A New, Simple and Exact Result for Calculating the Probability of Error for Two-dimensional Signal Constellations

In most signaling schemes in digital communication, the boundaries of the decision regions are straight lines. For some of these schemes such as quadrature amplitude modulation and its special cases where the boundaries meet at right angles, simple and exact expressions are known for the probability of error on an AWGN channel. In other cases where the boundaries do not meet at right angles, the exact expressions contain integrals ofspecial functions which areoften difficult to evaluate. Examples of these are M-ary phase shift keying (MPSK) and the wide variety of QAM and AMPM constellations of signal points, among many others. Here by expressing the double integral for the probability of error (PE) in a different way than has been done before, we show that the PE can be expressed as the sum of asmall number of simple, single integrals, all with the same integrand containing only elenientary functions and all having a finite range. Applied to MPSK, this approach gives just one simple, finite integral for PE, and the values of PE so obtained are in excellent agreement with those tabulated by Lindsey and Simon in their book Teleconimunicarion Sysrems Engineering. Numerical results for these and other signal constellations of interest will be presented in the paper. This approach is applicable to any set of signal points having polygonal decision regions. An interesting related result is that this approach gives a new and simple integral expression for the Gaussian tail probability function Q(x). and by the same token, a similar result for the complementary error function erfc(x).