Midterm-2008 - Detection and Estimation Midterm, 2008 1....

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Detection and Estimation Midterm, 2008 1. (35%) Assume we have observed N samples x[0], x[1], …, x[N-1], whose model is expressed as 1 - N ..., 1, , 0 ], [ ) 2 cos( ] [ 0 = + = n n w f n A n x π . ( Case 1 ) Assume the frequency f 0 is known, and the noise w[n] is white Gaussian with unknown variance σ 2 . In this case, we are interested in estimating the parameter vector θ 1 = [A, σ 2 ] T . (a) Find the Fisher information matrix I( θ 1 ) . (12 pts) (b) Assume A ˆ and 2 ˆ σ are unbiased estimators of A and σ 2 , respectively. Find the lower bounds for ) ˆ var( A and ) ˆ var( 2 . ( 8 p t s ) (c) Assume the signal noise ratio is defined as 2 2 ) 2 ( α A = and ˆ is an unbiased estimator of . Find the lower bound of ) ˆ var( . (5 pts) ( Case 2 ) Assume the amplitude A and the frequency f 0 are unknown. The noise w[n] is white Gaussian with known but varying variance var(w[n]) = σ n 2 . In this case, we are interested in estimating the parameter vector θ 2 = [A, f 0 ] T
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Midterm-2008 - Detection and Estimation Midterm, 2008 1....

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