Channel Capacity and Constellation Optimization of M -PAM Input AWGN Channels with Non-equiprobable Symbols

In this letter, we show how to compute optimal constellations which maximize the capacity of M -PAM input AWGN (MIAWGN) channels with non-equiprobable symbols. We find for binary signaling that the optimal constellations, which minimize the bit error rate, will also maximize the capacity. The rate capacity of MIAWGN channels with non-equiprobable symbols and optimal constellations approaches to H(X) when Eb/N0 becomes large, where Eb/N0 is bit energy-to-noise density ratio and H(X) is the entropy of the source symbols. At the low rate regions (i.e., R < H(X)/2) the capacity loss due to non-equiprobable symbols is not significant if the constellations are optimized for the symbol probabilities. However, if we select wrong constellations, the capacity loss could be significant for M = 2 and 4. For binary signaling with symbol probabilities of 0.3 and 0.7, 0.72 dB increase of Eb/N0 is required if we use optimal constellations for equiprobable symbols (i.e., {xi = ±1}). WhenM becomes large, the shaping effect helps rate maximization and the constellation mismatch has little impact on capacity.