Average of product of two Gaussian Q-functions and its application to performance analysis in Nakagami fading

The average of the product of two Gaussian Q-functions having arguments as different scaled versions of the same Nakagami distributed fading gain magnitude is derived in closed-form. Both the cases when (1) the fading parameter is an integer, and (2) the fading parameter is an odd multiple of one-half, are considered. The results are applied to evaluating in closed-form (a) the symbol error probability (SEP) of rectangular quadrature amplitude modulation with maximal-ratio combining in independent Nakagami fading, and (b) the outage probability of a two-user synchronous code-division multiple-access system having signal-to-noise ratio (SNR) imbalance among users in Nakagami fading. It is found in application (a) that with increase in the degree of fading, the robustness to decision distance imbalance between in-phase and quadrature signals increases. In addition, there exists an optimum decision distance ratio at which the SEP has a minimum. In application (b), we find that the outage probability increases sharply with SNR imbalance factor at low SNR imbalance and tends to reach saturation as the imbalance increases.