ECSE 521A Prof. Bajcsy Assignment 2 - parts 1 & 2 Due: October 21 McGill University Fall 2015 Problems 1: Channel Capacity Theorem QuickBucks Telecom Ltd. has purchased an optical Internet link between Montreal and New York to transmit long data packets for a commercial customer. The link can be modeled as a binary erasure channel with probability of bit erasure = 0.05. The company wants to transmit at a rate of 0.975 bits per channel use and a new, improved coding scheme is to be designed every year, so that the probability of packet decoding error in year 2n 1 n (n=0,1,2,...) does not exceed . 2 (a) Can the company fulfill its design objective in a long run? (Please justify your answer by calculation and/or reference to result/results from class.) (b) Would your answer change if the channel had better quality and = 0.01. Please fully explain your answer. (c) For = 0.01, could the company design just one fixed coded system and satisfy the quality of service in a long term? Again, please explain. Problem 2: Soft Decision Demodulation (a) Consider a 4-QAM digital modulation scheme that maps messages {00, 01, 10, 11} onto signal points in the signal space shown below (A=2). Assuming AWGN channel with No = 0.6, perform soft-decision demodulation when r = (0.75, 0.26) . "01" "00" A -A "11" A -A "10" (b) Redo part (a) when the transmitted bits are independent but NOT equiprobable, i.e., p0 = 0.9 and p1=0.1. (c) Use Matlab, sketch the optimal hard-decision regions for the unequal priors from (b) and system parameters from (a), when the receiver performs hard decision demodulation. Show your derivation and submit an appropriate printout with decision regions. PROBLEM 3: Channel Capacity Using Matlab, plot the capacity of the following channels as a function of the channel parameter/s: (a) Binary erasure channel with erasure probability . (b) Binary symmetric channel with the error probability p. (c) For your capacity curve from (b), explain what happens when p=0.5 and p>0.1. (d) Discrete-time Gaussian noise channel with power limit P and noise variance 2 . Plot C as a function of SNR = P/ 2 in dB. (e) Explain what happens to the capacity value and growth in (d), when the channel SNR gets very small (SNR << 0 dB) and very large (SNR > 50 dB). (f) Continuous-time AWGN channel with bandwidth W, noise power spectral density No/2 and signal power limit P. (g) Will the capacity in (f) go to infinity when W approaches infinity and transmit power P is limited? Justify your answer from the plot by deriving appropriate limit. PROBLEM 4: Basics of Information Measures Consider random variables X and Y with joint probability mass function P ( X = 0, Y = 0) = 1/ 2 , P ( X = 1, Y = 0) = P ( X = 1, Y = 1) = 1/ 8 P ( X = 0, Y = 1) = 1/ 4 Determine the following: (a) H(X,Y) (b) H(X) & H(Y) (c) D(Px || Py) & D(Py || Px) (d) H(X|Y) & H(X|Y) (e) I(X;Y) PROBLEM 5: Evaluation of Channel Capacity Consider a binary non-symmetric channel with input X and output Y, described by PY | X (Y = 0 | X = 0) = 1 PY | X (Y = 1| X = 0) = PY | X (Y = 0 | X = 1) = PY | X (Y = 0 | X = 1) = 1 , where , [0,1] are fixed parameters describing the specific channel. (a) Sketch the channel model based on this given probabilistic description, clearly labelling the channel inputs, channel outputs and transition probabilities. If PX ( X = 1) = p , p [0,1] , write down the expression for the mutual information I(X;Y) between the channel input X and the channel output Y . The expression should be a function of p, while containing , as parameters. (b) For a specific values of = 0, = 0.5 , plot the mutual information from part (a) as a function of p. From the plot, determine the channel capacity of this channel and the value of p for which the maximum happens. (c) Observe that in part (b) the mutual information is a concave function of P(X=0)=p. Show (e.g., using differentiation), that the concavity property is also true for any specific values of parameters , . (d) Using the max function in Matlab, write a script that finds the channel capacity of this given channel for any specified , . Using this script and the mesh function, plot the capacity of this channel as a function of , [0,1] . (HINT: Use sufficiently small step-size for , , e.g., 0.01, and to avoid problems with log(0), start with = = 1010 .)