Expurgating the union bound to error probability: a generalization of the Verdu-Shields theorem

The union bound is known as the most useful tool for the computation of error probability in digital communication schemes (both coded and non-coded). Given a signal set or codebook C={x₀,x ₁,...,x_M-1}, the conditional error probability P(e|x₀) is upper bounded by a sum of pairwise error probabilities (PEP). Here we consider the additive white Gaussian channel, where C is a discrete and finite set of points in the N-dimensional real Euclidean space. In this case, the PEPs are the probabilities of the noise taking a signal on the other side of the hyperplane bounding the decision regions of x₀ and x_i . An important step toward the reduction of the union bound to its minimal form was made by Verdu, who derived a theorem, commonly referred to as the Verdu-Shields theorem showing how terms can be removed from the upper bound to error probability in the case of binary antipodal transmission over the Gaussian channel with intersymbol interference. In this paper we derive a generalization of the Verdu-Shields theorem which provides a sufficient condition for expurgating a given error sequence x _k from the union bound