Further Results on the Dirac Delta Approximation and the Moment Generating Function Techniques for Error Probability Analysis in Fading Channels

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 KEYWORDSMoment generating function method, Dirac delta approximation, Gaussian quadrature approximation.

Closed‐form solutions to SEP integrals over generalized fading distributions η−μ and κ−μ using approximate computing

“…Firstly, the solution to integral containing Q k (a √ ) is valid only for k ≤ 2, since it considers the integrals containing Q(a √ ) and Q 2 (a √ ) as special cases of Q(a √ )Q(b √ ) integral; therefore, the SEP of neither DE-QPSK nor TQAM (as derived in Qureshi et al 29 ) can be calculated, as it requires integral solutions upto k = 4 and k = 6, respectively. Secondly, the solution is specifically derived for the exponential-based approximation to Gaussian Q function of Sadhwani et al 16 From the above discussions, we can conclude that though there are many approaches in literature till date 24,25,28 to derive the closed-from solution to integrals, but the solution set is either incomplete 25,28 or involves complex mathematical operations 24 or uses double approximations. 24,25 In this paper, we derive closed-form solution to SEP integrals over − and − using the exponential-based approximations to the Gaussian Q function.…”

“…Secondly, the solution is specifically derived for the exponential-based approximation to Gaussian Q function of Sadhwani et al 16 From the above discussions, we can conclude that though there are many approaches in literature till date 24,25,28 to derive the closed-from solution to integrals, but the solution set is either incomplete 25,28 or involves complex mathematical operations 24 or uses double approximations. 24,25 In this paper, we derive closed-form solution to SEP integrals over − and − using the exponential-based approximations to the Gaussian Q function. The expressions are generic in nature, which to best of the author's knowledge, are unique and new.…”

“…From the above discussions, we can conclude that though there are many approaches in literature till date to derive the closed‐from solution to integrals, but the solution set is either incomplete or involves complex mathematical operations or uses double approximations . In this paper, we derive closed‐form solution to SEP integrals over η − μ and κ − μ using the exponential‐based approximations to the Gaussian Q function.…”

“…Though, the study of Benitez and Casadevall 18 is used in Lopez-Benitez 24 and Annamalai et al 25 to obtain the solution to integral containing Q k (a √ ) over these fading distributions, but these use double approximation; while in Lopez-Benitez, 24 the solution is first represented in form of infinite series using the approximation in Benitez and Casadevall 18 and then truncated to arrive at the closed-form solution; on the other hand, in Annamalai et al, 25 the approximation of Benitez and Casadevall 18 along with Dirac-delta approximation results in closed-from solution to integrals. Furthermore, the solution in Lopez-Benitez 24 is mathematically complex as it requires computing parabolic cylinder function D v (·), 26, (9.240) while there is no solution for the integral containing Q(a √ )Q(b √ ) in Annamalai et al 25 A class of approximations that represent Q(·) as sum of exponentials [13][14][15][16] are especially useful when we need to derive the closed-form solutions to SEP integrals over generalized fading distributions − and − using approximate computing. In Sadhwani et al, 27 a similar approach is used to derive certain expressions over these fading distributions but are specific to hexagonal quadrature amplitude modulation (HQAM) and therefore cannot be used for other modulation schemes.…”

“…Similarly, previous works also approximate Q ( x ) in terms of both and , thereby failing to lead to solutions to SEP integrals over complex fading. Though, the study of Benitez and Casadevall is used in Lopez‐Benitez and Annamalai et al to obtain the solution to integral containing over these fading distributions, but these use double approximation; while in Lopez‐Benitez, the solution is first represented in form of infinite series using the approximation in Benitez and Casadevall and then truncated to arrive at the closed‐form solution; on the other hand, in Annamalai et al, the approximation of Benitez and Casadevall along with Dirac‐delta approximation results in closed‐from solution to integrals. Furthermore, the solution in Lopez‐Benitez is mathematically complex as it requires computing parabolic cylinder function D v (·),, (9.240) while there is no solution for the integral containing in Annamalai et al…”