131A_1_week10sol

131A_1_week10sol - 2. Let Y=X+N, where X and N are...

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• 2. Let Y=X+N, where X and N are independent zero- mean Gaussian rv’s with different variances. – a. Find the mmse linear estimator for Y 22 2 [] [ ] [ ] 0 (, ) [ ] () XN X XX XY N EY EX EN Var Y Var X Var N EXY COV X Y E XY σσ σ ρ =+= + = == = = + , 2 ˆ ([ ] ) [ ] Y X XY X YX E X E Y σσ σ =− + mmse linear estimator: 2 [( ) ] [ ] N mse E Y X E N =−= =
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b. Find the estimator that when observing X=x predicts the value of y that maximizes f Y (y|x) 2 2 () 2 (|) [ | ] [ ] 1 2 N Y yx Y N F yx PY yX x Px N y PN y x fy x e σ πσ =≤ = = + = ⇒= We know the peak value of the Gaussian pdf occurs at y=mean=x. So the estimator is ˆ YX =
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• d. Find the mmse linear estimator for X , 22 2 2 ˆ ([ ] ) [ ] X XY Y XX X X XY Y Y X N E Y E X YY Y σ ρ σσ σσ σ =− + == = + 2 2 N NX XN ˆ [( ) ] [( ) ] [| ] 0 if 0 if if X mse E X X E X Y VAR X Y + = = + →∞ Mean Square Error of this estimator for X:
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e. Find the estimator that when observing Y=x predicts the value of y that maximizes f X (x|y) 2 2 2 2 2 22 2 2 2 () 2 2 , 2( ) , ( ) 2 11 (, ) () ( | ) 1 (,) ) 1 (|) ( ) / ( ) ) 1 ex ) N X XN NX N X yx x XY X Y y YX Y y x X fx yf x f y x e e fy f
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This note was uploaded on 02/09/2011 for the course EE 131A taught by Professor Lorenzelli during the Spring '08 term at UCLA.

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131A_1_week10sol - 2. Let Y=X+N, where X and N are...

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