Tight error bounds for nonuniform signaling over AWGN channels

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_a) and bit error (P_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 _b/N₀. The new algorithmic bound and the greedy bound are particularly impressive as they agree with the simulation results even during very severe channel conditions