# A Novel Fractional Fourier Transform-Based ASK-OFDM System for Underwater Acoustic Communications
## 6 Numerical Results  
This section presents numerical results for the proposed system. We investigate both the ordinary FFT-OFDM well studied for UWA systems with QPSK (4-QAM) and 16 QAM, and compare it to the novel FrFT-OFDM systems with the following parameters: \(N=64\) subcarriers; cyclic prefix (nCP) = 8, delay taps (nTap) = 2 and modulation scheme is \(M\)-ary ASK. The propagation environment is a frequency selective UWA channel in the presence of CFO, where the Rician \(k\)-factor is in the order of 10 for UWA channel model. Also, a wide range of fractional order (\(\alpha\)) has been tried to verify the simulation results; specifically, \(\alpha=1.0\) (FFT), \(\alpha=1.576\), \(\alpha=1.580\), \(\alpha=1.582\), \(\alpha=1.584\) and \(\alpha=1.62\). Figure 4 shows a BER performance comparison of a FrFT-based OFDM system assuming \(M\)-ary ASK and \(M\)-ary QAM modulation techniques in UWA channel with CFO = 0.05. The plot shows that both 2-ary FrFT-ASK-OFDM and 4-ary FrFT-QAM-OFDM have identical BER performance. Similarly, 4-ary FrFT-ASK-OFDM has similar BER performance as 16-ary FrFT-QAM-OFDM. Thus, \(\sqrt{(M)}\)-ary FrFT-ASK-OFDM achieves comparable BER performance to their counterpart of M-ary FrFT-QAM-OFDM. All simulations have been tested for 10,000 samples. Figure 5 shows a BER comparison of 2-ASK FrFT and the FFT-based OFDM system over the frequency selective UWA channel in the presence of different CFOs. As can be seen, the proposed FrFT-ASK-OFDM system outperforms the corresponding FFT-ASK-OFDM in the UWA channel. Figure 6 shows the BER performance of the FrFT-OFDM system at different values of fractional order (\(\alpha\)) in the same channel conditions with CFO = 0.15. It is visible that at \(\alpha=\pi/2\), both the FrFT-ASK-OFDM system and the FFT-ASK-OFDM system behave in an identical manner. Also, the minimum BER performance is achieved for a value of \(\alpha=1.584\) and the performance degrades with changing this value. Figure 4: BER performance comparison of M-ary FrFT-ASK-OFDM and M-ary FrFT-QAM-OFDM in the frequency-selective UWA channel with CFO = 0.05. Figure 7 depicts the BER performance comparison of 2-ASK, 4-ASK, 8-ASK and 16-ASK FrFT-OFDM in the UWA channel with CFO = 0.05, which shows the tradeoff between BER performance and data rate. Figure 5: BER for 2-ASK FrFT-based OFDM over the frequency-selective UWA channel with CFO = 0.1, 0.15, and 0.2. Figure 6: BER for 2-ASK FrFT-based OFDM over the frequency-selective UWA channel with CFO = 0.15 for different values of \(\alpha\). A summary of the achievable BER for QPSK, 16-QAM, 4-ASK and 16-ASK for different SNRs is given in Table 1 for CFO = 0.05. As can be seen, the BER ranges from 0.001 to 0.094. Recent BER results presented in the literature ranged from 0.005 to 0.25 as in [35]. We aimed to calculate the spectral efficiency and data rate based on realistic parameters reported in the literature, and compare them for the cases under investigation. For \(T\) = 85.33 ms, \(T_{g}\) = 25 ms, \(N_{d}\) = 672, and \(N\) = 1024, the spectral efficiencies and data rates are as follows. For QPSK, the spectral efficiency \(\alpha_{Q}\) = 1.014, and for 16-QAM \(\alpha_{Q}\) = 2.02821. The corresponding spectral efficiencies for 4-ASK and 16-ASK are 2.024 and 4.0485, respectively. The corresponding data rates assuming long range \(B=5\) kHz, medium range \(B=10\) kHz, and small range \(B=50\) kHz are summarized in Table 2. As can be seen, for the same modulation order, ASK-OFDM achieves approximately double the data rate compared to QAM-OFDM. \begin{table}
\begin{tabular}{c c c c c} \hline  & **QPSK** & **16-QAM** & **4-ASK** & **16-ASK** \\ \hline SNR = 10 dB & 0.021 & 0.039 & 0.039 & 0.094 \\ SNR = 17 dB & 0.006 & 0.013 & 0.013 & 0.037 \\ SnR = 25 dB & 0.001 & 0.004 & 0.004 & 0.016 \\ \hline \end{tabular}
\end{table} 
Table 1: BER comparison of OFDM systems under investigation at different SNRs for CFO = 0.05.