Using a geometrical approach inspired by the method of Craig for the one-dimensional Gaussian integral (one-dimensional Q-function, 1DQ) and similar to that of Simon and Divsalar, we propose a new form for the two-dimensional Gaussian integral (two-dimensional Q-function, 2DQ). Unlike that of Simon and Divsalar, our alternate form is applicable to any set of arguments and is not limited to nonnegative arguments. This rewrites the infinite-range double-integral expression in the naive form of 2DQ in terms of simple finite-range single integrals. The expressions given are simpler than those obtained by writing the 2DQ for negative arguments in terms of a sum of a number of 2DQ terms with nonnegative arguments. In addition, for a fixed required accuracy, the numerical complexity of the method is smaller than the rational function approximations of Donnelly.

New Bonferroni-type lower bounds on the word error probability of uncoded systems are developed. The new family of bounds is based on a recent Bonferroni inequality proposed by Cohen and Merhav. These novel tight bounds are developed for optimal maximum a posteriori (MAP) coherent detectors with nonuniform signalling over additive white Gaussian noise channel. The results are compared to the state-of-the-art KAT lower bounds and it is shown that the superiority of one bound to another is dependent on the signal constellation, the amount of nonuniformity of the Bernoulli source to be communicated, and the SNR range of interest. For instance, for smaller deviations from the uniform case, which are in fact more plausible, and at low SNRs, the new bounds are tighter than KAT lower bounds for all the constellations studied.