hw6

# hw6 - IST 1 Introduction to Information Prof M Eros...

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IST 1 Handout #35 Introduction to Information Wednesday, 5/9/7 Prof. M. Effros Homework Set #6 Due: Wednesday, May 16 1. Determine if each of the following channels is weakly symmetric, or not. Find the capacity of each weakly symmetric channel. (a) For 0 < p < 1, P Y | X = 1 p p 0 0 p 1 p 0 0 0 0 1 p p 0 0 p 1 p . (b) For 0 < p,q < 1, p negationslash = q , P Y | X = 1 - p 2 p 2 q 2 1 - q 2 p 2 1 - p 2 q 2 1 - q 2 1 - p 2 p 2 1 - q 2 q 2 p 2 1 - p 2 1 - q 2 q 2 . (c) For 0 < p,q < 1, p negationslash = q , P Y | X = 1 - p 2 p 2 q 2 1 - q 2 p 2 1 - p 2 q 2 1 - q 2 1 - q 2 q 2 1 - p 2 p 2 q 2 1 - q 2 1 - p 2 p 2 . 2. Consider the composition of two BSCs as follows (0 p,q 1). 0 1 0 1 0 1 q q 1 q 1 q 1 p p p 1 p Z Y 0 1 X 0 1 (a) What is the capacity of the channel with input X and output Z and transition probabilities governed by the given composition? (b) Fix p = 0 . 2. Plot a graph of the capacity versus q [0 , 1].

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