# \(\alpha\)-Fluctuating Nakagami-\(m\) Fading Model for Wireless Communications
## 5 Simulation and Results' Analysis
### \(\alpha\)-Fluctuating Nakagami-m Model Performance
Figure 5: AoF comparison for various channel parameters. Figure 7 presents the average bit error rate (ABER) as a function of SNR for both coherent and non-coherent modulations, evaluated under different \(\alpha\) values with \(m=0.5\) and \(m_{s}=2\). The figure includes results for widely used modulation schemes such as BPSK, 16-QAM, DBPSK, and 16-FSK, allowing for a comprehensive assessment of system performance under various modulation strategies. The solid lines depict the exact analytical solutions, while the dashed lines represent the high-SNR asymptotic approximations. Additionally, numerical integration results (circle markers) and Monte Carlo simulations (cross markers) validate the theoretical expressions. The plot shows that coherent modulations consistently outperform non-coherent modulations with the same constellation size, with BPSK and 16-QAM achieving lower error rates than DBPSK and 16-FSK, respectively, at the same SNR levels. Increasing \(\alpha\) results in a steeper decline in ABER, confirming its role in mitigating severe fading. The performance gap between coherent and non-coherent schemes widens at higher SNRs, emphasizing the benefits of coherent detection in hyper-Rayleigh conditions. The asymptotic approximations provide reliable error rate estimates beyond \(\bar{\gamma}=25\) dB for small \(\alpha\) (i.e., \(\alpha\approx 1\)) and for large \(\alpha\) beyond 10 dB, reducing computational complexity while maintaining accuracy. The results indicate that selecting an appropriate \(\alpha\) value and modulation scheme is crucial for optimizing communication link quality in WSN operating under Fluctuating Nakagami-\(m\) fading conditions. Figure 8 depicts the ergodic capacity as a function of SNR for different \(\alpha\) values illustrating the impact of fading severity on spectral efficiency. The capacity results are compared with the theoretical upper bound for an additive white Gaussian noise (AWGN) channel, represented by a dot-dashed black line. The solid lines correspond to the exact analytical capacity expressions, while the dashed lines depict the high-SNR asymptotic approximations. The figure shows that higher \(\alpha\) values yield greater capacity, confirming that less severe fading allows for improved spectral efficiency. At \(\gamma=10\) dB, the capacity for \(\alpha=1\) is approximately \(1.5\) bits/\(\mathrm{s}/\mathrm{Hz}\), whereas for \(\alpha=3\), it exceeds \(3\) bits/\(\mathrm{s}/\mathrm{Hz}\), demonstrating a twofold increase in achievable data rates. The capacity curves exhibit logarithmic growth with SNR, aligning with theoretical predictions for fading channels. However, at low \(\alpha\) values, the capacity saturates even at high SNRs, indicating that deep  
Figure 6: Outage probability for various \(\alpha\) with \(m=1\), \(m_{s}=2.5\), and \(\gamma_{th}=0\) dB: solid lines—the derived analytic solutions (6), dashed blue lines—the derived asymptotic expression (16), circle markers—numeric integration, black cross markers—Monte Carlo simulation. fades significantly degrade achievable rates. The Monte Carlo simulation results (black cross markers) and numerical integration results closely follow the analytical predictions, validating the derived expressions. The figure highlights that the \(\alpha\)-Fluctuating Nakagami-\(m\) model effectively captures the impact of fading on channel capacity, providing a useful framework for optimizing transmission strategies. Figure 8: Capacity for various \(\bar{\gamma}\) and fading conditions: solid lines—the derived analytic solution (32), dashed blue lines—the derived asymptotic expression (34), circle markers—numeric integration (31), black cross markers—Monte Carlo simulation, black dot-dashed line—AWGN channel.