Jensen-cotes upper and lower bounds on the Gaussian Q-function and related functions

• Authors: Giuseppe Thadeu Freitas de Abreu
• Venue: IEEE Transactions on Communications
• Year: 2009

We present new families of lower and upper bounds on Q-functions. First, we consider the Craig form of the Gaussian Q-function Q(x) and shown that its integrand ϕ(θ; x) can be partitioned into a pair of complementary convex and concave segments. This property is then exploited in conjunction with the Jensen inequality and the Newton-Cotes' quadrature rule to produce a complete family of upper and lower bounds on Q(x), which can be made arbitrarily tight by finer segmentation. The basic idea is then utilized to derive families of upper and lower bounds also for the squared Gaussian Q-function Q2(x), the 2D joint Gaussian Q-function Q(x, y, p), and the generalized Marcum Q-function QM(x, y). The bounds are shown to be tighter than alternatives found in the literature, and in some cases the lower bounds provided find no equivalent in current literature. The generality of the principle is the elegant point of the method and the resulting Jensen-Cotes bounds are easy to implement and evaluate since only elementary transcendental functions are involved. As an example of application to the analysis of communication systems, we consider the bit error rates (BER's) of decode-and-forward (DF) cooperative relaying schemes with coherent and differential phase-shift keying (PSK) modulations, which have been shown to have an intricate dependence on the Gaussian Q-function, complicated by cross-products, irrational functional arguments and multiple numerical integrations. In that example the bounds substantially reduce the complexity required to evaluate the expressions, retaining tightness despite multiple numerical integrations with infinite limits.