# Optimal Design of Switchable Wearable Antenna Array for Wireless Sensor Networks
## 2 Models and Methods
### Inverse Problem Formulation  
We formulate the inverse problem in the following way: \"Find the position of array elements and phase shift values such that the highest value of radiation pattern is directed towards signal while the  
Figure 5: The radiation pattern of the wearable antenna array obtained with FDTD based simulations and the cylindrical model of the body compared to results obtained with the simplified model given by Equation (2) for various values of \(G_{0}\).  
Figure 6: Antenna array radiation pattern \(G(\varphi)\) with four elements located on the \(\psi_{a}\): \([-20^{\circ},20^{\circ},160^{\circ},\)\(200^{\circ}]\), phasing vector \(\alpha\) = [0,0,0,0].  
lowest value towards interferences, for all the considered positions of signal and interference.\" To this end, an optimization problem is set up, the degrees of freedom of which are the angular position vector \(\psi(I)\), \(\psi(2)\), \(\psi(3)\), \(\psi(4)\) of the antennas composing the four-element circular array and the value of non-zero phase shift \(\alpha_{const}\) that can be applied in the feeding network; so, a five-dimensional design space is considered. Accordingly, a possible objective function \(f\) to be maximized should express the SIR which is the ratio of the array gain towards the signal \(G_{s}\) to the array gain towards the interference \(G_{i}\) that could be obtained against all possible combinations of \(n\)-\(th\) antenna feeding phases, \(n=[1.4]\), given the positions of signal and interference. For this reason, the radiation pattern of the array has to be calculated for \(P\) combinations of antenna phasing vector, to obtain the objective function components \(G_{s}\) and \(G_{i}\), namely  
\[P=\left(2^{4}-1\right)=15\]  
This computation has to be performed for each signal position \(\varphi_{s}\) and interference position \(\varphi_{i}\). We consider \(S\) combinations of signal and interference angles separated by \(45^{\circ}\) in a \(360^{\circ}\) angle, namely  
\[S=\left(2^{\frac{3\omega_{0}}{25}}-8\right)=248\]  
The number of calls to the radiation pattern computations at each iteration of an optimization algorithm is a reasonable estimate of the cost per iteration \(C\) that can be calculated as:  
\[C=S*P=3720\]  
In other words, the total number of radiation pattern computations depends on the number of optimization algorithm iterations--3720 in every single iteration. For this reason, the use of a cost-effective analytical model of array radiation pattern like, e.g., Equation (2) is mandatory because performing 3720 simulations based on FDTD in each optimization step would be impractical due to the computational time. The conceptual flow-chart presenting the algorithm for the calculation of objective function components is shown in Figure 8.  
In the following sections, we present four diverse versions of the objective function, which we proposed and examined. This approach follows from our experience that for complex problems in  
Figure 8: Optimization loop flow-chart showing the algorithm for the calculation of objective function components.  
electromagnetics, the capability of stochastic optimization methods for an effective exploration and exploitation of the objective space strongly depends on the objective function formulation. We intended to identify a problem-specific function that offers the fastest convergence.