# Optimal Design of Aperiodic Reconfigurable Antenna Array Suitable for Broadcasting Applications
## 3 PSO with Velocity Mutation  
There are many PSO variants introduced so far in order to improve the original one. In this variant (PSOvm), the main idea is to apply perturbation to the velocities of the particles that did not improve their fitness value in the previous iteration. In this way, the swarm is protected from premature convergence and achieves greater exploration. PSOvm is based on the _constriction coefficient_ PSO (CCPSO) and uses the _gbest_ model. In the _gbest_ model, one of the components used to update the velocity of a particle is influenced by the best position found by the whole swarm. The other terms are the best position found so far by the particle and its velocity at the previous iteration. By denoting the swarm size as \(N\) in a problem with \(Q\) optimization variables (\(Q\)-dimensional search space), the equations that update the velocity and position of the \(n\)th particle (\(n=1,...,N\)), when it improves its fitness at the \(i\)th iteration, can be written as:  
\[v_{nq}(i+1)=k\big{\{}v_{nq}(i)+\phi_{1}R[p_{nq}(i)-x_{nq}(i)]+\phi_{2}R[g_{q}( i)-x_{nq}(i)]\big{\}} \tag{9}\]  
\[x_{nq}(i+1)=x_{nq}(i)+v_{nq}(i+1)\;, \tag{10}\]  
where \(v_{nq}\) and \(x_{nq}\) are, respectively, the \(q\)th velocity component and the \(q\)th position coordinate (\(q=1,...,Q\)) of the \(n\)th particle, \(p_{nq}\) and \(g_{q}\) are the \(q\)th coordinates of the best positions found at the  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **Lower Limit** & **Upper Limit** & **Parameter** & **Lower Limit** & **Upper Limit** \\ \hline \(W\) & 105 mm & 245 mm & \(t_{w}\) & 4.2 mm & 16.8 mm \\ \(L\) & 91 mm & 211 mm & \(s\) & 26.5 mm & 61.9 mm \\ \(d_{0}\) & 214 mm & 343 mm & \(g\) & \(W_{cur}/40\) & \(W_{cur}/10\) \\ \(r_{i_{r}}\)\(i=1,2,3\) & 0 mm & 86 mm & \(A_{n}\), \(n=1,\ldots,8\) & 0.1 & 1 \\ \(t_{l}\) & 0 mm & 107 mm & \(P_{n}\), \(n=1,\ldots,8\) & 0\({}^{\circ}\) & 360\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Optimization Parameter Limits.  
Figure 2: Geometry of the proposed aperiodic microstrip antenna array (AMAA).  
end of the \(i\)th iteration by the \(n\)th particle and the whole swarm, respectively, \(R\) represents random numbers uniformly distributed within the interval \((0,1)\), \(\phi_{1}\) and \(\phi_{2}\) are, respectively, the _social_ and the _cognitive_ coefficient, and are both equal to 2.05, and finally \(k=0.73\) is the _constriction coefficient_. The above equations are the original ones used in CCPSO. However, in PSOvm, if a particle fails to improve its fitness at the previous iteration, then its velocity is updated by using:  
\[\begin{split} v_{n\eta}(i+1)=k\big{\{}(0.6+0.1m)(2R-1)v_{n\eta}( i)+\phi_{1}R[p_{n\eta}(i)-x_{n\eta}(i)]\\ +\phi_{2}R[g_{d}(i)-x_{n\eta}(i)]\big{\}},\quad m=1,...,6\,,\end{split} \tag{11}\]  
where \(m\) is the number of consecutive iterations with no fitness improvement for the \(n\)th particle. Practically, the previous velocity of a particle is multiplied by random numbers uniformly distributed in the interval \((-0.7,+0.7)\) when \(m=1\). The boundaries of this interval increase by 0.1 for each consecutive fitness improvement failure. If the \(n\)th particle does not improve its fitness six consecutive times, then its velocity is updated by applying (9) and the mutation process stops. This also happens when the particle manages to improve its fitness. The term \((0.6+0.1m)(2R-1)\) as well as the maximum value of \(m\) were found after testing PSOvm with several well known multi-dimensional mathematical fitness functions [16].