_{1}and R_{2}. - Show that each estimator is unbiased, and compute their error covariance matrix.
- Compute the vector Cramer-Rao bound, and show that the estimators are efficient.

- Compute the
maximum likelihood estimates of A
- We observe K iidrvs r
_{1}...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 s^{2}I. m is:

H_{0}: [A,0,B]; H_{1}: [0,A,B]; H_{2}: [-A,0,B]; H_{3}: [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
- 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).

*thought*(uh-oh!)):

- Compute the log-likelihood log p
- 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.

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