# Two Interval Upper-Bound \(Q\)-Function Approximations with Applications
## 1 Introduction
In Section 6, we explore some new benefits of \(Q\)-function approximations, such as applications in designing quantization with equiprobable cells for Gaussian sources. Finally, Section 7 concludes on our research results. As already stated, in numerous areas of science and engineering, we encounter the Gaussian \(Q\)-function. This paper also aims to explore the benefits of approximating the \(Q\)-function by a closed form expression when quantization issues are in question. In particular, we consider the quantization of variables from a Gaussian source, where the probability that the variable belongs to a particular cell is the difference of two \(Q\)-functions. With the goal to achieve equiprobable quantization cells, in [24], a quantization noise was injected. Unlike [24], in this paper, the equiprobable quantization cells and quantizer design are specified in a different way. To design the quantizer for the assumed Gaussian source having equiprobable quantization cells, the system of integral equations with arguments of \(Q\)-functions as unknown variables has to be solved. With the application of the composite upper-bound approximation proposed in this paper, we manage to avoid solving the systems of integral equations, which makes our proposal useful. In brief, the main contributions of this paper are:  
* We propose an analytical approach for determining the composition of upper-bound \(Q\)-function approximations specified on disjoint intervals of input argument valuesso as to provide the highest accuracy within the intervals by utilizing the disposable upper-bound approximations of the \(Q\)-function;
* We determine particular analytical forms of two such compositions, named interval upper-bound \(Q\)-function approximations;
* We demonstrate that a further increase of the accuracy, achieved in the case with two upper-bound approximations composing the interval approximation, can be obtained by forming a composite interval approximation of the \(Q\)-function that assumes another extra interval and by specifying the third form for the upper-bound \(Q\)-function approximation;
* We show that in the whole domain of the function argument, our novel interval upper-bound \(Q\)-function approximations provide noticeable improvements in terms of the accuracy in comparison with \(Q\)-function approximations of similar analytical forms from the literature selected for comparison purposes;
* We analyze the versatility of our proposal in determining the performance (ABER), when the transmission in Nakagami-\(m\) fading channel is observed for the assumed BPSK and DE-QPSK modulation formats and provide an additional justification of the benefits of using the proposed composite improved interval approximation of the \(Q\)-function (our second upper-bound composite interval approximation of the \(Q\)-function) in designing wireless communication systems for an accurate estimation of the expected QoS (quality of service);
* We demonstrate the usefulness of our proposal in issues occurring in designing scalar quantizers for the Gaussian source. In Table 1 we present the symbols used in paper and their brief description.