Midterm Solution 2008 - Detection & Estimation Spring,...

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Detection & Estimation Spring, 2008 Midterm Exam Solution 1. (a) = = 4 1 0 2 0 2 1 2 0 0 )) 2 (cos( 1 ) ( σ π θ N n f N n I (b) = 1 0 2 0 2 )) 2 (cos( ) ˆ var( N n n f A N 4 2 2 ) ˆ var( (c) 4 4 1 0 2 0 2 2 2 )) 2 (cos( ) ˆ var( α N A n f A N n + = (d) = = = = = 1 0 2 0 2 2 2 2 1 0 2 0 1 0 2 0 1 0 2 0 2 2 ) 2 ( sin 4 ) 4 sin( ) 4 sin( ) 2 ( cos ) ( N n n N n n N n n N n n nf A n nf A n nf A n nf I 2. (a) = 1 0 2 ] [ N n n x n (b) = 1 0 2 ] [ N n n x n , = 1 0 ] [ N n n nx (c) = 1 0 2 ] [ N n n x n , = 1 0 ] [ N n n nx , = 1 0 ] [ N n n x (d) = 1 0 2 ] [ N n n x n , = 1 0 ] [ N n n nx , = 1 0 ] [ N n n x , = 1 0 2 ] [ N n n x
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3. (a) = = 1 0 1 0 2 ] [ 1 ˆ N n n N n n e n x e A λ , A A E = ) ˆ ( , = = 1 0 2 2 ) ˆ var( N n n e A σ (b) = = 1 0 1 0 2 ] [ 1 ˆ N n n N n n e n x e A , = = 1 0 2 2 ) ˆ var( N n n e A (c) = = 1 0 1 0 2 ] [ 1 ˆ N n n N n n e n x e A , A A E = ) ˆ ( , = = 1 0 2 2 ) ˆ var( N n n e A (d) = = 1 0 1 0 2 ] [ 1 ˆ N n n N n n e n x e A , = = 1 0 2 2 ) ˆ var( N n n e A (e) = = 1 0 1 0 2 ] [ 1 ˆ N n n N n n e n x e A , = = 1 0 2 2 ) ˆ ] [ ( 1 ˆ N n n e A n x N A A E = ) ˆ ( , 2 2 ) 1 1 ( ) ˆ ( N E = (f)
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Midterm Solution 2008 - Detection & Estimation Spring,...

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