s4color - ECE 580 / Math 587 SPRING 2011 Correspondence #...

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ECE 580 / Math 587 SPRING 2011 Correspondence # 10 March 15, 2011 Solution Set 4 29. i) Let M = z L 2 (Ω , P , IR n ) : z = i j =0 K j y j ; K j ∈ M nm . This is a closed linear subspace of L 2 (Ω , P ; IR n ) which is a Hilbert space. Hence, from the Projection Theorem, there exists a unique ˆ x M , expressed as ˆ x = ± i j =0 ˆ K j y j , with inf K j ∈M nm j =0 ,...,i x i j =0 K j y j = x ˆ x and a necessary and sufficient condition for ˆ x to be the minimizing solution is ( x ˆ x,z ) = 0 z M E [ x T QK y ] = i j =0 E [ y T j ˆ K T j QK y ] K ∈ M nm = 0 , 1 ,...,i Tr { E [ y x T QK ]} = i j =0 Tr { E [ y y T j ˆ K T j QK ]} K ∈ M nm . Since Q is positive definite, QK ∈ M nm whenever K ∈ M nm , and vice versa, and hence the earlier condition is Tr Λ ℓx i j =0 Λ ℓj ˆ K T j K = 0 K ∈ M nm = 0 , 1 ,...,i where Λ ℓx = E [ y x T ] ; Λ ℓj = E [ y y T j ] . Now, two random vectors are uncorrelated if their components (considered as random vari- ables) are uncorrelated. Furthermore, since E [ y ] = 0, we have Λ ℓj = ² Λ ℓℓ j = 0 otherwise . Hence, the condition now becomes Tr {( Λ ℓx Λ ℓℓ ˆ K T ) K } = 0 K ∈ M nm , ℓ = 0 ,...,i ;
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and assuming that Λ ℓℓ is invertible ( = 0 ,...,i ), we have a unique solution ˆ K = Λ T ℓx Λ - 1 ℓℓ , = 0 , 1 ,...,i. Note: Uniqueness of the ˆ K j ’s follows from the fact that Tr [ AB ] = 0 B A is a matrix with only zero entries (i.e., the zero matrix). If Λ
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This note was uploaded on 10/11/2011 for the course ECE 580 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.

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s4color - ECE 580 / Math 587 SPRING 2011 Correspondence #...

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