# Two Interval Upper-Bound \(Q\)-Function Approximations with Applications
## 1 Introduction  
The growing need for the mathematical characterization of complex phenomena in wireless transmission has occurred due to the recent rapid development of various wireless communication system services [1, 2]. Determining the performance of the wireless communication systems and observing how this performance depends on key transmission system parameter values are of importance for estimating the behavior of digital communication systems for a wide range of applied modulation types, detection types, and channel models (see for instance [1, 2, 3, 4, 5]). Signal transmission in a wireless medium is accompanied by various phenomena, among which the most important one is multipath fading. In the literature, various mathematical models of this fading phenomenon havebeen presented. As shown in [2], the Nakagami-\(m\) fading model provides the best fit for collected data in indoor and outdoor wireless environments. The average symbol error probability (ASEP) of a wireless communication system best quantifies the reliability or integrity of a received signal [1]. To analytically determine the ASEP for an assumed modulation format, it is necessary to average the expression for the conditional symbol error probability (SEP) over the probability density function (PDF) of the fading channel amplitude [1]. However, in most of the observed cases, the averaging is not straightforward because the averaging integral includes the Gaussian \(Q\)-function or directly related special functions: error function, erf(\(x\)), and/or complementary error function erfc(\(x\)) [1; 2; 3; 4; 5]. Thus, the motivation of our research was to examine the possibility of deriving a novel, simple, highly accurate \(Q\)-function approximation, which could be used efficiently not only for the SEP estimation of transmission over Nakagami-\(m\) fading channels, but also for the ASEP evaluation. Although this problem has been extensively reported in the literature [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20], the need for simpler and more accurate \(Q\)-function approximations still exists due to the continuous development of various wireless services. In regards to some of our previous research works [4; 21; 22; 23], herein, tighter bounds are observed in order to obtain better and more accurate results. The main goal of this paper was to simplify the selection of the approximation of the \(Q\)-function proposed in [6], where four different approximations were shown to approximate the \(Q\)-function for each argument value, which made such a choice cumbersome. Toward this goal, we first select two of these four approximations of the \(Q\)-function and determine analytically which of the two selected upper-bound approximations of the \(Q\)-function provides the best result in terms of accuracy for the widest interval of argument values. In other words, we determine the composition of the upper-bound \(Q\)-function approximations specified on the two disjoint intervals of input argument values so as to provide the highest accuracy within each of them. We show that a further increase of the accuracy can be obtained by forming a composite approximation of the \(Q\)-function that assumes another extra interval for the highest argument values and by specifying an additional form for the third upper-bound \(Q\)-function approximation. The rest of this paper is organized as follows: Section 2 describes the main motivation of this paper and gives a brief overview of the previous work in the field. In Section 3, the novel interval upper-bound \(Q\)-function approximations are specified. Section 4 provides the accuracy analysis of the novel interval upper-bound \(Q\)-function approximations and the approximations of the \(Q\)-function listed in Section 2. Section 5 demonstrates the applicability of the proposed \(Q\)-function approximations in the ASEP evaluation over Nakagami-\(m\) fading channels when BPSK and DE-QPSK modulation formats are assumed.