Evaluation of Error Probabilities for General Signal Constellations

Expressions for error probabilities have been derived for commonly used signal constellations such as QAM and MPSK. However, closed form expressions cannot be found for more complex signal constellations such as hexagonally packed signals. Saddle point integration (SPI) has been shown to be an efficient method of evaluating the one dimensional cumulative distribution function (CDF) of a random variable. SPI integrates the moment generating function (MGF) of the test statistic instead of probability density function (PDF). In this paper, the SPI method is used to evaluate Pe for general, complex signal constellations. The first step is to determine the decision regions for each symbol using an algorithm from the graphics literature for calculating Voronoi diagrams. Next, these decision regions are subdivided into subregions that can evaluated using SPI. The SPI method is very flexible and allows one to model many different system effects such as phase offset, DC offset, and colored noise.