The design of two-dimensional constellation mappings for the transmission of binary nonuniform memoryless sources over additive white Gaussian noise channels using standard M-ary PSK and QAM modulation schemes is investigated. The main application of this problem is the incorporation of an adaptive mapping assignment in modem devices that employ fixed PSK/QAM modulation schemes for the transmission of heterogenous data (such as multimedia information) containing various levels of nonuniformity. In general, the optimal mapping depends on both the probability distribution of the input signals and the signal-to-noise ratio (SNR) in the channel, in addition to the geometry of the signal constellation. We show that constellation mappings which follow the objective of minimizing the average symbol energy and, given this, maximizing the decoding probability of the most likely signals, can yield symbol-error-rate and bit-error-rate performance that is substantially better than Gray encoding maps. Gains as high as 3.5 dB in SNR E/sub b//N/sub 0/ are obtained for highly nonuniform sources. Finally, we note that the mappings techniques result in nonzero mean constellations and briefly consider their performances when they are converted to zero mean constellations by shifting. In this case, we observe that the shifted zero-mean Gray map outperforms our shifted maps for small- to medium-sized constellations (M/spl les/32), but not for larger sizes.

We consider a Bonferroni-type lower bound due to Kounias (1968) on the probability of a finite union. The bound is expressed in terms of only the individual and pairwise event probabilities; however, it suffers from requiring an exponentially complex search for its direct implementation. We address this problem by presenting a practical algorithm for its evaluation. This bound is applied together with two other bounds, a recent lower bound (the KAT bound) and a greedy algorithm implementation of an upper bound due to Hunter (1976), to examine the symbol error (P/sub a/) and bit error (P/sub b/) probabilities of an uncoded communication system used in conjunction with M-ary phase-shift keying (PSK)/quadrature amplitude (QAM) (PSK/QAM) modulations and maximum a posteriori (MAP) decoding over additive white Gaussian noise (AWGN) channels. It is shown that the bounds-which can be efficiently computed-provide an excellent estimate of the error probabilities over the entire range of the signal-to-noise ratio (SNR) E/sub b//N/sub 0/. The new algorithmic bound and the greedy bound are particularly impressive as they agree with the simulation results even during very severe channel conditions.