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hw8 - EE 378 Statistical Signal Processing Homework Set#8...

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EE 378 Handout #17 Statistical Signal Processing Wednesday, May 30, 2007 Homework Set #8 Due: Wednesday, June 6, 2007. Announcement: You can hand in the HW either after class or deposit it, before 5pm, in the Homework in box in the 378 drawer of the class file cabinet on the second floor of the Packard Building. 1. Let Y be a an N-length vector with independent elements (i.e. Y 1 ,... Y N are indepen- dent). Assume Y i , 1 i N , is Poisson with parameter θ . (a) Compute the Cramer-Rao bound for estimating θ based on Y . (b) Find ˆ θ ML , the maximum-likelihood estimate of θ based on Y . (c) Is ˆ θ ML unbiased? (d) Is ˆ θ ML consistent? (e) Is ˆ θ ML efficient? Now, instead assume Y i , 1 i N , is Laplacian with parameter θ , i.e. f ( y i ) = 1 2 θ e | y i | θ , and that ˆ θ = (1 /N + a/N 2 ) N i =1 | Y i | . (f) Is ˆ θ unbiased? (g) Find MSE ( ˆ θ ). (h) Find the value of a that minimizes MSE ( ˆ θ ). (i) Repeat parts (a) through (e) for the Laplacian case, with the given ˆ θ and the value a found in part (h). (j) There is an efficient estimator ˆ θ ( Y ) such that ˆ θ ( Y ) = θ + parenleftbigg 1 I( θ ) parenrightbigg parenleftbigg ln f θ ( Y ) ∂θ parenrightbigg . For both cases (the Poisson and the Laplacian), find the ˆ θ ( Y ) using this formula. Explain your results. 2. In order to estimate N 0 / 2, the amplitude of the power spectral density of W ( t ), a zero- mean, white Gaussian noise process, we use { θ i ( t ) , 0 t T } , 1 i n , a family of n real, orthonormal functions, i.e. integraldisplay T 0 θ i ( t ) θ j ( t ) dt, = δ ij , 1 i, j n. 1
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We calculate Y i = integraldisplay T 0 W ( t ) θ i ( t ) dt, 1 i n, and estimate N 0 / 2 based on Y 1 , ...Y n . (a) Find the maximum likelihood estimate of N 0 2 based on Y i , 1 i n . (b) Express the bias and the MSE of the estimator as a function of N 0 and n . (c) Compute the Cramer-Rao bound. Determine whether the estimator is efficient. (d) Define a function sequence { φ i ( t ) , 0 t T } , 1 i n , where φ i ( t ) = αφ i 1 ( t ) + θ i ( t ) , 1 i n, where θ 0 ( t ) = 0 and α is a real number. Repeat parts (a) through (c) with Y replaced by Z , where Z i = integraldisplay T 0 W ( t ) φ i ( t ) dt, 1 i n.
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