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hw5 - UCSD ECE153 Prof Young-Han Kim Handout#18 Thursday...

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UCSD ECE153 Handout #18 Prof. Young-Han Kim Thursday, April 28, 2011 Homework Set #5 Due: Thursday, May 5, 2011 1. Neural net. Let Y = X + Z , where the signal X U[ - 1 , 1] and noise Z ∼ N (0 , 1) are independent. (a) Find the function g ( y ) that minimizes MSE = E bracketleftbig (sgn( X ) - g ( Y )) 2 bracketrightbig , where sgn( x ) = braceleftBigg - 1 x 0 +1 x > 0 . (b) Plot g ( y ) vs. y . 2. Additive shot noise channel. Consider an additive noise channel Y = X + Z , where the signal X ∼ N (0 , 1), and the noise Z |{ X = x } ∼ N (0 , x 2 ), i.e., the noise power of increases linearly with the signal squared. (a) Find E ( Z 2 ). (b) Find the best linear MSE estimate of X given Y . 3. Estimation vs. detection. Let the signal X = braceleftbigg +1 , with probability 1 2 - 1 , with probability 1 2 , and the noise Z Unif[ - 2 , 2] be independent random variables. Their sum Y = X + Z is observed. (a) Find the best MSE estimate of X given Y and its MSE. (b) Now suppose we use a decoder to decide whether X = +1 or X = - 1 so that the probability of error is minimized. Find the optimal decoder and its probability of error. Compare the optimal decoder’s MSE to the minimum MSE.
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