# BDCOA: Wavefront Aberration Compensation Using Improved Swarm Intelligence for FSO Communication
## 2 BDCOA Based Optimum Control Signal
### Wavefront Aberration Using BDCOA
#### 2.4.3 Exploration Phase (Hunting and Attacking)  
In this phase, the coatis are separated into two different groups. The first group climbs the tree to frighten the prey while the second group waits under the tree for the falling of frightened prey. This kind of motion pattern allows the coati to thoroughly search the problem space. The coati's prey is referred to as Iguana. The location of the optimum member of the population is considered as the Iguana's location. Equation (9) specifies the first half of the coatis that climb the tree. If the Iguana's falls to the ground, its location is randomly changed based on Equation (10). Equation (11) specifies the second half of coatis that wait under the tree. The BM is incorporated in this phase, while the searching escapes from suboptimal solutions and searches in the other regions with extensive capacity to discover the global optimum value. This adaptive and dynamic process enhances the BDCOA capacity for effective exploring and exploitation of the searching space. Each step of the BM is discovered by the probability function denoted by normal Gaussian distribution with mean \((\delta=0)\) and unit variance \(\left(\sigma^{2}=1\right)\). The Probability Density Function of BM i.e., \(BM(y,\delta,\sigma)\) at a certain point \(y\) is denoted in Equation (12).  
\[X_{i}^{P1}:x_{i,j}^{P1}=x_{i,j}+rand...\left(\text{Iguana}_{j}-I...x_{i,j}\right)\,\,\,\,\text{for}\,\,i=1,2,\ldots,\left|\frac{N}{2}\right|\, \text{and}\,\,j=1,2,\ldots,m \tag{9}\]\[Iguana^{G}:Iguana_{i}^{G}=lb_{j}+rand\cdot\left(ub_{j}-lb_{j}\right) \tag{10}\]  
\[X_{i}^{P1}:x_{i,j}^{P1}=\left\{\begin{array}{ll}x_{i,j}+rand\cdot\left(Iguana_{j }^{G}-I\cdot x_{i,j}\right),&F_{Iguana}<F_{i}\\ x_{i,j}+BM\cdot\left(x_{i,j}-Iguana_{j}^{G}\right),&else\end{array}\right.\]  
\[\text{for }i=\left\lfloor\frac{N}{2}\right\rfloor+1,\,\left\lfloor\frac{N}{2} \right\rfloor+2,\ldots N\text{ and }j=1,2,\ldots,m \tag{11}\]  
\[BM(y,\delta,\sigma)=\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\left(-\frac{(y-\delta )^{2}}{2\sigma^{2}}\right)=\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{y^{2}}{2}\right) \tag{12}\]  
where, the new location computed for coati \(i\) is denoted as \(X_{i}^{P1},x_{i,j}^{P1}\) is in dimension \(j\), the new location's fitness value is denoted as \(F_{i}^{P1}\), real random value among \([0,\ 1]\) is denoted as \(r\), upper and lower values of variable \(j\) are denoted as \(ub_{j}\) and \(lb_{j}\), respectively, prey's location is denoted as \(Iguana\), \(Iguana\), is its \(i\)th dimension, \(I\) is an integer value that is either \(1\) or \(2\), prey's location in ground is denoted as \(Iguana^{G}\), its \(j\)th dimension is denoted as \(Iguana_{j}^{G}\), \(Iguana^{G}\)'s fitness value is denoted as \(F_{Iguana^{G}}\), and the floor function is denoted as \(\lfloor\cdot\rfloor\). If the fitness of \(X_{i}^{P1}\) of coati \(i\) has better fitness than the fitness of old location \(X_{i}\), it is unaltered. Otherwise, the old location is preserved as shown in Equation (13)  
\[X_{i}=\left\{\begin{array}{ll}X_{i}^{P1},&F_{i}^{P1}<F_{i}\\ X_{i},&else\end{array}\right. \tag{13}\]  
Each solution (coati) fitness is evaluated according to the RMS. It is computed by using Equations (14) and (15).  
\[F_{i}=\max\left\{\frac{1}{RMS}\right\} \tag{14}\]  
\[RMS=\iint\limits_{S}\left(R(r)-\overline{R(r)}\right)r\cdot dr \tag{15}\]  
where the residual phase is represented as \(R(r)=A(r)-C(r)\) and the normalized circle is denoted as \(S\).