Estimation vs. detection.Let the signal

+1,with probability1X=2

1,with probability1,2

and the noiseZUnif[2,2] be independent random variables. Their sumY=X+Zis observed.

(a) Find the LMS estimate ofXgivenY

(b) Find the (overall) MSE for the estimator you find in part (a).

(c) Now suppose we use a decoder to decide whetherX= +1 orX=1 so that the probability of error is minimized. Find the MAP decoder and its probability of error. Compare the MAP decoder's MSE to the least MSE.

2. Estimation vs. detection. Let the signal X _ +1: With probability l 1, with probability 5, and the noise Z N Unif[2, 2] be independent random variables. Their sum Y = X + Z is observed. (a) Find the LMS estimate of X given Y (b) Find the (overall) MSE for the estimator you nd in part (a). (c) Now suppose we use a decoder to decide Whether X = +1 or X = 1 so that the probability of error is minimized. Find the MAP decoder and its probability of error. Compare the MAP decoder's MSE to the least MSE