New exponential-type approximations for the erfc(.) and erfcp(.) functions with applications

In this article, we first develop very tight second and third-order exponential-type approximations for the Gaussian probability integral Q(.) and/or the complementary error function erfc(.) which are subsequently used to devise the conditional symbol error probability (CEP) formulas for several classes of digital modulation schemes that are both invertible and in a “desirable” exponential form. These invertible CEP formulas are of interest in the optimization of discrete-rate adaptive modulation designs and for computation of the symbol/bit error outage performance metric. We also present new non-invertible higher-order exponential-type approximations for the erfc(.) and erfc^p(.) functions based on Gauss-Chebyshev quadrature method. Our analytical framework also facilitates the derivation of simple and tight closed-form approximation formulas for the average symbol error rate performance metric of both cooperative and non-cooperative diversity systems in conjunction with a wide range of digital modulation schemes over generalized fading channels via the moment generating function (MGF) approach. The accuracies of our closed-form approximations have been validated with that of the “exact” formulas in an integral form and via computer simulations.