Universal Bounds on the Derivatives of the Symbol Error Rate for Arbitrary Constellations

The symbol error rate (SER) of the minimum distance detector under additive white Gaussian noise is studied in terms of generic bounds and higher order derivatives for arbitrary constellations. A general approach is adopted so that the recent results on the convexity/concavity and complete monotonicity properties of the SER can be obtained as special cases. Novel universal bounds on the SER, which depend only on the constellation dimensionality, minimum and maximum constellation distances are obtained. It is shown that the sphere hardening argument in the channel coding theorem can be derived using the proposed bounds. Sufficient conditions based on the positive real roots (with odd multiplicity) of an explicitly-specified polynomial are presented to determine the signs of the SER derivatives of all orders in signal-to-noise ratio. Furthermore, universal bounds are given for the SER derivatives of all orders. As an example, it is shown that the proposed bounds yield a better characterization of the SER for arbitrary two-dimensional constellations over the complete monotonicity property derived recently.