Electronics > QUESTIONS & ANSWERS > University of California, San DiegoECE 153hw5sol (All)
UCSD ECE153 Handout #27 Prof. Young-Han Kim Tuesday, May 6, 2014 Solutions to Homework Set #5 (Prepared by TA Fatemeh Arbabjolfaei) 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(sgn(X) − g(Y ))2 , where sgn(x) = (− +11 x x >≤ 0 0. (b) Plot g(y) vs. y. Solution: The minimum MSE is achieved when g(Y ) = E(sgn(X) | Y ). We have g(y) = E(sgn(X) | Y = y) = Z−∞ ∞ sgn(x)fX|Y (x|y) dx . To find the conditional pdf of X given Y , we use fX|Y (x|y) = fY |X(y|x)fX(x) fY (y) , where fX(x) = (0 otherwise 12 −1 ≤ X ≤. 1 Since X and Z are independent, fY |X(y|x) = fZ(y − x) ⇒ Y | {X = x} ∼ N(x, 1) . To find fY (y) we integrate fY |X(y|x)fX(x) over x : fY (y) = Z−∞ ∞ fY |X(y|x)fX(x) dx = Z−11 2√12π e− (y−2x)2 dx = 1 2 Z−11 √12π e− (x−2y)2 dx = 1 2 Z−∞1 √12π e− (x−2y)2 dx −Z1 ∞√12π e− (x−2y)2 dx = 12 (Q(−y − 1) − Q(−y + 1)) . Combining the above results, we get [Show More]
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