# A Novel Complex-Valued Blind Source Separation and Its Applications in Integrated Reception
## 5 Optimization Algorithm  
Traditional CVBSS commonly utilizes Newton's method or the gradient method as the optimization algorithm. However, these methods are sensitive to the initial value and can be greatly affected by the step size, often resulting in solutions becoming trapped in saddle points. On the other hand, swarm intelligence algorithms offer several advantages, such as high accuracy, fast convergence, and no need for parameter adjustment. In this study, we employ the grey wolf optimizer (GWO) [33; 34] as the optimization algorithm. Several GWO algorithms have been developed to improve its performance. Ref. [35] introduces chaos theory into the GWO algorithm to enhance its global convergence. Ref. [36] enhances the global search ability by improving the chaotic tent map used to initialize the wolf, and also uses a nonlinear convergence factor based on the Gaussian distribution curve to balance global and local searchability. The detailed steps of GWO are not extensively discussed. We introduce a new nonlinear convergence factor and chaotic perturbation to enhance the accuracy and convergence speed of the optimization process. This modified algorithm is referred to as the nonlinear chaotic grey wolf optimizer (NCGWO). By incorporating chaotic motion, which possesses spatial ergodicity and extrinsic randomness, the NCGWOis capable of exploring the entire solution space without repetition. Consequently, the overall quality-seeking capacity and the ability to find local superiority are significantly improved [37]. In GWO, the convergence factor \(\rho\) decreases linearly with the increase in iteration times. It is impossible to balance the global and local search capabilities, which is not conducive to the solution of the algorithm. In this paper, a new nonlinear convergence factor is used to improve the GWO. The expression of the convergence factor equation is  
\[\rho=\exp(iter-((Max\_iter+gen)/(Max\_iter-iter))) \tag{39}\]  
where _Max_iter_ is the maximum number of iterations, and _iter_ is the current iteration. The equation of the Statistical Parametric Map (SPM) is as follows:  
\[\gamma(\vartheta+1)=\left\{ \begin{array}{c}\text{mod}\left(\frac{\gamma( \vartheta)}{\eta}+\mu\sin(\pi\gamma(\vartheta))+rand,1\right),0\leq\gamma( \vartheta)<\eta\\ \text{mod}\left(\frac{\gamma(\vartheta)/\eta}{0.5-\eta}+\mu\sin(\pi\gamma( \vartheta))+rand,1\right),\eta\leq\gamma(\vartheta)<0.5\\ \text{mod}\left(\frac{(1-\gamma(\vartheta))/\eta}{0.5-\eta}+\mu\sin(\pi(1- \gamma(\vartheta)))+rand,1\right),0.5\leq\gamma(\vartheta)<1-\eta\\ \text{mod}\left(\frac{(1-\gamma(\vartheta))}{\eta}+\mu\sin(\pi(1-\gamma( \vartheta)))+rand,1\right),1-\eta\leq\gamma(\vartheta)<1\end{array} \right. \tag{40}\]  
where \(\eta\) and \(\mu\) are the control parameters, respectively. \(\gamma\) is the sequence value of the SPM map. \(\vartheta\) is the sequence number. _rand_ is a random number. When \(\eta\in(0,1)\), \(\mu\in(0,1)\), the system is in a chaotic state. As shown in Figure 3, the SPM map is uniformly distributed and has a uniform probability density distribution. It indicates that the SPM map has good ergodicity. The following are the steps of the chaotic perturbation operator. Step 1: Set the number of chaotic perturbations \(R\) and iterate to generate a sufficient number of chaotic sequences. Randomly select \(R\) values in the above chaotic sequence to obtain the chaotic sequence \(\hat{\mathbf{C}}=\{\hat{c}_{1},\hat{c}_{2},\cdots,\hat{c}_{R}\}\). Step 2: Let \(\{\mathbf{X}_{tt},h=1,2,\cdots H\}\) represent the current wolves, \(X_{h}=\{X_{h1},X_{h2},\cdots,X_{lu},\cdots,X_{hUL}\}\) represent the \(i\)-th wolf, \(U=4\)\(\mathbf{C}_{K}^{2}\). \(\mathbf{gX}=\{gX_{1}^{*},gX_{2}^{*},\cdots,gX_{u}^{*},\cdots,gX_{UL}^{*}\}\) represents the position where the optimal wolf is located. After chaotic perturbation \(R\) times, \(R\) chaotic wolves are obtained within the neighborhood of the optimal wolf, denoted as \(\{\mathbf{gX}_{r},r=1,2,\cdots R\}\). Equation (41) represents the position of each wolf.