ITCSolution5

# ITCSolution5 - Zheng Gao (a) The exponential density, f (x)...

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Zheng Gao 35702962 * y i 0 / 8.1 Differential entropy. Evaluate the differential entropy h(X) = - I flnf for the following: (a) The exponential density, f(x) = helh, x 1 0. (b) The Laplace density, = :~e-"l"' (c) The sum of X, and X,, where XI and X2 are independent normal random variables with means pi and variances a? i=1,2. Solution: = -1nA + 1 nats = log,: bits J (b) h(X) = - J-: f ~e-~I"ln~h-"~dx h = -1n-f 1 nats 2 2e = logzT bits X = f X2 - N(pI + pz,(r: + 6;) h(X) = :log, [2ne(o: + o:)] bits J 8.3 Uniformly distributed noise. Let the input random variable X to a channel be uniformly distributed over the interval - 1/2 I x 5 1/2. Let the output of the channel be Y = X + 2, where Z the noise random variable is uniformly distributed bver the interval - a/2 < #4 5 a/Z. (a) Find [(X; Y) as a function of a. (b) For a = 1 find the capacity of the channel when the input X is peak-limited; that is, the range of X 1 1 limited to -- < x 5 - . What probability distribution on X maximizes the mutual 2- 2 information I(X; Y)? (c) (Optional) Find the capacity of channel for all values of a, again assuming that the range of X

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## This note was uploaded on 02/24/2011 for the course EE 634 taught by Professor Pados during the Spring '09 term at SUNY Buffalo.

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ITCSolution5 - Zheng Gao (a) The exponential density, f (x)...

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