# Two Interval Upper-Bound \(Q\)-Function Approximations with Applications
## 5 Application to Fading Channels Performance Analysis
Similarly, the PDF of \(\gamma\) could be obtained by introducing a change of variables in the expression for the fading amplitude PDF, \(p_{x}(x)\), yielding [19]:  
\[p_{\gamma}(\gamma)=p\Bigg{(}\sqrt{\frac{\Omega}{\tilde{\gamma}}}\Bigg{)}\ |J|, \tag{18}\]  
and  
\[|J|=\frac{dx}{d\gamma}=\frac{1}{2\sqrt{\gamma\frac{\tilde{\gamma}}{\Omega}}}. \tag{19}\]  
A comparison of the exact and approximated values of the ASEP for the DE-QPSK modulation are presented in Tables 6-8. The exact values of the ASEP were calculated by means of numerical integration using Equation (1). For determining the approximated values, Equations (10) and (11) were applied as well as the corresponding ones from the literature, carefully chosen for comparison purposes. From Tables 6-8 and Figure 5, it can be concluded that the approximate values of the ASEP for DE-QPSK over Nakagami-\(m\) fading channels could be efficiently and precisely evaluated for different values of the parameter \(m\) by utilizing the proposed approximations. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \(E_{\text{n}}N_{\text{0}}\) & Exact & \(Q^{\text{cm,2-3}}(x)\) & \(Q^{\text{cm,2-3}}(x)\) & \(Q^{\text{[44}}(x)\) & \(Q^{\text{[11]}}(x)\) & \(Q^{\text{[12]}}(x)\) & \(Q^{\text{[13]}}(x)\) & \(Q_{\text{[17]}}(x)\) & \(Q_{\text{[27]}}(x)\) \\ \hline \(-10\) & \(7.1831\times 10^{-1}\) & \(7.1870\times 10^{-1}\) & \(7.1880\times 10^{-1}\) & \(7.0980\times 10^{-1}\) & \(7.2608\times 10^{-1}\) & \(6.7508\times 10^{-1}\) & \(7.2013\times 10^{-1}\) & \(7.1873\times 10^{-1}\) & \(7.2865\times 10^{-1}\) \\ \hline \(0\) & \(4.8137\times 10^{-1}\) & \(4.8220\times 10^{-1}\) & \(4.8344\times 10^{-1}\) & \(4.9996\times 10^{-1}\) & \(5.0492\times 10^{-1}\) & \(5.0609\times 10^{-1}\) & \(4.822\times 10^{-1}\) & \(4.8340\times 10^{-1}\) & \(4.8760\times 10^{-1}\) \\ \hline \(10\) & \(0.4325\times 10^{-1}\) & \(0.4308\times 10^{-1}\) & \(0.4344\times 10^{-1}\) & \(0.4570\times 10^{-1}\) & \(0.4655\times 10^{-1}\) & \(0.5122\times 10^{-1}\) & \(0.4352\times 10^{-1}\) & \(0.4357\times 10^{-1}\) & \(0.4350\times 10^{-1}\) \\ \hline \(20\) & \(3.0977\times 10^{-4}\) & \(3.1053\times 10^{-4}\) & \(3.1074\times 10^{-4}\) & \(3.2642\times 10^{-4}\) & \(3.3358\times 10^{-4}\) & \(3.6930\times 10^{-4}\) & \(3.1288\times 10^{-4}\) & \(3.1196\times 10^{-4}\) & \(3.1095\times 10^{-4}\) \\ \hline \(30\) & \(1.0814\times 10^{-6}\) & \(1.0838\times 10^{-6}\) & \(1.0845\times 10^{-4}\) & \(1.1389\times 10^{-6}\) & \(1.1642\times 10^{-6}\) & \(1.2888\times 10^{-6}\) & \(1.0928\times 10^{-6}\) & \(1.0889\times 10^{-6}\) & \(1.0851\times 10^{-6}\) \\ \hline \(40\) & \(3.4546\times 10^{-9}\) & \(3.4621\times 10^{-9}\) & \(3.4642\times 10^{-9}\) & \(3.6378\times 10^{-9}\) & \(3.7190\times 10^{-9}\) & \(4.1169\times 10^{-9}\) & \(3.4911\times 10^{-9}\) & \(3.4783\times 10^{-9}\) & \(3.4664\times 10^{-9}\) \\ \hline \end{tabular}
\end{table} 
Table 6: Values of the exact and approximated ASEP of the DE-QPSK for parameter \(m=2.5\). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \(E_{\text{n}}N_{\text{0}}\) & Exact & \(Q^{\text{cm,2-3}}(x)\) & \(Q^{\text{cm,2-3}}(x)\) & \(Q^{\text{[44}}(x)\) & \(Q^{\text{[11]}}(x)\) & \(Q^{\text{[12]}}(x)\) & \(Q^{\text{[13]}}(x)\) & \(Q_{\text{[17]}}(x)\) & \(Q_{\text{[27]}}(x)\) \\ \hline \(-10\) & \(7.1831  
By using the approximations of the \(Q\)-function given by Equations (10) and (11), the ABER measures were bounded more closely than by using the closed-form \(Q\)-function approximations selected from the literature, in the whole range of fading conditions. Figure 5 presents the values of the absolute relative error of the ASEP DE-QPSK approximations for a Nakagami-\(m\) channel and \(m=2.5\), to point out the smallest possible deviation of the approximated ASEP DE-QPSK values from the exact ones in case of applying the proposed \(Q\)-function approximations.