# Berkeley ELENG 226A - EE 226 Problem Set (10 pages)

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## EE 226 Problem Set

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- Pages:
- 10
- School:
- University of California, Berkeley
- Course:
- Eleng 226a - Random Processes in Systems

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EE226 Random Processes in Systems Fall 06 Problem Set 5 Due Oct 24 Lecturer Jean C Walrand GSI Assane Gueye This problem set essentially reviews estimation theory and the Kalman filter Not all exercises are to be turned in Only those with the sign F are due on Tuesday October 24th at the beginning of the class Although the remaining exercises are not graded you are encouraged to go through them We will discuss some of the exercises during discussion sections Please feel free to point out errors and notions that need to be clarified Exercise 5 1 Let Y H X Z where X is some Gaussian random vector in Rn with zero mean and covariance matrix KX H is a non singular n n known matrix and Z is a gaussian random noise with zero mean and non singular covariance matrix KZ uncorrelated to X a Find the MMSE estimator of X given Y b Explain what happens in the case where KZ 0 c Repeat part b when both H and KZ are singular Solution Hint a Since the matrix H is invertible we can obtain a simpler equation Y H 1 Y X H 1 Z X Z where Z is again Gaussian N 0 H 1 KZ H 1 T note the covariance matrix of Z is not singular Since we have only Gaussian random vectors the MMSE of X given Y is given by using formula 1 1 X KX KX H 1 KZ H 1 T H Y b Note that if KZ 0 then the covariance of Z is singular So without loss of generality we can discuss the transformed observation Y The singularity of the covariance matrix indicates that the noise lies in a lower dimensional space then the signal X which lies in Rn The components of X that are in the subspace of the noise will be perturbed but the components that are orthogonal to the noise can be detected without error A more rigorous analysis using eigenvalue decomposition will involve projection into the subspace of the noise and into the orthogonal to that subspace c If H is singular we cannot make anymore transformation of the observation but still we can do the same analysis The effect of multiplying by H is to reduce the signal into a lower

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