# Two Interval Upper-Bound \(Q\)-Function Approximations with Applications
## 5 Application to Fading Channels Performance Analysis
By using \(Q^{com\_2}(x)\) and \(Q^{com\_1}(x)\), very tight approximations (lower values of the absolute error of BER) were obtained in a wide range of argument values compared to cases utilizing the \(Q\)-function approximations from the literature selected for comparison purposes and used for the ASEP values computation. Our proposed \(Q\)-function approximations can also be applied for the ASEP evaluation for the assumed differentially encoded quadrature phase-shift keying (DE-QPSK) modulation format and Nakagami-\(m\) fading channels. The ASEP of DE-QPSK is given by [31]  
\[P_{b}=\int_{0}^{\infty}\overline{P_{b}}\cdot f_{\gamma}(\gamma)d\gamma, \tag{15}\]  
where \(f_{\gamma}(\gamma)\) is the Nakagami-\(m\) PDF of the signal-to-noise ratio (SNR) per symbol, distributed as specified in [32]:  
\[f_{\gamma}(\gamma)=\frac{m^{m}}{\Gamma(m)\gamma^{m}}\gamma^{m-1}\exp\left\{-m \frac{\gamma}{\tilde{\gamma}}\right\}, \tag{16}\]  
and  
\[\overline{P_{b}}=4Q(\sqrt{\gamma})-8Q^{2}(\sqrt{\gamma})+8Q^{3}(\sqrt{\gamma })-4Q^{4}(\sqrt{\gamma}), \tag{17}\]  
Figure 4: Absolute relative error of determining ASEP for BPSK modulation: comparison of the applied proposed approximations and the selected ones from the literature. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \(E_{\mathbf{a}}N_{\mathbf{b}}\) (dB) & Exact (m = 6) & \(Q^{com\_1}\)(\(\mathbf{q}\)) & \(Q^{com\_1}\)(\(\mathbf{q}\)) & \(Q^{[\mathbf{46}}\) (\(\mathbf{q}\)) & \(Q^{[\mathbf{21}}\)(\(\mathbf{q}\)) & \(Q^{[\mathbf{21}}\)(\(\mathbf{q}\)) & \(Q^{[\mathbf{21}}\)(\(\mathbf{q}\)) & \(Q^{[\mathbf{21}}\)(\(\mathbf{q}\)) & \(Q_{[\mathbf{21}}\)(\(\mathbf{q}\)) & \(Q_{[\mathbf{21}}\)(\(\mathbf{q}\)) \\ \hline \(-10\) & \(3.313\times 10^{-1}\) & \(3.325\times 10^{-1}\) & \(3.326\times 10^{-1}\) & \(3.307\times 10^{-1}\) & \(3.497\times 10^{-1}\) & \(2.945\times 10^{-1}\) & \(3.333\times 10^{-1}\) & \(3.325\times 10^{-1}\) & \(3.484\times 10^{-1}\) \\ \hline \(0\) & \(0.913\times 10^{-11}\) & \(0.916\times 10^{-1}\) & \(9.205\times 10^{-2}\) & \(0.983\times 10^{-1}\) & \(0.997\times 10^{-1}\) & \(1.080\times 10^{-1}\) & \(0.916\times 10^{-1}\) & \(0.921\times 10^{-1}\) & \(0.924\times 10^{-1}\) \\ \hline \(10\) & \(3.814\times 10^{-4}\) & \(3.820\times 10^{-4}\) & \(3.821\times 10^{-4}\) & \(3.996\times 10^{-4}\) & \(4.109\times 10^{-4}\) & \(4.551\times 10^{-4}\) & \(3.894\times 10^{-4}\) & \(3.843\times 10^{-4}\) & \(3.821\times 10^{-4}\) \\ \hline \(20\) & \(3.803\times 10^{-9}\) & \(3.806\times 10^{-9}\) & \(3.807\times 10^{-9}\) & \(3.945\times 10^{-9}\) & \(4.053\times 10^{-9}\) & \(4.334\times 10^{-9}\) & \(3.940\times 10^{-9}\) & \(3.828\times 10^{-9}\) & \(3.807\times 10^{-9}\) \\ \hline \(30\) & \(5.089\times 10^{-15}\) & \(5.093\times 10^{-15}\) & \(5.094\times 10^{-15}\) & \(5.273\times 10^{-15}\) & \(5.417\times 10^{-15}\) & \(5.771\times 10^{-15}\) & \(5.285\times 10^{-15}\) & \(5.122\times 10^{-15}\) & \(5.094\times 10^{-15}\) \\ \hline \(40\) & \(5.244\times 10^{-21}\) & \(5.248\times 10^{-21}\) & \(5.249\times 10^{-21}\) & \(5.433\times 10^{-21}\) & \(5.581\times 10^{-21}\) & \(5.944\times 10^{-21}\) & \(5.447\times 10^{-21}\) & \(5.278\times 10^{-21}\) & \(5.249\times 10^{-22}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison of ASEP values over Nakagami-\(m\) fading channels for parameter \(m=6\), obtained by applying our \(Q\)-function approximations and selected ones from the literature. where \(\tilde{\gamma}\) represents the average SNR per symbol. As is well-known from the telecommunication theory, the relation between the received carrier amplitude under fading influence \(x\), its mean squared value \(\Omega\), and the instantaneous signal-to-noise power ratio (SNR) per symbol, \(\gamma\), is given with \(\gamma=x^{2}\tilde{\gamma}/\Omega\).