Tight Bounds on the Mutual Information of the Binary Input AWGNChannelM

Tight Bounds on the Mutual Information of the Binary Input AWGN Channel M. R. Shane and R. D. Wesel1 Department of Ele tri al Engineering University of California, Los Angeles e-mail: fshane,weselg ee.u la.edu Abstra t | We derive tight upper and lower bounds on the mutual information of the binary input AWGN hannel with power onstraint P . These bounds, while more omplex than previous bounds, have the advantage of being very tight for all values of SNR, and are thus suitable for general purpose use. I. Introdu tion To determine the apa ity of many ommuni ation hannels of interest, numeri al integration is required. However, analyti bounds may be useful if they are suÆ iently tight. For the binary input AWGN hannel, several bounds exist (see [1℄ and the referen es therein), but they are only a urate for limited ranges in SNR. II. AWGN Channel Capa ity and Shaping Gain Consider the AWGN hannel with poweronstrained input X, noise Z N (0; N), and output Y = X +Z. The apa ity of this hannel is a hieved when the distribution of the input X is also Gaussian, and is given by (1) in nats where P is the power onstraint on the Gaussian input Xg N (0; P ). C = I(Xg;Yg) = 1 2 ln 1 + P N (1) An input Xng with non-Gaussian distribution is suboptimal from a apa ity point of view. We an write the mutual information of the AWGN hannel with input Xng as the di eren e between (1) and its maximum shaping gain I(Xg;Yg) I(Xng ;Yng). For Xg and Xng transmitting at the power onstraint, the shaping gain an be expressed as the relative entropy between fg and fng, whi h are the distributions of outputs Yg and Yng, respe tively. Thus the mutual information of the hannel with input Xng is given by I(Xng;Yng) = 1 2 ln 1 + P N D(fng jjfg). III. Binary Input Mutual Information Bounds Let Xb represent the binary input with iid symbols pP . It an be shown that the relative entropy between the two output distributions fg(y) and fb(y) eventually yields: D(fbjjfg) = Z 1 1 fb(y) ln 1 2 + 1 2 ln 1 + P N P 2N dy + 2 Z 1 0 fb(y)pP N ydy + Z 1 1 fb(y) P 2N(P +N)y2dy + 2 Z 1 0 fb(y) ln(1 + exp( 2pPy=N))dy: (2) 1This work was supported by a fellowship from The Aerospa e Corporation, the Xetron Corporation, and Conexant. −25 −20 −15 −10 −5 0 5 10 15 −0.03 −0.025 −0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02