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

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.

- Random variables m and n have distributions n~N(0,s) and Pr[m=± 1]=½. a is a constant.
- Under H
- Show there is a UMP test. Specify it and compute its power function P
_{D}. - Now let random variable m have an arbitrary pdf p
_{m}(M).

Specify a necessary and sufficient condition for p_{m}(M) for a UMP test to exist.

_{0}, we observe r=n. Under H_{1}, we observe r=ma+n.

- Show there is a UMP test. Specify it and compute its power function P
- We observe random variable r, which is known to have a Poisson distribution.

Under H_{0}, the average arrival rate is 3.

Under H_{1}, 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.

- 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 s^{2}and correlation coefficient p.

- Under H
- Compute the sufficient statistic L in terms of s
^{2},p, signals x,y and observation R. - Define the
*Fisher discriminant*as d^{2}=(E[L|H_{1}]-E[L|H_{0}])^{2}/Var[L|H_{0}].

Choose x,y to maximize the Fisher discriminant d^{2}, subject to x and y having energy E. - Compute and sketch an expression for the optimal value of d
^{2}as a function of p.

Compute this expression for p=0 and ± 1, and*interpret*these results.

_{1}, we observe r=x+n. Under H_{0}, we observe r=y+n.

- Compute the sufficient statistic L in terms of s
- 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.

- Let K be an NXN symmetric positive semi-definite matrix with (i,j)
^{th}element K_{ij}.

K has eigenvalues e_{1}...e_{N}and associated eigenvectors v_{1}...v_{N}.

Prove the finite-dimensional version of Mercer's theorem (to be used shortly):

K_{ij}=e_{1}v_{1}(i)v_{1}(j)+e_{2}v_{2}(i)v_{2}(j) +e_{3}v_{3}(i)v_{3}(j)+...+e_{N}v_{N}(i)v_{N}(j).

"Football is the worst of America: Violence punctuated by committee meetings"-George Will.