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hw5 - EE 278 Statistical Signal Processing Homework#5 Due...

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EE 278 October 21, 2009 Statistical Signal Processing Handout #8 Homework #5 Due: Wednesday October 28 1. Additive-noise channel with path gain. Consider the additive noise channel shown in the figure below, where X and Z are zero mean and uncorrelated, and a and b are constants. a b X Z Y = b ( aX + Z ) Find the MMSE linear estimate of X given Y and its MSE in terms only of σ X , σ Z , a , and b . 2. Shot Noise Channel. Consider an additive noise channel with input signal X U(0 , 1) and output signal Y = X + Z , where the noise Z |{ X = x } ∼ N (0 , ax ), for some constant a > 0, i.e., the noise variance is proportional to the signal. Observing Y , find the minimum MSE linear estimate of X . Your answer should be in terms only of a . 3. Worst noise distribution. Consider an additive noise channel with signal X ∼ N (0 , P ) and noise Z with zero mean and variance N . Assume X and Z are independent and the output is Y = X + Z . Specify a distribution of Z that maximizes the minimum MSE of estimating X given Y , i.e., the distribution of the worst noise Z that has the given mean and variance. You need to justify your answer. 4. Camera measurement. (Bonus) The measurement from a camera can be expressed as Y = AX + Z , where X is the object position with mean μ and variance σ 2 X , A is the occlusion indicator function and is equal to 1 (if the camera can see the object) with probability p , and 0 (if the camera cannot see the object) with probability (1 - p ), and Z is the measurement error with mean 0 and variance σ 2 Z . Assume that

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