International Journal of Computer Networks & Communications (IJCNC) Vol.5, No.1, January 2013 DOI : 10.5121/ijcnc.2013.5102 21 FURTHER RESULTS ON THE DIRAC DELTA APPROXIMATION AND THE MOMENT GENERATING FUNCTION TECHNIQUES FOR ERROR PROBABILITY ANALYSIS IN FADING CHANNELS Annamalai Annamalai 1 , Eyidayo Adebola 2 and Oluwatobi Olabiyi 3 Center of Excellence for Communication Systems Technology Research Department of Electrical & Computer Engineering, Prairie View A&M University, Texas 1 aaannamalai@pvamu.edu, 2 eyidayoadebola@yahoo.com, 3 engr3os@gmail.com ABSTRACT In this article, we employ two distinct methods to derive simple closed-form approximations for the statistical expectations of the positive integer powers of Gaussian probability integral [ ( )] p E Q γ β γ Ω with respect to its fading signal-to-noise ratio (SNR) γ random variable. In the first approach, we utilize the shifting property of Dirac delta function on three tight bounds/approximations for Q(.) to circumvent the need for integration. In the second method, tight exponential-type approximations for Q(.) are exploited to simplify the resulting integral in terms of only the weighted sum of moment generating function (MGF) of γ. These results are of significant interest in the development of analytically tractable and simple closed- form approximations for the average bit/symbol/block error rate performance metrics of digital communications over fading channels. Numerical results reveal that the approximations based on the MGF method are much more versatile and can achieve better accuracy compared to the approximations derived via the asymptotic Dirac delta technique for a wide range of digital modulations schemes and wireless fading environments. KEYWORDS Moment generating function method, Dirac delta approximation, Gaussian quadrature approximation. 1. INTRODUCTION The Gaussian Q-function is defined as 2 1 1 () erfc( ) exp( / 2) , 0 2 2 2 x x Qx y dy x π ∞ = = - ≥ ∫ (1) which corresponds to the complement of the cumulative distribution function (CDF) of a normalized (zero-mean, unit variance) Gaussian random variable. This mathematical function plays a vital role in the analysis and design of digital communications since the conditional error probability (CEP) of a broad class of coherent modulation schemes can be expressed either in terms of Q(x) alone or as a weighted sum of its integer powers (e.g., see Table 1, [1, Eqs. (8.36)- (8.39)], [2, Chapter 4]). In addition, system performance measures such as the average symbol, bit or block error probabilities in fading channels typically involve taking the statistical expectation of Q(x) and its integer powers with respect to the random variable that characterizes