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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 2 2 2 2 2 2 2 [ ] [ ] [ ] 0 [ ] [ ] [ ] [ ] ( , ) [ ] ( ) X N X X X X Y X Y X Y X X N E Y E X E N Var Y Var X Var N E XY COV X Y E XY σ σ σ σ σ ρ σ σ σ σ σ σ σ σ σ = + = = + = + = = = = = + , 2 ˆ ( [ ]) [ ] Y X Y X X Y X Y X Y X E X E Y X X σ ρ σ σ σ σ σ σ = + = = mmse linear estimator: 2 2 2 [( ) ] [ ] N mse E Y X E N σ = = =
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