A 4-Term Exponential-Quadratic Approximation for Gaussian Q or Error Functions Accurate to $1.65\times 10^{-4}$

Abstract

Integrals on [0, ∞) where the integrand is of the form Qn(a√x) p(x), where Q is the Gaussian Q function, p(·) a Gamma PDF, n a positive integer and a > 0; or of the form erfn(ax + b) xr exp(-c2x2 + 2dx), where erf(x) is the error function, with integers r ≥ 0, n > 0, arise in performance modelling of communication and machine learning systems. Such integrals cannot be evaluated analytically in general, but they are reducible to a set of key integrals whose integrand is erfn(ax + b) N(x; m, s) where N() is a Gaussian PDF with mean m and variance s. Seeking an efficient and accurate evaluation method, we develop a new 4-term exponential quadratic approximator (EQA) for the error function that includes both linear and quadratic terms in its exponents. The EQA minimises a sum-of-squares cost function with two “spline-type” constraints, i.e., constraints on the function value and its first derivative. This constrained optimisation problem is reduced to an unconstrained one by inverting a 4-D linear system, then solved by gradient descent. The resulting approximator has a maximum absolute error of 1.65 × 10-4 on the real line, and outperforms many other exponential sum approximators for erf(x) on x ∈ [0, 1.5] and for Q(x) on x ∈ [0, 2]. Moreover, due to its functional form, the EQA leads to an analytical formula for the set of key integrals, which, in the n = 1 case, is accurate to 3 to 4 significant figures while being orders of magnitude more efficient than Monte Carlo integration. The EQA can equally be used to obtain closed forms for the average symbol error probability of various modulation schemes on Rayleigh fading channels.