## 3 Results and Discussion
### Optimization  
After 1421 iterations, the objective function containing Equation (17) was minimized to 0.57679, with \(w\) equal to 0.00100002. The other achieved parameters are shown in Table 1. The value of \(w\) was just slightly higher than its lower bound. Although this suggests that it was too bounded, a more wide search range introduces local solutions that affect the convergence. Initially, the GA was not converging to good solutions, then some default options for the stopping criteria were changed to obtain more evaluations: (a) the number of generations was set to be 100 times more than the default, which was 100 times the number of variables; (b) the stall generations, which is 50 at default, was changed to 500; and (c) the function tolerance, which is \(10^{-6}\) by default, was changed to \(10^{-9}\). The measured signals and the signals simulated by using the optimized values are compared in Figure 12. In the simulation, we used \(\varepsilon_{\text{r(MUT)}}=2.61\), \(\varepsilon_{\text{r(floor)}}=5.31\) and \(\sigma_{\text{(floor)}}=0.02\) S/m [31]. For simplicity, dispersion was not considered. We note a few interesting observations from the results. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **AIR** & **MUT** & **MET** \\ \hline \(c\) & 1.9213 & 1.9500 & 1.2155 \\ \(i\) & 956 & 917 & 849 \\ \(d\) & 1.1932 & 1.2025 & 1.6187 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Optimized parameters for the three configurations. Figure 12: Measured and simulated complex signals for the three configurations, using the optimized values of Table 1. Magnitudes are on the left and phases are on the right. The configurations from top to bottom are AIR, MUT and MET. From Table 1, we see that the constant \(c^{\rm(MET)}\) is quite different from \(c^{\rm(AIR)}\) and \(c^{\rm(MUT)}\). It is consistent with the fact that less energy reaches the receiver when the line of sight is blocked by the metal, particularly in the higher frequencies, as seen in Figure 12.  
We suggest that the \(c\) values for AIR and MUT were quite similar as the MUT is a low-loss material and does not cause much attenuation to the signal. One can see in Figure 12 that the magnitude of the MUT has more variability, with a few values discrepant from the reference, for instance, two in the region of 0.9-0.95 GHz and others just above 1.5 GHz. The indexes \(i\) of the samples were consistent with the empiric observations that the last samples were less subjected to errors. The \(i^{\rm(AIR)}\) and \(i^{\rm(MUT)}\) were closer to the upper bound than to the lower bound, whilst the \(i^{\rm(MET)}\) was a little lower, but still had a high value. The \(d\) values for adjusting the phases were compatible with the electrical path traveled by the wave. This path was slightly longer than the physical path, as the wave propagated slowly in the coaxial cable and connectors. The antennae could also introduce some variation in phase, which was not considered here. A further observation was that \(d^{\rm(MUT)}\) was about 0.01 m longer than \(d^{\rm(AIR)}\), which was consistent with the nominal thickness of the sample (1 cm) that occupied the center of the setup in the MUT configuration. The \(d^{\rm(MET)}\) was a little higher than the estimated multi-path, suggesting that most of the power took a less direct path. Due to the frequency-dependent directivity, we expect that the power received at the lower frequencies is higher than that received at the higher frequencies. It is worth remembering that a 2D model can limit the accuracy of an analysis. This may be more significant in the MET configuration, as the line of sight was blocked and the three dimensional nature of the reflected and diffracted electromagnetic fields was more difficult to approximate with a simpler model. We can observe in Figure 12 that the initial phase (at 0.85 GHz) of the MET was in agreement with part of the results presented in Figure 8, that is, a positive value, in contrast to values from AIR and MUT, which always had negative initial values.