## 3 Rate-Adaptive Scheme of Joint Control of Photoelectric Domain  
### Rate-Adaptive Control of Electrical Domains  
When the transmitter and receiver communicate, the receiver estimates the SNR and feeds back the next set of modulation information to the transmitter to ensure the optimal performance of the system. A block diagram of the UWOC adaptive control system is \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value/Unit** \\ \hline \(c(\lambda)\) & 0.0145 m\({}^{-1}\) \\ \(\phi_{1/2}\) & 5\({}^{\circ}\) \\ \(\phi\) & 0\({}^{\circ}\) \\ \(\psi\) & 0\({}^{\circ}\) \\ \(D_{t}\) & 50 mm \\ \(D_{r}\) & 50 mm \\ \(\eta_{t}\) & 0.1289 \\ \(\eta_{r}\) & 0.9500 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Key parameters of underwater channel model. shown in Figure 1. When the system performs rate-adaptive control, it is assumed that the channel will change when the distance between the transmitter and receiver changes. In the simulation, the transmission power is set to 80 mW. When the system sends a signal, the received signal can be expressed as \(y(t)=x(t)h+n(t)\) according to (4). Under the assumption that the noise and transmitted signal are statistically independent, the received signal power is expressed as follows [8]:  
\[\sigma_{y}^{2}=|h|^{2}P_{x}+\sigma_{\omega}^{2}, \tag{6}\]  
where \(h\) represents the underwater channel gain and \(P_{x}\) represents the transmitting signal power. When the system does not send signals, only the noise is captured at the receiver [23], and then the variance, that is, the noise's power, can be measured. The expression is as follows:  
\[\sigma_{y}^{2}=\sigma_{\omega}^{2}=\frac{1}{N}\sum\limits_{k=1}^{N}|y_{n}(k)|^ {2}. \tag{7}\]  
Therefore, the estimated SNR can be expressed as follows:  
\[\widetilde{SNR}=\frac{\sigma_{y}^{2}-\sigma_{\omega}^{2}}{\sigma_{\omega}^{2} }=\frac{\sigma_{y}^{2}}{\sigma_{\omega}^{2}}-1. \tag{8}\]  
When the receiver obtains link state information at this time, it can select the appropriate modulation mode under the premise of satisfying the set BER threshold, thereby extending the dynamic range of the communication distance and enhancing the system's flexibility. When the distance between the transmitter and receiver increases, leading to a decrease in the SNR, the system adaptively selects a lower-order modulation scheme. This change sacrifices the rate and improves the BER performance. In order to make up for this rate loss, CPS technology is combined with the above method; by reducing the probability of constellation points with high energy in the outer circle and increasing the probability of constellation points with low energy in the inner circle [24], the average power of the constellation diagram is effectively reduced. Under the premise of ensuring consistent power, the signal after probability shaping has a higher fault tolerance rate and longer communication distance. The specific reasons for this will be explained below. The signal probability distribution that best matches the AWGN channel is MB distribution, which can be expressed as follows [25]:  
\[P_{X}(x_{k})=\frac{\exp(-v(\text{Re}(x_{k})^{2}+\text{Im}(x_{k})^{2}))}{\sum \limits_{i=1}^{M}\exp(-v(\text{Re}(x_{i})^{2}+\text{Im}(x_{i})^{2}))}, \tag{9}\]  
where \(x_{i}\) represents the \(i\)-th constellation point of the complex MQAM and \(v\) is a probability shaping factor that determines the probability distribution of the constellation points. The size of information entropy is determined by the probability of the occurrence of each symbol, and the unit is bit/symbol. The expression is as follows:  
Figure 1: Block diagram of UWOC adaptive control system. \[H(X)=E[I(X)]=\sum_{i=1}^{M}p(x_{i})I(x_{i})=-\sum_{i=1}^{M}p(x_{i})\log_{2}p(x_{i}), \tag{10}\]  
therefore, when \(v\) decreases, the degree of probability shaping decreases; conversely, when \(v\) increases, the degree of shaping increases, and the information entropy decreases.