## 3 Methodology
### The Advanced Whale-Based Optimization Algorithm for Signal Detection in Underwater MI-MIMO Systems  
The WOA is an optimization algorithm that emulates the hunting behavior of whales, which involves searching, circling, tracking, and attacking prey to maximize their search range. It has advantages such as faster convergence, less parameter control, and a more relaxed objective function requirement [15]. However, the traditional WOA can quickly converge to a local saddle point, which affects the optimization results. Therefore, in this paper, we propose an advanced whale optimization algorithm based on the conventional WOA. The initial population of the WOA is randomly distributed, which leads to significant uncertainty and randomness in the first generation of individuals, affecting the optimization results. We utilize the Halton sequence to optimize the initial population, evenly distributing the initial population variation to address this issue. In the traditional WOA, the parameter \(\overset{\rightarrow}{A}\) is essential for determining whale exploration and is calculated by \(\overset{\rightarrow}{a}\), playing a linear role in the search process. In this paper, cc is redefined under different iterations to expand the search range to present a nonlinear expression. Combined with the Gaussian walk strategy, individuals can jump out of local saddle points when they fall into local optima. Figure 3: N \(\times\) N MIMO Schematic diagram of the equivalent circuit. The algorithm adds adaptive weight to update positions to expand the search range. The improved WOA has the same meaning as the original WOA, which is named AWOA. The steps of the AWOA are as follows:  
**Step 1**: Set the initial values of each parameter, such as the population size \(G\_number\) and the maximum iteration number \(MAX\_z\), and initialize the whale population \(W_{i}\) (\(i\) = 1,2... \(n\)) using the Halton sequence. **Step 2**: Calculate and update the optimal selection of individual parameter values in the population, including \(\stackrel{{\rightarrow}}{{a}}\), \(\stackrel{{\rightarrow}}{{A}}\), \(\stackrel{{\rightarrow}}{{C}}\), \(l\), \(p\), and \(\beta\). Here, \(\stackrel{{\rightarrow}}{{a}}\) is a parameter that decreases linearly from 2 to 0 with the number of iterations, \(\stackrel{{\rightarrow}}{{A}}\) is a parameter that controls the whale's action during prey encirclement, \(\stackrel{{\rightarrow}}{{C}}\) is a coefficient vector for the whale to swim towards the target, \(l\) is a uniformly distributed random number between 1 and \(-1\) that plays a role in the whale's spiral swimming, \(p\) is a decision parameter that mainly determines the whale's following action, and \(\beta\) is a weight parameter that expands the search range. The variation patterns of \(\stackrel{{\rightarrow}}{{a}}\), \(\stackrel{{\rightarrow}}{{A}}\), \(\stackrel{{\rightarrow}}{{C}}\), and \(\beta\) are as follows:  
\[\stackrel{{\rightarrow}}{{a}}= \begin{cases}2-\tan(2\times\pi \times\frac{t}{Max\_z})&t\leq\frac{1}{2}Max\_z\\ \sin(((t-\frac{1}{2}Max\_z)/Max\_z)\times\pi)&t>\frac{1}{2}Max\_z\\ \end{cases}  \tag{8}\]  
\[\stackrel{{\rightarrow}}{{A}}=2\cdot\stackrel{{ \rightarrow}}{{a}}\cdot\stackrel{{\rightarrow}}{{r}}- \stackrel{{\rightarrow}}{{a}} \tag{9}\]  
\[\stackrel{{\rightarrow}}{{C}}=2\cdot\stackrel{{ \rightarrow}}{{r}} \tag{10}\]  
\[\beta=2-\cos(0.5\times(t/Max\_z)\times\pi+1) \tag{11}\]  
where \(t\) is the number of current iterations, \(Max\_z\) is the maximum number of iterations, and \(\stackrel{{\rightarrow}}{{r}}\) represents a random number between \(-1\) and \(1\). **Step 3**: Use the Gaussian wandering strategy to jump out of the local optimum. **Step 4**: If \(p<0.5\) and \(\left|\stackrel{{\rightarrow}}{{A}}\right|<1\), execute the whale algorithm roundup phase.