Question: 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

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.

Estimation vs. detection.Let the signal+1,with probability1X=21,with probability1,2and the noiseZUnif[2,2] be independent random

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

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