## 3 System Model
### Joint Channel and Power Assignment  
In this study, the algorithm aims to achieve biologically friendly communication. It first obtains the positions and channel state information of marine organisms through perception. Then, using a game-theoretic approach, it allocates system power to enhance the communication quality while maximizing the system capacity. Assume that there are L sub-channels in the underwater acoustic channel, Subuser \(k\in[1,K]\), one of the sub-channels \(l\in[1,L]\), and assuming that the other interfering user is m, then the interfering user \(m\in[1,M]\), \(a_{lk}\) indicates whether channel \(l\) is assigned to user \(k\), specifically defined as:  
\[\alpha=\left\{ \begin{array}{ll}1,\ k\ occupies\ l\\ 0,\ otherwise\end{array} \right. \tag{1}\]  
Figure 2: Algorithm flow. Compared with the signal-to-noise ratio, the Signal-to-Interference-plus-Noise-Ratio (_SINR_) can better reflect the communication quality of the communication system, which is defined as:  
\[SINR=\frac{P_{lk}h_{lk}}{L_{0}Ba_{lk}+\sum\limits_{m\
eq k,\ m=1}^{M}P_{m}h_{mk}} \tag{2}\]  
Among them, \(P_{lk}\) is the transmitting power of cognitive user \(k\) on channel \(l\), \(P_{m}\) is the transmitting power of other interfering node \(m\), \(L_{0}\) represents the power spectral density of noise, \(B\) represents the bandwidth of a sub-channel, and \(h_{lk}\) represents the channel gain of the cognitive user \(k\) communicating on the sub-channel \(l\). The channel gain can be obtained by sending the pilot signal from the receiving node. According to [PERSON]'s theory, the channel capacity \(C_{k}\) of the \(K_{th}\) cognitive user can be expressed as follows:  
\[C_{k}=\sum\limits_{l=1}^{N}a_{lk}Blog_{2}\left(1+\frac{P_{lk}h_{lk}}{L_{0}Ba_ {lk}+\sum\limits_{m\
eq k,\ m=1}^{M}P_{m}h_{mk}}\right) \tag{3}\]  
Therefore, the capacity of the multi-user underwater acoustic channel can be expressed as follows:  
\[C_{k}=\sum\limits_{k=1}^{K}\sum\limits_{l=1}^{N}a_{lk}Blog_{2}\left(1+\frac{P _{lk}h_{lk}}{L_{0}Ba_{lk}+\sum\limits_{m\
eq k,\ m=1}^{M}P_{m}h_{mk}}\right) \tag{4}\]  
To achieve biologically friendly communication, the transmission power of underwater sensor nodes acting as secondary users need to be restricted by the auditory threshold of marine mammals. The National Marine Fisheries Service (NMFS) stipulates [21] that when the auditory threshold of marine mammals reaches a threshold value of 180 dB, it is classified as permanent injury Level \(A\). When the auditory threshold reaches a threshold value of 160 dB, it is classified as Level \(B\), indicating behavioral effects. To ensure that the life of marine mammals is not affected, the upper limit of the transmission power of underwater sensor nodes is calculated as follows:  
\[P_{lim}=A(D_{min}^{k},f_{n})\cdot 10^{[0.1(\delta-170.77)]} \tag{5}\]  
where \(f_{n}\) is the central frequency used by marine mammals to communicate in the channel, and \(d_{k}\) is the distance between cognitive user nodes that communicate with each other. After the variable \(a_{lk}\) is scaled to a continuous variable from 0 to 1, it can be regarded as a coefficient. Then, in Formula (4), we can solve the expression of \(a_{lk}\) by taking the partial derivative of \(a_{lk}\) and making its derivative equal to 0, as shown in Equation (6):  
\[\frac{\partial C_{k}}{\partial a_{lk}}=0\Rightarrow ln(\frac{P_{lk}h_{lk}}{a _{lk}})-\frac{P_{lk}h_{lk}}{a_{lk}}=\varepsilon \tag{6}\]  
where \(\varepsilon\) is a constant, and if \(\frac{P_{lk}h_{lk}}{a_{lk}}\) in Equation (6) is taken as a global variable \(u\), Equation (6) can be simplified to:  
\[lnu-u=\varepsilon\Rightarrow a_{lk}=\frac{P_{lk}h_{lk}}{\varepsilon^{ \varepsilon+u}} \tag{7}\]  
The channel allocation ratio is represented by \(a_{lk}\), and a user with the maximum \(a_{lk}\) value is selected to join a channel each time, which is as follows:  
\[a_{max}=max(\frac{P_{lk}h_{lk}}{\varepsilon^{\varepsilon+u}}) \tag{8}\]The cyclic execution of Formula (8) enables cognitive users to select the optimal channel access in turn.