# Designing Advanced Multistatic Imaging Systems with Optimal 2D Sparse Arrays
## 4 Simulation  
A simulator based on PO principles is used to model the presented system [41]. This simulator allows a comprehensive simulation of the electromagnetic response of the imaging system when exposed to an arbitrary OUT. In a first step, the simulator calculates the induced electric currents within the OUT by considering the interaction between the incident electromagnetic wave and the specific material properties and geometric characteristics of the OUT. This complex calculation is based on the solution of [PERSON]'s equations, which allows the derivation of the spatial distribution of the induced currents on the surface of the OUT. The simulator then determines the electromagnetic field received by each individual receiver, resulting from the field scattered by the OUT. This step is performed independently for each transmitting antenna and over a range of discrete frequencies (\(\mathrm{f_{f}}\)). In the context of radiation modeling, both the transmitting and receiving antennas are abstracted as ideal spherical sources. The OUTs are introduced into the simulator as computer-aided design (CAD) models, encapsulating their complex geometries. The detailed operation of the simulator is defined in [41]; however, for a simplified scenario involving a single \(\mathrm{t_{x}}_{m}\), one OUT, and one \(\mathrm{r_{x}}_{n}\), the process can be summarized as follows:  
1. **Incident field calculation:** The process begins with a point source (\(\mathrm{t_{x}}_{m}\)) emitting spherical electromagnetic waves, which propagate uniformly in all directions. These waves consist of electric (\(\vec{E}_{inc}\)) and magnetic (\(\vec{H}_{inc}\)) fields traveling at the speed of light. To determine the direction of incidence, the Poynting vector (\(\hat{p}_{inc}\)) is calculated using the following equation: \[\hat{p}_{inc}=Re\Bigg{\{}\frac{\vec{E}_{inc}\times\vec{H}_{inc}^{*}}{\|\vec{E }_{inc}\times\vec{H}_{inc}^{*}\|}\Bigg{\}}\] (4)
2. **Equivalent currents calculation:** The next step involves calculating the equivalent currents on the surface of the metallic OUT, represented as \(\vec{j}_{out}\), which is responsible for radiating fields. These currents are determined as follows: \[\vec{j}_{out}=2\hat{n}_{out}\times\vec{H}_{inc}\] (5)
3. **Scattered field calculation:** The simulator models the OUT as a collection of triangular patches with specific points and vectors, allowing for the calculation of the scattered magnetic field (\(\vec{H}_{outn}\)) at the receiver (\(\mathrm{r_{x}}_{n}\)) point using PO nearfield equations. This involves several substeps: 1. Calculate the position vector (\(\vec{R}_{outn}\)) between the receiver and the center of each triangular patch, along with the distance (\(R_{outn}\)) and direction (\(\hat{R}_{outn}\)). 2. Compute the magnetic field (\(\vec{H}_{outn}\)) at the receiver due to each triangular patch using the formula: \[\vec{H}_{outn}=\frac{e^{-jkR_{outn}}}{4\pi R_{outn}^{3}}(1+jkR_{outn})\Big{(} \vec{j}_{out}\times\vec{R}_{outn}\Big{)}I_{outn}\] (6)Calculate the integral term (\(I_{outn}\)) using the given expression and constants:  
\[I_{outn}=2A_{out}e^{-j\frac{\kappa_{outn}+\hat{\kappa}_{outn}}{3}}\left[\frac{ \alpha_{outn}e^{i\beta_{outn}}-\beta_{outn}e^{i\alpha_{outn}}+\beta_{outn}- \alpha_{outn}}{(\alpha_{outn}-\beta_{outn})\alpha_{outn}\beta_{outn}}\right] \tag{7}\]  
where  
\[\alpha_{outn}=k\vec{\mu}_{out_{12}}\cdot\left(\hat{\delta}_{outn}-\hat{\rho}_{ i_{out}}\right) \tag{8}\]  
\[\beta_{outn}=k\vec{\mu}_{out_{13}}\cdot\left(\hat{\delta}_{outn}-\hat{\rho}_{ i_{out}}\right) \tag{9}\]  
Compute the electric field (\(\vec{E}_{outn}\)) at the receiver due to each triangular patch using the formula:  
\[ \begin{split}&\vec{E}_{outn}=-\frac{j\eta e^{-jkR_{outn}}}{4\pi kR _{outn}^{3}}\left[\vec{\mu}_{m}\left(-1-jkR_{outn}+(kR_{outn})^{2}\right)\right. \\ &+\hat{R}_{outn}\left(3+3 jkR_{outn}-(kR_{outn})^{2}\right)\left( \hat{R}_{outn}\cdot\vec{\mu}_{m}\right)\right]\end{split}