## 2 Formulation of the Problem
The equivalent wall conductivity may change with the frequency, and it does not depend on the equivalent surface choice. Because of discretization, applying the superposition of the effects, (2) can be written as:  
\[\stackrel{{\rightarrow}}{{E}}=\sum_{i=1}^{N_{\alpha\eta}} \left(\stackrel{{\rightarrow}}{{E}}_{IRR}^{i}+\stackrel{{ \rightarrow}}{{E}}_{TE}^{i}+\stackrel{{\rightarrow}}{{E}}_{TM} \right),  
\[\overline{E}^{i}{}_{TM}(x,y,z)= \frac{32}{a\ b^{c}\ \delta p}\sum\limits_{m,n,p}\frac{1}{k_{m}^{2} \omega_{y}-\beta_{TM}^{2}}\sin\left(k_{y}\Delta_{y}\right)\sin(k_{z}\Delta_{z}) \left\{M_{e}^{i}\cos(k_{x}x_{H})\sin(k_{y}y_{H})\cos(k_{z}z_{H})\right. \tag{9}\] \[-j\omega\mu\frac{k^{2}}{k_{m}^{2}\omega_{y}}\sin(k_{x}x_{H})\left[ \frac{l_{e\alpha}^{i}\sin\left(k_{y}y_{H}\right)\cos(k_{z}z_{H})}{k_{y}\ k_{z}}\right.\] \[\left.-\frac{l_{e\alpha}^{i}\cos\left(k_{y}y_{H}\right)\sin(k_{z }z_{H})}{k_{z}^{2}}\right]\right\}\] \[\left\{ \begin{array}{l}k_{x}\cos(k_{x}x)\sin(k_{y}y)\sin(k_{z }z)\hat{x}\\ k_{y}\sin(k_{x}x)\cos(k_{y}y)\sin(k_{z}z)\hat{y}\\ k_{z}\sin(k_{x}x)\sin(k_{y}y)\cos(k_{z}z)\hat{z}\end{array} \right.,\]  
where \(\delta_{k}=\left\{ \begin{array}{l}1,\ k\
eq 0\\ 2,\ k=0\end{array} \right.\) and \(k_{x}=\frac{m\pi}{a},k_{y}=\frac{m\pi}{b},k_{x}=\frac{p\pi}{c}\). Similar expressions can be found for the faces oriented along the y and z directions. For a better readability of the manuscript, they are reported in the Appendix A. The algorithm to reconstruct the radiated emission in free space of an EUT placed into an RC is detailed in [17] and here briefly reported in the flow chart of Figure 3, for the sake of completeness. Two set of data are required. The first one is the value of the electric field measured at the samples' positions. The second one is the \(Z_{ni}\) matrix that represents the analytical transfer function between each equivalent source and the electric field at sample position. Four equivalent sources are placed in each sub surface in which the whole equivalence surface is discretized. Figure 3: Flow chart of the algorithm for the reconstruction of the radiated emissions. At first, the algorithm evaluates the field due to each equivalent current in all sample positions. Then the distance between this field and the reference field is computed at the samples' positions according to (10):  
\[d_{i}=\frac{\sum_{n=1}^{N_{s}}\lvert E_{n}^{res}-Z_{ni}I_{i}\rvert}{\sum_{n=1}^{ N_{s}}\lvert E_{n}^{res}\rvert},\quad i=1,\dots,N_{eq} \tag{10}\]  
where \(I_{i}\) represents the i-th equivalent source (\(J_{eq}^{i}\) or \(M_{eq}^{i}\)) and each element of the matrix \(Z_{ni}\) represents the electric field generated by the i-th equivalent source and normalized to \(I_{i}\), calculated at the n-th sample point. Subsequently, the \(i^{*}-th\) equivalent source, which minimizes \(d_{i}\), is chosen and the residual electric field, computed at each iteration, is (11):  
\[E_{n}^{res}=E_{n}^{res}-Z_{ni^{*}}I_{i^{*}},\;n=1,\dots,N_{s} \tag{11}\]  
The relative error checked to decide when terminating the iteration is (12):  
\[err=\frac{\sum_{n=1}^{N_{s}}\lvert E_{n}^{res}\rvert}{\sum_{n=1}^{N_{s}}\lvert E _{n}^{ref}\rvert}. \tag{12}\]  
Iterations terminate when err is lower than a desired threshold.