## 3 Methodology
### Underwater Magnetic Induction MIMO Communication System Model
#### 3.1.2 The Model of Underwater MI-MIMO  
Magnetic induction communication offers several advantages in underwater environments. However, its low channel capacity poses a limitation. MIMO is employed to increase channel capacity and enhance signal transmission reliability. The use of multiple receiving and transmitting antennas enables the MI-MIMO system to mitigate the effects of crosstalk signals between coils and improve the efficiency of the underwater magnetic induction channel. Figure 2 shows the schematic diagram of the MI-MIMO system with an N \(\times\) N antenna array. In this figure, \(T_{XI}\) and \(T_{XN}\) represent the number of transmit antennas, while \(T_{RI}\) and \(T_{RN}\) refer to the number of receive antennas. CS indicates the crosstalk signal between each coil. \(L\) denotes the transmission distance, and \(d\) represents the distance between coils. The strength of the crosstalk signal between coils is directly proportional to the distance between them. The dotted line between the coils represents the transmitted signal. Let the channel parameter matrix be denoted by \(H\), where the element \(h_{i,j}(k)\) in the matrix represents the channel response from the \(i\)-th transmitter's antenna to the \(j\)-th receiver's antenna. The expression for \(H\) is given by (3):  
\[H=\left[ \begin{array}{cccc}h_{1,1}(k)&h_{1,2}\ (k)&\cdots&h_{1,N_{T}}(k)\\ h_{2,1}(k)&h_{2,2}(k)&\cdots&h_{1,N_{T}}(k)\\ \vdots&\vdots&\ddots&\vdots\\ h_{N_{R},1}(k)&h_{N_{R},2}(k)&\cdots&h_{N_{R},N_{T}}(k)\end{array} \right] \tag{3}\]  
Let the vector of the transmitted signal be represented by \(X\ =\ [x_{1},x_{2},\cdots x_{N_{T}}]^{T}\), and the corresponding vector at the receiving end be denoted by \(Y\). The expression for \(Y\) is given by Equation (4):  
\[Y=HX+N \tag{4}\]  
Here, \(N=\left[n_{1},n_{2},\cdots n_{N_{R}}\right]\) represents the N*1-dimensional Gaussian white noise vector generated by the MI-MIMO system itself, where the mean value of each element is 0, and the variance is \(\delta^{2}\). Thus, Equation (4) can be written as (5):  
\[Y=\left[ \begin{array}{cccc}h_{1,1}(k)&h_{1,2}\ (k)&\cdots&h_{1,N_{T}}(k)\\ h_{2,1}(k)&h_{2,2}(k)&\cdots&h_{1,N_{T}}(k)\\ \vdots&\vdots&\ddots&\vdots\\ h_{N_{R},1}(k)&h_{N_{R},2}(k)&\cdots&h_{N_{R},N_{T}}(k)\end{array} \right] \left[ \begin{array}{c}x_{1}\\ x_{2}\\ \vdots\\ x_{N_{T}}\end{array} \right]+\left[ \begin{array}{c}n_{1}\\ n_{2}\\ \vdots\\ n_{N_{R}}\end{array} \right] \tag{5}\]  
The equivalent circuit schematic of the MI-MIMO system depicted in Figure 3 is derived by applying appropriate circuit transformations to the original MI-MIMO schematic. Figure 2: N \(\times\) N MI-MIMO system. In Figure 3, the circuit elements \(Z_{t}\), \(Z_{tn}\), \(Z_{r}\), and \(Z_{rn}\) represent the impedances of the transmitters and receivers, respectively. The load impedances of the receiver circuits are denoted by \(Z_{L}\) and \(Z_{Ln}\), while \(C\) represents the capacitance of each circuit. The circuit voltage is represented by \(V\), and \(V_{tn}\) means the voltage across the mutual inductance signals M\({}_{1}\), M\({}_{2}\), M\({}_{3}\), M\({}_{4}\), M\({}_{5}\), and M\({}_{6}\) between the coils. The interference between the coils on the same side can be eliminated by adjusting the coupling strength and coil spacing. \(L_{1}\), \(Ln\), \(L_{lr}\), and \(L_{nr}\) denote the transmitting and receiving coils. When the MI-MIMO system has \(N_{T}\) transmitting coils and \(N_{R}\) receiving coils, the path loss can be expressed as (6) [22; 32]:  
\[P_{MI-MIMO}=\frac{4\pi^{2}f^{2}M_{i}(f)^{2}Z_{L}\Big{(}Z_{t}+4\pi^{2}f^{2}M_{i} (f)^{2}/(Z_{r}+Z_{L})\Big{)}}{\Big{(}Z_{r}+Z_{L}+4\pi^{2}f^{2}M_{i}(f)^{2}/Z_{ t}\Big{)}^{2}Z_{t}^{2}} \tag{6}\]  
The channel capacity is expressed as (7) [22; 32], where \(f\) represents the signal frequency.