Performance bounds of non-uniform signaling over AWGN channels

We present two bounds (one lower bound and one upper bound) on the probability of a union of a finite number of events. The bounds-which are expressed in terms of only the individual event probabilities and the pairwise event probabilities-are applied to examine the symbol error (P_s) and bit error (P_b) probabilities of an uncoded communication system used in conjunction with M-ary PSK/QAM modulations and maximum a posteriori (MAP) decoding over AWGN channels. It is shown that both bounds provide a good, and often excellent, estimate of the error probabilities over the entire range of the signal-to-noise ratio E_b/N₀. The upper bound on P _s and the lower bound on P_b are particularly impressive as they agree with the simulation results even during very severe channel conditions