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by: Deondre Ullrich


Deondre Ullrich
Rice University
GPA 3.54


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Class Notes
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This 1 page Class Notes was uploaded by Deondre Ullrich on Monday October 19, 2015. The Class Notes belongs to ELEC 535 at Rice University taught by Staff in Fall. Since its upload, it has received 27 views. For similar materials see /class/224979/elec-535-rice-university in Electrical Engineering & Computer Science at Rice University.

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Date Created: 10/19/15
14 The Data Processing Theorem The Data Processing Theorem compares the Kullback Leibler distance relative entropy be tween the probability distributions of two inputs to a system to the distance between the cor responding output distributions The Data Processing Theorem says the ratio of the output distance to the input distance must be less than or equal to one Suppose Xn is a sequence of statistically independent Gaussian random variables having some mean and variance 0 The system simply adds Nn white Gaussian noise statistically independent of the input having zero mean and variance 0 to produce the output Yn Xn N Here we explore the Data Processing Theorem when the input distributions dilTer only in their means a If Xn has mean me or m1 what is the ratio of the output to the input KullbackLeibler distances b The Data Processing Theorem does not suggest whether feedback can increase or decrease the ratio of distances Suppose YnaYn71XnNn7 indicating the previous output modi es the value of each current output For Yn to be stationary the coef cient must satisfy lal lt 1 What is the ratio of KullbackLeibler distances between the output and input values at some time index no Can the ratio be greater than one c V More generally we can study Y AXN which represents amore general expression for feedback Assume that A is an invertible matrix and that X and N are Gaussian random vectors statistically independent of each other with covariance matrices 031 and UJZVI respectively As before N has zero mean and the mean of X assumes two possible values What is the ratio of KullbackLeibler distances between the output and input vectors Comment on your answer in light of your result for part b


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