Novel Composite Approximation for the Gaussian Q-Function

This paper, by using Borjesson’s and Benitez’s approximation of Q-function, presents a novel and improved composite approximation of Q-function with wide applicability. The presented approach is very general and can be implemented on any observed interval. Based on the proposed approximation of Q-function, the average bit error rate is assessed by observing the transfer over Nakagami-m fading channel. The simplicity of the proposed approximation form in conjunction with yet another feature utmost accurateness appeared to be a better choice than the suggested approximations of similar complexity in terms of analyticity. The paper emphasizes the wide implementation possibilities in numerous tasks of communication theory and functional analysis that include Q-function.


II. Q-FUNCTION APPROXIMATION ACCURACY ANALYSIS
Gaussian Q-function Q(x) has a major role in performance analysis of different communication schemes where the noise is represented as a Gaussian random variable [12]. In particular, the problem comes down to finding the closedform formula that approximates Q-function given with (1). Q-function is defined as a complementary function of the cumulative distribution of Gaussian random variable X with zero mean and unit variance [1]- [11]   exp .
x erfc x erf x t dt Borjesson and Sundberg [2] proposed the definite integral and solid approximation of function Q(x) stressing the simplicity and applicability of their proposal     where a and b are scalar fitting parameters having optimal values that amount to a = 0.339 and b = 5.510. Chiani, Dardari, and Simon presented approximation of the Q-function in a form of the sum of exponential functions [3]. This approximation has a relatively simple analytical form, but is less accurate Karagiannidis and Lioumpas [5] proposed a new, simple, and tight approximation for the Gaussian Q-function which proved highly accurate for smaller values of arguments where values of A and B were found numerically as 1.98 and 1.135, respectively, for the argument region (x  [0, 20]). The Prony (sum-of-exponentials) approximation of Qfunction proposed by Loskot and Beaulieu [13] in terms of two exponential functions is given with which can make the exact prediction of the Gaussian Qfunction, in particular for the small values of argument x. However, it should be noted that its exactness can have a proportionally decreasing tendency as the argument x's values rise up. In [15], another approximation of the Gaussian Q-function with two exponential terms was proposed, denoted as Qa-Sofotasios(x) The second-order exponential function presented in [11], denoted as Qa-Benitez(x) (11)

III. MAIN RESULT
The most significant attainment of this paper is the new composite Q-function approximation, which is elaborated in particular in the forthcoming part.
In search of the simplest approximation with high accuracy, the approximation of Q-function proposed in [16] can be used as a starting point 34 ELEKTRONIKA IR ELEKTROTECHNIKA, ISSN 1392-1215, VOL. 26, NO. 5, 2020 where n denotes the approximation order and 10 [ , , ..., ] nn a a a   unknown parameters. Equation (12) can be presented in a closed-form as in [16]     10 , , , ..., , where f(x) denotes non-linear correlation between {x, an, an-1, …, a0} and Qm(x). For a considered n, values of Ω are calculated with MSE minimization in order to achieve the best approximation of Q-function for the observed range x where k denotes k th sample, N denotes the number of samples. Q(xk) denotes accurate values of Q-function, while Qm(xk) denotes values calculated with the proposed approximation. Equation (14) can be stated as This new approximation of Q-function, as a result of MSE application, can be defined as: Examining a characteristic that accuracy in approximating the Q-function is dependable on the fitting parameters resulted in the idea to construct this particular composite function, which is also highly dependable on the argument's range.
The In order to optimize Q-function approximation in terms of observed parameter range, we provided two forms of QMSE approximation presented with (16). The first approximation form Q1(x) yields better results for the small range of parameter x ( 0 0.7 x  ), while for the higher range of this parameter (x > 0.7), Q2(x) approximation is more suitable. It can be noticed that the better Q-function approximation for the small range of parameter x has been obtained with the second-order approximation, so there was no need for further consideration.

IV. NUMERICAL RESULTS
In this section, we are comparing the accuracy of the proposed approximation of the Q-function with the already existing approximations of the Q-function available in the literature. In the literature, different empirical and analytical methods of approximation have been presented, providing different compromises between the accuracy of the Q-function approximation and the analytical complexity. Also, some of the available approximations, despite adequate analytical complexity, continue to provide insufficient accuracy. Particular methods can prove appropriate for small arguments, but not for large arguments and likewise. In other words, in this section, we will show the validity of the proposed composite approximation of the Q-function by comparing its characteristics with the already existing approximations of the Q-function. Table I exemplifies the comparison or Q-function and its approximations. Evidently, the improvement in accuracy is achieved throughout the range of values by using this new proposed approximation (17) with the assistance of MSE.
In order to compare as described in numerous papers on the related subject, we have decided to select a common interval of argument x ranging [0, 6], since we have determined the dependences of the absolute relative errors of approximating the Q-function on the argument x, as presented in Fig. 1.
After the evaluation of optimization parameters, it has been examined how the proposed approximation is achieved with the use of MSE denotes Gaussian Q-function. In Fig. 2, the proposed approximation is compared with the approximations proposed by Karagiannidis and Borjesson, observed through the prism of the absolute relative error values as a function of argument x.  As depicted in Fig. 2, all previously proposed approximations have low accuracy compared to the proposed approximation. The proposed approximation offers a significant improvement in accuracy compared to Karagiannidis and Borjesson proposed approximations throughout the range, and particularly for smaller values of argument x. Figure 3 and Figure 4 show that the proposed method has the lowest values of the absolute relative error, i.e., improvement in accuracy has been achieved throughout the range.   Implementation of the composite method of Q-function approximation can also be extended to the calculation of the average error rate for DE-QPSK. Figure 7 provides ASER values for DE-QPSK modulation format in the presence of the Nakagami-m fading channel. Figure 8 provides values of the absolute relative error of the proposed approximation ASER for DE-QPSK.
Based on Fig. 7 and Fig. 8, it can be concluded that ASEP values for DE-QPSK implemented modulation format in the presence of the Nakagami-m fading channel for different values of parameter m can be efficiently and accurately evaluated with the use of the proposed composite method. By using the proposed approximation, the performance measures for ASER have been evaluated more accurately than by using any other known Q-function approximations in closed-form throughout the range of input values.

V. CONCLUSIONS
This paper provides a novel and improved composite Qfunction approximation. The comparison with other known closed-form Q-function approximations, which has been provided at Table I and at Figs. 1-4, verifies the improvement in accuracy throughout the domain of function argument values. Moreover, the comparison between ASEP values has been provided and presented at Figs. 5-8. The comparison has been presented by the use of a novel and improved composite Q-function approximation, along with other well-known Q-function approximations, for cases when BPSK and DE-QPSK are being observed. The proposed approximation can also be useful in tackling numerous problems in communication theory.

CONFLICTS OF INTEREST
The authors declare that they have no conflicts of interest.