## 5 Simulation Results  
The performance of the proposed system is evaluated by simulating the coded CPM-OFDM transceiver over a realistic underwater acoustic channel. In this work, we evaluated the performance of a binary CPM-OFDM system. Other simulation parameters are shown in Table 2.  
To generate a realistic UWA channel, we used a Bellhop simulator [3; 7]. We generated seven typical UWA channel profiles for seven transmitter-receiver distance scenarios. The channel scenarios for four of these distances are shown in Figure 3, while a typical power delay profile is shown in Figure 4. One of the reasons for these varying profiles is the multipath effect of the channel, which increases with increasing distance [66]. When a signal travels over unguided media such as a radio channel or underwater acoustic channel, the antenna (in the case of radio transmission) and hydrophone (in the case of underwater channels) transmits the signal in more than one direction, thus, creating several copies of the signal. The farther away the receiver, the more the transmitted signals get reflected from obstacles in the environment and the more multipaths are created. \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameters** & **Values** \\ \hline Number of subcarriers & 256 \\ CP Length & 64 \\ Modulation index & 23 rational values \\ Bandwidth & 10 kHz \\ OFDM symbol duration & 25.6 ms \\ Channel Encoder & BCH(63,10), BCH(63,16), BCH(31,6), BCH(31,11) \\ Tx–Rx distance & [100, 250, 400, 550, 700, 850, 1000] m \\ Transmitter depth & 100 m \\ Receiver depth & 100 m \\ Speed of sound in water & 1500 m/s \\ Speed of sound in bottom & 1200 m/s \\ Coherence time & 60 s \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Simulation parameters. To simulate the proposed system, 5000 OFDM blocks were transmitted. A BCH(63,10) encoder is used for encoding the data. With reference to Equation (4), the numerator \(K_{i}\) is varied between 1 and 15 to choose the values of the modulation index \(h\). To maintain the rationality of the modulation index, we varied the denominator \(p\) between 2 and 16. As a result, we had 23 values of \(h\) in total that are shown in Table 3. We determined that this set is adequate to evaluate the intended system's performance, and that any other numbers outside this range do not result in any better performance. Furthermore, choosing non-rational values of \(h\), such as 1/3 results in a constellation with too many points that increases the complexity of the Viterbi decoder manifolds, is without any benefit. Figure 3: Channel profiles for four typical channel distances. Figure 4: Power delay profile of the UWA channel generated using Bellhop for the channel of Figure 3c. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Numerator (\(K_{\sharp}\))** & **Denominator (\(p\))** & **Modulation Index (\(h\))** \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 3: Values of modulation indices. In Figure 5, the BER of the proposed system is plotted against the channel distance. Each of the 23 curves of this figure is plotted for a fixed value of \(h\). The legend on the right-hand side of the figure enlists the \(h\) values used for each curve. The curves have been plotted starting from the best BER to the worst BER and the \(h\) values have been listed in the legend in the same order. For example, the first curve is for \(h=7/16\), which gives the best BER performance and is the first value in the legend. It is observed that there is no error when transmitting the signals over a channel distance of 100 and 250 m for the first five values of \(h\). Hence, these should be considered as error-free transmissions. It is also noted that the values of \(h\) that give superior performance are  
\[\left\{\frac{7}{16},\frac{9}{16},\frac{5}{8},\frac{3}{5},\frac{3}{8},\frac{2}{5 },\frac{5}{16}\right\}\]  
In the above set, only the sixth value of \(h\), i.e., \(2/5\), will have five phase states, as the value in the numerator of \(h\) is even.