ECE 563 Fall 2004 Take - home Final

Instructions: I expect you to work on the problems by yourself. You can refer to any textbook (or the technical literature in general), but not confer with any person. 1. Short Questions (a) How many bits are needed to specify a selection of k objects from n objects? (n and k are assumed to be known and the selection of k objects is unordered.) (b) Either prove that the following code is uniquely decodable or give an ambiguous concatenated sequence of codewords: c 0 = 101 c 1 = 0011 c 2 = 1001 c 3 = 1110 c 4 = 00001 c 5 = 11001 c 6 = 11100 c 7 = 010100 (c) Consider the memoryless AWGN channel y = x + z where z is zero mean Gaussian random variable with variance σ 2. The transmit signal x has an average power constraint of P. With no other constraints on the input, the capacity of the channel is C = 1 2 log 2 1 + P σ 2. Now suppose x is restricted to the binary alphabet − √ P , + √ P. i. Find an expression for the capacity of this restrictive channel and denote it byˆC (you may not be able to find a closed form expression forˆC but you should be able to identify the optimal input distribution exactly). ii. When isˆC close to C? For small signal-to-noise ratios (defined as the ratio P/σ 2) or large ones? Commentary: This justifies the usual engineering practice of using simple binary modulation on the AWGN channel in a certain SNR regime.