Invertible Exponential-Type Approximations for the Gaussian Probability Integral Q(x) with Applications

In this article, we present very tight 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. Our framework also facilitates the derivation of simple and tight closed-form approximation formulas for the average symbol error rate performance metric for a wide range of digital modulation schemes over generalized fading channels via the moment generating function approach. The accuracies of our new closed-form approximations have been validated with that of the exact expressions in an integral form.