Some new twists to problems involving the Gaussian probability integral

Using an alternate form of the Gaussian probability integral discovered a number of years ago, it is shown that the solution to a number of previously considered communication problems can be simplified and, in some cases, made more accurate (i.e., exact rather than bounded). These problems include the evaluation of: 1) bit-error probability of uncoded phaseshift keying (PSK) with Costas loop tracking; 2) word-error probability of antipodal modulation in the presence of fading; 3) bit-error probability of coded M -ary PSK (MPSK) over the memoryless fading channel with given channel-state information; 4) conditional symbol-error probability of MPSK in the presence of carrier synchronization error; and 5) average error probability for the binary additive white Gaussian noise (AWGN) intersymbol interference channel. Also obtained is a generalization of this new alternate form to the case of a two-dimensional Gaussian probability integral with arbitrary correlation which can be used to evaluate the symbol-error probability of MPSK with I–Q unbalance.