A Representation for the Symbol Error Rate Using Completely Monotone Functions

The symbol error rate of an arbitrary multidimensional constellation in the absence of coding impaired by additive white Gaussian noise is characterized as the product of a completely monotone function with a nonnegative power of the signal-to-noise ratio, when the minimum distance detector is used. This representation is also shown to apply to cases when the impairing noise is compound Gaussian. Using this general result, it is proved that the symbol error rate is completely monotone if the rank of its constellation matrix is either one or two. Further, a necessary and sufficient condition for the complete monotonicity of the symbol error rate of a constellation of any dimension is also obtained. Applications to stochastic ordering of wireless system performance are also discussed.