## 2 Methodology
### Optimization
#### 2.5.1 The Path Length  
An estimate of the effective path can be made by summing the elements of the path weighted by their relative velocities, as in Equation (3) for AIR, Equation (4) for MUT and Equation (5) for MET, where \(P\) stands for the perceived path, \(L_{\text{cable}}\) is the length of the coaxial cable, \(v_{\text{cable}}\) is the relative velocity in the coaxial cable, \(L_{\text{air}}\) is the length of the free-space path (antenna to MUT), \(v_{\text{air}}\) is the relative velocity in free space, \(L_{\text{mut}}\) is the thickness of the MUT, \(v_{\text{mut}}\) is the relative velocity in the MUT, \(L_{\text{adapter}}\) is the length of the adapter in the Rx, \(v_{\text{adapter}}\) is its relative velocity and \(h\) is the height of the antenna above the floor. \[p^{\text{(AIR)}}=\frac{L_{\text{cable}}}{v_{\text{cable}}}+2\frac{L_{\text{ air}}}{v_{\text{air}}}+\frac{L_{\text{mut}}}{v_{\text{air}}}+\frac{L_{\text{adapter}}}{v_{\text{adapter}}}, \tag{3}\]  
\[p^{\text{(MUT)}}=\frac{L_{\text{cable}}}{v_{\text{cable}}}+2\frac{L_{\text{ air}}}{v_{\text{air}}}+\frac{L_{\text{mut}}}{v_{\text{mut}}}+\frac{L_{\text{adapter}}}{v_{\text{adapter}}}, \tag{4}\]  
\[p^{\text{(MET)}}=\frac{L_{\text{cable}}}{v_{\text{cable}}}+2\sqrt{\left(\frac{ L_{\text{air}}}{v_{\text{air}}}\right)^{2}+\left(\frac{h}{v_{\text{air}}} \right)^{2}}+\frac{L_{\text{mut}}}{v_{\text{air}}}+\frac{L_{\text{adapter}}}{v _{\text{adapter}}}. \tag{5}\]  
The lengths were directly measured by inspection: \(h=0.45\) m, \(L_{\text{cable}}=0.20\) m, \(L_{\text{air}}=0.48\) m, \(L_{\text{mut}}=10.18\) mm and \(L_{\text{adapter}}=30\) mm. \(v_{\text{air}}=1\) is a known constant, and the velocities in the cable and adapter were obtained from datasheets and were found to be \(v_{\text{cable}}=0.7\) and \(v_{\text{adapter}}=0.7\), respectively; \(v_{\text{mut}}\) is unknown. Using the above values for the parameters, we obtained approximately \(P^{\text{(AIR)}}=1.230\) m and \(P^{\text{(MET)}}=1.626\) m, which are about 5% more than the absolute lengths. In turn, \(P^{\text{(MUT)}}\) depended on \(v_{\text{mut}}\) and could not be estimated beforehand without knowing the properties of the material, as the velocity depended on it, as shown in Equation (6), where \(c_{0}\) is the velocity of the wave in vacuum [1]. The result from the optimization permitted this calculation. \[v=\frac{1}{\sqrt{\mu\epsilon}}=\frac{c_{0}}{\sqrt{\mu_{r}\epsilon_{r}}}. \tag{6}\]  
Assuming a non-magnetic material under test, that is, \(\mu_{r}=1\), its real permittivity can be calculated by [1]:  
\[\epsilon_{r}^{\prime}=\frac{1}{v_{\text{mut}}^{2}}. \tag{7}\]  
The imaginary part \(\epsilon_{r}^{\prime\prime}\) can be approximated by Equation (8), where \(\Delta A\) is the attenuation in dB and \(\lambda_{0}\) is the wavelength in free space [33]. \[\epsilon_{r}^{\prime\prime}\approx\frac{\Delta A\lambda_{0}\sqrt{\epsilon_{r} }}{8.686\pi L_{\text{mut}}}. \tag{8}\]  
It is also possible to interpret the problem as a system with the form in Equation (9), where \(\mathbf{x}\) are the coefficients to be determined and are related to the relative velocities in the three media, that is, the guided propagation (\(x_{1}\)), the free-space propagation (\(x_{2}\)) and the propagation in the material (\(x_{3}\)). \[\mathbf{Ax}=\mathbf{b}, \tag{9}\]where  
\[\mathbf{A}=\left[ \begin{array}{ccc}\left(L_{\text{cable}}+L_{\text{adapt}}\right)& 2L_{\text{air}}+L_{\text{mut}}&0\\ \left(L_{\text{cable}}+L_{\text{adapt}}\right)&2L_{\text{air}}&L_{\text{mut}} \\ \left(L_{\text{cable}}+L_{\text{adapt}}\right)&2\sqrt{L_{\text{air}}^{2}+h^{2} }+L_{\text{mut}}&0\end{array} \right], \tag{10}\]  
and  
\[\mathbf{b}=\Big{[} \begin{array}{ccc}d^{\text{(AIR)}}&d^{\text{(MUT)}}&d^{ \text{(MET)}}\end{array} \Big{]}^{T}. \tag{11}\]  
The rows in \(\mathbf{A}\) correspond to the configurations AIR, MUT and MET, respectively, whilst the columns represent the propagation in cables and adapters (first column), air (second column) and tested material (third column).