A new, simple and exact result for calculating the probability of error for two-dimensional signal constellations

The author presents a simple integral expression for calculating the exact probability of a symbol error for an arbitrary array of signal points. The integrand contains only elementary functions and the range of integration is finite. The approach is introduced by applying it to M-ary phase shift keying (MPSK). The special case of M=2 gives novel and possibly useful expressions for calculating the Gaussian tail probability function and the related complementary error function. The approach is outlined for polygonal decision regions, and results are given for 16-point signal constellations. A method of obtaining, not exact, but even simpler and highly accurate expressions for symbol error probability when the latter is less than a few hundredths is presented