# Two Interval Upper-Bound \(Q\)-Function Approximations with Applications
## 7 Conclusions  
In this paper, two novel interval upper-bound approximations of the \(Q\)-function were proposed. The first one had a simpler analytical form and, therefore, could be applied in a simpler manner, whereas the second one was more accurate. In addition, by comparing both approximations with the \(Q\)-function approximations of similar analytical forms from the literature, we ascertained noticeable improvements in terms of the accuracy achieved by the proposed interval upper-bound approximations of the \(Q\)-function in the whole domain of the function argument.  
In addition, comparisons of the estimated ASEP values for Nakagami-\(m\) fading channels for several values of parameter \(m\) were given in the paper for two modulation types, BPSK and DE-QPSK, to demonstrate the usefulness of the proposed \(Q\)-function approximations. Furthermore, we showed that the closed-form approximation of the \(Q\)-function could be used in the quantization design for variables from Gaussian sources, whereby using the closed-form \(Q\)-function approximation avoided solving the integral equations. Another advantage of the new composite approximation formulas was a very small increase in computational complexity. That made them suitable for various applications including the analysis of the performance of various communication channels.  
In brief, for the observed applications, we showed the superiority of our proposal over the \(Q\)-function approximations previously used in the literature, considered in the paper for comparison purposes. Due to the relatively small absolute relative errors that the novel \(Q\)-function approximations provide, we can anticipate that they can find applications in a variety of problems encountered not only in communication theory, but also in a number of analyses involving the estimation of the \(Q\)-function.  
Conceptualization and methodology, Z.P. and A.M.; software and validation, A.M.; formal analysis, J.N. and M.D.P.; writing--original draft preparation, A.M. and N.K.; writ  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(R\) (Bits/Sample) & \(x_{i}\), \(i=0,1,\ldots,N/2\) & \(y_{i}\), \(i=1,\ldots,N/2\) & \(D\) & **SQNR (dB)** \\ \hline
2 & 0 & & 0.1458 & 8.36 \\  & 0.6787 & 0.3205 & & \\  & +\(\infty\) & 1.1555 & & \\ \hline
3 & 0 & & 0.0578 & 12.38 \\  & 0.3207 & 0.1582 & & \\  & 0.6788 & 0.4924 & & \\  & 1.1551 & 0.8923 & & \\  & +\(\infty\) & 1.5375 & & \\ \hline
4 & 0 & & 0.0235 & 16.30 \\  & 0.1581 & 0.0788 & & \\  & 0.3203 & 0.2383 & & \\  & 0.4923 & 0.4053 & & \\  & 0.6789 & 0.5830 & & \\  & 0.8923 & 0.7810 & & \\  & 1.1553 & 1.0148 & & \\  & 1.5371 & 1.3227 & & \\  & +\(\infty\) & 1.8646 & & \\ \hline \hline \end{tabular}
\end{table}
Table 9: Scalar quantization designed to have equiprobable cells when quantizing a Gaussian variable with zero mean value and unit variance ing--review and editing, J.N., Z.P., M.D.P. and A.J.; visualization, A.M. and N.K. All authors have read and agreed to the published version of the manuscript.  
This research was supported by the Science Fund of the Republic of Serbia, 6527104, AI-Com-in-AI.  
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.