final - EE 278 Statistical Signal Processing Handout#18...

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EE 278 Handout #18 Statistical Signal Processing Friday, Aug. 14, 2009 Final Exam This is a three hour exam. The total number of points is 100. The exam is open book and open notes. Hoewever, computer/internet use is not permitted. You can quote any results from the lecture notes or other class handouts without providing their derivations. However, you must make it clear what results you are using and why they apply. Please begin each problem solution on a new page. 1
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1. (13 points) Suppose that we have access to two noisy observations of a zero mean unit variance random variable X . Y 1 = X + Z 1 Y 2 = X + Z 2 where Z 1 and Z 2 are independent from X and have the following characteristics: E ( Z 1 ) = E ( Z 2 ) = 0; E ( Z 2 1 ) = E ( Z 2 2 ) = 1; E ( Z 1 Z 2 ) = ρ. Our goal is to estimate X linearly from Y 1 and Y 2 . (a) (9 points) Find the MMSE-optimal linear estimator of X given Y 1 and Y 2 and its MSE. Your answer will depend on ρ . (b) (4 points) For which value of ρ does the estimator perform the best? For which value of ρ does it perform the worst? 2. (15 points) For n 1, let X n be an exponentially distributed random variable, X n Exp( λ/n ), with λ > 0. Define the sawtooth function s ( x ) = x − ⌊ x , where
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