# CPM-OFDM Performance over Underwater Acoustic Channels
## 5 Simulation Results  
The performance of the proposed system was evaluated using simulations where a CPM-OFDM transceiver is communicating over a UWA channel. The simulation parameters used are shown in Table 1.  
We selected the values of the modulation index \(h\) by varying the numerator \(p\) from 1 to 15. We chose the values of the denominator \(q\) by varying it from 2 to 16 such that \(p/q\) remains rational. This gave us a total of 23 values of \(h\). Any other values outside this set do not give any better performance, and we found this set to be sufficient to assess the performance of the intended system. Moreover, selecting higher values of \(p\) and \(q\) would result in higher number of points in the constellation, increasing the decoder complexity. The error performance of a typical CPM-OFDM system for these 23 values of \(h\) is shown in Figure 7. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline Number of Subcarriers & 512 \\ CP Length & 128 (typically, 1/4 th the no. of subcarriers) \\ Modulation Index & 23 rational values \\ Bandwidth & 10 kHz \\ Average Path Gains & [0; \(-1.5\); \(-2.5\); \(-7.0\)] dB \\ Path Delays & [0; 1; 2.5; 5.4] ms \\ Sampling Frequency & 10,000 samples per second \\ OFDM Symbol Duration & 51.2 ms \\ No. of OFDM Symbols Transmitted & 160,000 \\ No. of Simulation Runs & 20 \\ Tx-Rx Distance & 500 m \\ Transmitter Depth & 50 m \\ Receiver Depth & 50 m \\ Doppler Frequency & 10 Hz \\ Atmospheric pressure & 1.01325 \(\times\) 10\({}^{5}\) Pa \\ Salinity & 35 parts/1000 \\ Density & 10\({}^{3}\) kg/m\({}^{3}\) \\ Water temperature & 25 \({}^{\circ}\)C \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters. It is evident that the best BER is achieved when \(h\) is \(7/16\), which has been highlighted with a yellow color. The performance of \(h=9/16\) is very similar to the case when \(h=7/16\) and is highlighted with an orange color. In both these cases, the number of points in the constellation will be \(2q\), i.e., \(32\). If a smaller number of points are desired in the constellation to decrease the decoder complexity, then \(h\) values with a smaller \(q\) could be selected, such as \(h=3/8\) that gives \(16\) points in the signal constellation at a slightly inferior BER. Another choice could be \(h=3/5\) that would give only \(10\) points in the signal constellation at the cost of BER. From the two tables, it is observed that in Category A, there is only one value of \(h\) that has an even value of \(p\) and it is \(h=2/5\). Although it gives the minimum number of points in the signal constellation, its BER is much inferior when compared to the best case of \(h=7/16\) or \(h=9/16\). The BER performance of the proposed scheme with varying transmitter-receiver distances is shown in Figure 8. Although performance degradation as the distance increases is obvious, CPM-OFDM still gives acceptable BER up to \(1.5\) km. Figure 9 illustrates the BER performance as a function of channel bandwidth. The lowest BER is noted for a channel \begin{table}
\begin{tabular}{c c c c c c c c} \hline \multicolumn{2}{c}{\(h=plq\)} & \multicolumn{6}{c}{**SNR (dB)**} \\ \multicolumn{2}{c}{\(p\)} & \(q\) & **0** & **5** & **10** & **15** & **20** & **25** & **30** \\ \hline
7 & 8 & 0.37788 & 0.303382 & 0.219266 & 0.132345 & 0.06243 & 0.028404 & 0.015389 \\
7 & 10 & 0.341236 & 0.188574 & 0.059911 & 0.00778 & 0.000159 & \(9.77\times 10^{-9}\) & 0 \\
7 & 16 & 0.329276 & 0.162725 & 0.037263 & 0.001883 & \(3.67\times 10^{-6}\) & 0 & 0 \\
9 & 10 & 0.388748 & 0.325557 & 0.250039 & 0.169949 & 0.104257 & 0.060448 & 0.033613 \\
9 & 16 & 0.329202 & 0.16274 & 0.037281 & 0.001882 & \(3.87\times 10^{-6}\) & 0 & 0 \\
11 & 16 & 0.337094 & 0.180739 & 0.052517 & 0.001882 & \(3.87\times 10^{-6}\) & 0 & 0 \\
13 & 16 & 0.357658 & 0.239519 & 0.113663 & 0.030164 & 0.002452 & \(9.13\times 10^{-6}\) & 0 \\
15 & 16 & 0.412149 & 0.367034 & 0.30793 & 0.232537 & 0.151276 & 0.082633 & 0.039155 \\ \hline \end{tabular}
\end{table} 
Table 2: \(Cont.\)