## 3 Methodology
### The Advanced Whale-Based Optimization Algorithm for Signal Detection in Underwater MI-MIMO Systems
\[\stackrel{{\rightarrow}}{{D}}=\left|\stackrel{{ \rightarrow}}{{C}}\times\stackrel{{\rightarrow}}{{X}}_{aim}- \stackrel{{\rightarrow}}{{X}}(t)\right| \tag{12}\]  
\[\stackrel{{\rightarrow}}{{X}}(t+1)=\beta\times\stackrel{{ \rightarrow}}{{X}}_{aim}-\stackrel{{\rightarrow}}{{A}}\times \stackrel{{\rightarrow}}{{D}} \tag{13}\]  
In this context, \(\stackrel{{\rightarrow}}{{D}}\) denotes the distance vector between a specific whale's position and the best whale position at iteration \(t\), where \(t\) is the current iteration number. \(\stackrel{{\rightarrow}}{{X}}_{aim}\) refers to the best position vector found so far in the optimization process, while \(\stackrel{{\rightarrow}}{{X}}(t)\) and \(\stackrel{{\rightarrow}}{{X}}(t+1)\) denote the position vectors of the whale at iterations \(t\) and \(t+1\), respectively. If the value of p falls within the range of 0.5 to 1, the whale is in the encircling phase and approaching the prey. In this case, a random search action is executed to expand the search range. \[\stackrel{{\rightarrow}}{{D}}=\left|\stackrel{{ \rightarrow}}{{C}}\times\stackrel{{\rightarrow}}{{X}}_{m}- \stackrel{{\rightarrow}}{{X}}(t)\right| \tag{14}\]  
\[\stackrel{{\rightarrow}}{{X}}(t+1)=\beta\times\stackrel{{ \rightarrow}}{{X}}_{m}-\stackrel{{\rightarrow}}{{A}}\times \stackrel{{\rightarrow}}{{D}} \tag{15}\]  
where \(\stackrel{{\rightarrow}}{{X}}_{m}\) is a randomly selected whale from the population. When \(p>0.5\), the whale algorithm enters the spiral predation phase. \[\stackrel{{\rightarrow}}{{D}}^{\prime}=\left|\stackrel{{ \rightarrow}}{{X}}_{aim}-\stackrel{{\rightarrow}}{{X}}(t)\right| \tag{16}\]\[\overset{\rightarrow}{X}(t+1)=\beta\times\overset{\rightarrow}{D}^{\prime}\times e ^{bl}\times\cos(2\pi l)+\overset{\rightarrow}{X}_{aim} \tag{17}\]  
where \(\overset{\rightarrow}{D}^{\prime}\) represents the distance vector between the whale and the prey, and \(b\) is a constant used to define the logarithmic spiral shape, usually set to 1. **Step 5**: Compute the fitness value after updating the location and population. **Step 6**: Check if the maximum number of iterations is reached. If the stopping criterion is satisfied, output the best solution; otherwise, increase t by 1 and update the parameters to continue the next iteration. For MIMO signal detection, ML detection is the best-performing algorithm within the search range of \(M^{N}\), where \(M\) is the modulation order, and \(N\) is the number of transmitting antennas. However, in practical applications with large modulation orders and sending antenna numbers, the computation resources required for ML detection are significant, necessitating the design of suboptimal detection algorithms. Unlike the principle of the ML detection algorithm, the AWOA considers the transmit signal vector as the solution space. The whale randomly selects a signal vector \(X=[x_{1},x_{2},\ldots x_{N_{T}}]\) and evaluates it using the fitness function to find the transmit signal with the smallest fitness value. This is the best detection signal for the MIMO system [44], as described in (18):  
\[ \begin{array}{rl}\hat{\mathbf{X}}&=\underset{X\in S}{argmin}\left\{|\mathbf{ Y}-\mathbf{H}\mathbf{X}|^{2}\right\}\\ &=\underset{X\in S}{argmax}\left[2X^{\mathbf{T}}H^{\mathbf{T}}\mathbf{Y}-X^{ \mathbf{T}}H^{\mathbf{T}}H\mathbf{X}\right]\end{array}  \tag{18}\]  
\(SS\) represents the solution space of all possible signals, \(\hat{\mathbf{X}}\) is the estimated transmit signal vector, \(Y\) is the received signal, and \(\mathbf{H}\) is the channel matrix. Therefore, Equation (18) is utilized as the fitness evaluation function in the AWOA for signal detection. To facilitate a more comprehensive understanding of MIMO signal detection using AWOA, we designed a schematic diagram to illustrate the actions of the whales, as depicted in Figure 4.  
Figure 4: The whale optimization algorithm in MI-MIMO signal detection. As shown in Figure 4, the AWOA uses three strategies to search for the best signal vector in the solution space at iteration \(t\).