## 3 Rate-Adaptive Scheme of Joint Control of Photoelectric Domain
### Rate-Adaptive Control of Electrical Domains
The probability distributions of PS\(-\)16 QAM and PS\(-\)64 QAM signals with different \(v\) values are shown in Figure 2.  
The average relative power of the signal is \(\overline{P}=\sum\limits_{i}|X_{i}|^{2}P(X_{i})\). As shown in Figure 2, the CPS technology increases the probability of constellation points that are closer to the origin and decreases the probability of constellation points that are far away from the origin, thus saving power. In order to ensure that the average power of the signal remains consistent before and after shaping, the constellation diagram of the reshaping signal is amplified by multiplying the coefficient \(\alpha\). The expression is as follows:  
\[E[|\alpha\cdot X|^{2}]=E[|X_{0}|^{2}]. \tag{11}\]  
Take the PS\(-\)16 QAM signal as an example, and let \(v=0.2\). The probability distribution of each constellation point can be obtained according to (9), and then the average relative power calculated after shaping is 4.7. The average relative power of the UN\(-\)16 QAM signal is 10. According to (10), we obtain \(\alpha\approx 1.459\), that is, the amplitude of each symbol of PS\(-\)16 QAM increases by a factor of 1.459. As shown in Figure 3, the Euclidean distance between the points of each constellation becomes larger after shaping and the error tolerance is higher, so the communication distance is longer under the desired BER threshold. As shown in Figure 4, the constellation diagrams at the receiver and the BER performance before and after shaping are shown. Specifically, (a) and (c) are the constellation  
Figure 1.0 Comparison of amplitude changes of symbols before (**a**) and after (**b**) shaping for 16 QAM. diagrams of UN\(-\)16 QAM and PS\(-\)16 QAM when the SNR = 10 dB. Correspondingly, (b) and (d) are the constellation diagrams when the SNR = 15 dB. It can be seen from the simulation diagram that the constellation points of the shaped signals are concentrated in the inner circle, which reduces the average transmission power of the system and reduces the power demand. Figure 4e shows a comparison of the BER performance before and after shaping. It can be seen that at the FEC threshold, when comparing the same modulation orders, the SNR is improved by approximately 4 dB after shaping when the entropy loss is 1. This indicates that the shaped signals significantly enhance the BER performance. Figure 4f shows a comparison of signals at different shaping levels, and the results indicate that a higher degree of shaping leads to a better performance. The probability of each symbol calculated by (9) is the ideal distribution \(P_{A}\). In practice, the number of occurrences of each symbol is an integer and the sum is the total number of symbols. For the same \(P_{A}\), there may be a variety of symbol numbers with different values, and then there are different probability combinations. To identify the actual probability distribution that is closest to the ideal probability distribution, ref. [26] uses Kullback-Leibler (KL) divergence to measure the degree of proximity between the two. The shaping signal can be obtained by the above process. In order to transmit more information at the same SNR, CPS is combined with the uniform QAM rate-adaptive algorithm; by switching between the uniform QAM signal and the shaping signal, the rate and spectral efficiency switching mechanism achieve a higher granularity, and the communication distance of the shaped signal is longer. The specific simulation results are shown in Section 4.