problem set4 - ELEC 531 STATISTICAL SIGNAL PROCESSING...

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ELEC 531: STATISTICAL SIGNAL PROCESSING Department of Electrical and Computer Engineering Rice University Fall 2007 Problem Set 4 Due: September 27, 2008 Problem 4.1 Let the observations be of the form Y = H θ + N where θ and N are statistically independent Gaussian random vectors. θ ∼ N ( 0 , K θ ) N ∼ N ( 0 , K n ) The vector θ has dimension M ; the vectors Y and N have dimension L . (a) Derive b θ MMSE , the minimum mean-squared error estimator of θ . (b) Show that this estimator and the optimum linear estimator b θ LMMSE are equal. (c) Find an expression for the mean-squared error for these estimators. Problem 4.2 Consider the communication system shown below. The message X is a N (0 , σ 2 X ) random variable. The transmitter output is hX , and the receiver input is Y = hX + N, where N in an N (0 , σ 2 N ) random variable that is statistically independent of Y . RCVR XMTR + X N hX Y X Figure 1: Suppose the transmitter is subject to intermittent failure, i.e. h is a random variable taking values 0 and 1 with probabilities 1 - p and p , respectively. Assume
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