Tight error bounds for nonuniform signaling over AWGN channels

We consider a Bonferroni-type lower bound due to Kounias 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 exp onentially complex search for its direct implementation. We address this pr oblem by presenting a practical algorithm for its evaluation. This bound is applied together with two other bounds, a recent lower bound (the KAT boun d) and a greedy algorithm implementation of an upper bound due to Hunter, t o examine the symbol error (P-s) and bit error (P-b) probabilities of an un coded communication system used in conjunction with M-ary phase-shift keyin g (PSR)/quadrature amplitude (QAM) (PSK/QAM) modulations and maximum a post eriori (MAP) decoding over additive white Gaussian noise (AWGN) channels. I t is shown that the bounds-which can be efficiently computed-provide an exc ellent estimate of the error probabilities over the entire range of the sig nal-to-noise ratio (SNR) E-b/N-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.