# Closed-Form Performance Analysis of the Inverse Power Lomax Fading Channel Model
## 4 Simulation and Results' Analysis
### Experimental Verification
The analysis demonstrated that the IPL model can be successfully applied in all the considered scenarios. Thus, it supports the flexibility of the model and the versatility of its applicability. The dependence of the outage probability \(P_{\text{out}}(\gamma)\) on the average signal-to-noise ratio \(\gamma\) is illustrated in Figure 10 for various channel parameters. The subplots demonstrate the impact of \(\beta\) for a fixed small \(\alpha=0.1\) (see Figure 10a), the effect of \(\alpha\) for a fixed small \(\beta=2.1\) (see Figure 10b), and the impact of the threshold \(\gamma_{\text{th}}\) for fixed \(\alpha=0.5\) and \(\beta=2.1\). It can be seen that the derived high-SNR approximation for the OP is extremely accurate for a small power-transformation parameter, irrespective of the value of the shape parameter. Moreover, it is observed that a small \(\alpha\) has a greater impact on the OP than \(\beta\) and \(\gamma_{\text{th}}\). Reasonable values of the OP (e.g., \(P_{\text{out}}\sim 10^{-4}\) with \(\beta=2.1\)) are achieved around 23.6 dB for \(\alpha=0.1\) and 11 dB for \(\alpha=2.1\). If \(\alpha=0.1\) is preserved, the same change in \(\beta\) (i.e., from \(\beta=2.1\) to \(\beta=4.1\)) reduces \(\gamma\) (required to achieve \(P_{\text{out}}\sim 10^{-4}\)) by only about 5 dB (from 23.6 dB to 18.7 dB). Figure 11 displays the average bit error rate (ABER) as a function of \(\gamma\) for different modulations and channel parameters, including the impact of the constellation size for M-PSK (see Figure 11a) and M-QAM (see Figure 11b) modulations with \(\alpha=1\), \(\beta=3\), and the effect of fading parameters for QAM-4 (see Figure 11c). The modulations were chosen to cover cases with both low-dimensional constellations (e.g., BPSK, QPSK, and QAM-4) and high-dimensional constellations (e.g., PSK-8 for PSK modulations, QAM-128, and QAM-1024). It is clear that reasonable ABER values (\(\tilde{P}_{\text{er}}\sim 10^{-6}\)) are successfully achieved for a moderate \(\tilde{\gamma}\) (i.e., \(\gamma\sim 15\)-30 dB) only for a small-to-moderate \(M\) (not higher than 128). The impact of fading parameters (see Figure 11c) was analyzed for five distinct sets of parameters: either a small \(\alpha\) or small \(\beta\) (for a large second parameter), both parameters being either small or large, and an intermediate scenario (with both a moderate \(\alpha\) and \(\beta\)). It can be seen that in the case of a small power-transformation parameter value, \(\tilde{P}_{\text{er}}\sim 10^{-6}\) cannot be achieved for any reasonable average SNR; thus, such a scenario (corresponding to deep fading) demonstrates an extremely poor link quality. In contrast, if both parameters are relatively high (\(\alpha=3,\beta=10\)), the required SNR can be as small as 15.5 dB. Again, as demonstrated in the previous plots, the high-SNR approximation becomes less accurate with a higher \(\alpha\). Figure 12 presents the ergodic capacity \(C\) of the IPL channel as a function of the average SNR \(\tilde{\gamma}\) for different sets of channel parameters \(\alpha\) and \(\beta\). As a reference line, the ergodic capacity for a Rayleigh channel is also presented. It can be noted that small values of \(\alpha\) correspond to cases where the capacity is smaller than that of a Rayleigh channel (i.e., the channel is in the HRR in the EC sense). For large values of the power-transformation parameter, the capacity is greater than Rayleigh's, even for a small \(\beta\), indicating that the impact of \(\alpha\) on the EC is more pronounced. Figure 9: AoF vs. channel parameters \((\alpha,\beta)\). Figure 11: ABER vs. \(\bar{\gamma}\) for modulations and channel parameters: (**a**) the impact of constellation size for M-PSK with \(\alpha=1,\beta=3\), (**b**) the impact of constellation size for M-QAM with \(\alpha=1,\beta=3\), and (**c**) the impact of fading parameters for QAM-4.