# Two Interval Upper-Bound \(Q\)-Function Approximations with Applications
## 5 Application to Fading Channels Performance Analysis  
Let us now exploit the advantages of the application of the proposed \(Q\)-function approximations in the ASEP evaluation over Nakagami-\(m\) fading channels when a BPSK modulation is assumed. To numerically evaluate the ASEP, we need to average the expression for the conditional SEP over the Nakagami-\(m\) fading channel amplitude for the assumed modulation format, i.e., to evaluate the following expression:  
\[P_{b}=\int_{0}^{\infty}Q\bigg{(}\sqrt{\frac{2E_{b}}{N_{0}}}x\bigg{)}f_{x}(x)dx, \tag{12}\]  
where \(f_{x}(x)\) is the PDF of the Nakagami-\(m\) fading channel amplitude. \(E_{b}\) and \(N_{0}\) denote the average bit energy and the one-sided noise power spectral density, respectively. Figure 3: The comparison of the absolute relative errors of the two novel interval upper-bound approximations of the \(Q\)-function with the one from [13]. Figure 2: The comparison of the absolute relative errors of the first interval upper-bound approximation of the \(Q\)-function and selected ones from the literature [7; 11; 12; 13; 14]. The Nakagami-\(m\) distributed channel envelope has a PDF given by [6]:  
\[f_{x}(x)=\frac{2}{\Gamma(m)}\frac{m^{m}}{\Omega^{m}}x^{2m-1}\exp\bigg{\{}-m\frac{ x^{2}}{\Omega}\bigg{\}}. \tag{13}\]  
\(\Omega\) denotes the average signal power, \(m\) stands for the inverse normalized variance of \(x\), describing the fading severity [6]. \(\Gamma(m)\) represents the special Gamma function defined with [16]:  
\[\Gamma(m)=\int_{0}^{\infty}x^{m-1}\exp\{-x\}dx. \tag{14}\]  
The Nakagami-\(m\) fading model fits data obtained in scenarios with multipath scattering and presents large delay-time spreads with different clusters of reflected waves [1]. This model provides the best fits for channel data within satellite-to-indoor and satellite-to outdoor radio communication and also fits data within observed land-mobile and ionospheric radio links [28]. As a general fading model, the Nakagami-\(m\) fading model can be reduced to the Rayleigh fading model (by setting the parameter \(m\) value to \(m=1\)), and the one-sided Gaussian fading model (for \(m=1/2\)) [29]. The Nakagami-\(m\) fading model closely approximates the Nakagami-\(q\) model as well for \(m\leq 1\), and it could approximate the Rician model for \(m>1\)[30]. After substituting our expressions for the \(Q\)-function approximations, Equations (10) and (11), along with (20) into (12), we could efficiently evaluate values for the ASEP over Nakagami-\(m\) fading conditions. Comparisons of the ASEP values over Nakagami-\(m\) fading channels for various values of parameter \(m\), obtained by applying our \(Q\)-function approximations and selected ones from the literature are provided in Tables 3-5. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \(\mathbf{E_{n}N_{0}}\) **(dB)** & **Exact (\(m=4\))** & \(\mathbf{Q^{cm,2}(c)}\) & \(\mathbf{Q^{cm,3}(c)}\) & \(\mathbf{Q^{[4]}}\) **(c)** & \(\mathbf{Q^{[21]}(c)}\)  
From Tables 3-5, it can be noticed that the values of the ASEP for a BPSK modulation and a Nakagami-\(m\) fading channel could be efficiently and accurately evaluated by using the proposed composite interval approximation \(Q^{com\_1}\) for all assumed values of parameter \(m\) in a wide range of the observed argument values. Furthermore, it can be concluded that by using \(Q^{com\_2}\) composite interval approximation, an additional improvement in the bounding average bit error rate (ABER) values in the whole range of fading conditions was provided. To point out an excellent match of the values of the ASEP for a BPSK modulation, calculated with the proposed composite approximations \(Q^{com\_2}(x)\) and \(Q^{com\_1}(x)\), and the exact ones, in Figure 4, we present the values for the absolute relative error of the BER for \(m=2\).