# Designing Smart Electromagnetic Environments for Next-Generation Wireless Communications
## 2 Mathematical Formulation  
Let us consider the wireless scenario depicted in Figure 1 where a primary source \(\underline{I}_{inc}(\underline{r})\), \(\underline{r}\in\mathbb{V}\) (\(\underline{r}=\{r,\,\theta,\,\phi\}\)), that models a BTS antenna array composed of \(N\) radiators with reconfigurable amplitude and phase weights \(\underline{w}=\{w_{n};\,n=1,\,\ldots,\,N\}\), is installed in front of \(K\) scatterers (e.g., buildings) with known permittivity, \(\varepsilon(\underline{r})\), and conductivity, \(\sigma(\underline{r})\), distributions. The electric field \(\underline{E}_{inc}(\underline{r})\) generated in far-field from the BTS is given by  
\[\underline{E}_{inc}(\theta,\phi)=\sum_{n=1}^{N}\underline{E}_{n}(\theta,\phi )w_{n}e^{i\underline{E}_{inc}}\hat{a} \tag{1}\]  
where \(\underline{E}_{n}(\theta,\phi)\), \(n=1,\ldots,N\) is the embedded/active far-field pattern of the \(n\)-th antenna including the local mutual coupling interactions [13], \(k=\frac{2\pi}{\lambda}\) is the wavenumber, \(\lambda\) being the working wavelength, \(\underline{r}_{n}\) (\(n=1,\ldots,N\)) is the position of the \(n\)-th antenna of the BTS array, and \(\hat{\underline{a}}=\sin\theta\cos\phi\hat{\underline{x}}+\sin\theta\sin\phi \hat{\underline{y}}+\cos\theta\hat{\underline{z}}\) is the unit vector identifying an arbitrary direction (\(\theta,\phi\)). By assuming the absence of EM coupling between the primary, \(\underline{J}_{inc}(\underline{r})\), and the equivalent sources, \(\underline{J}_{eq}(\underline{r})\), induced from the incident field \(\underline{E}_{inc}(\underline{r})\) generated by the BTS on the scatterers lying in the environment \(S\), the total electric field \(\underline{E}_{tot}(\underline{r})=\underline{E}_{inc}(\underline{r})+ \underline{E}_{scatt}(\underline{r})\)[21] is measured at \(M\) sampling positions \(\underline{r}_{m}\in\Omega\), being \(\Omega\) an external observation domain (Figure 1) where the end-user terminals are located. Moreover, \(\underline{E}_{scatt}(\underline{r})\) is the electric field generated by \(\underline{J}_{eq}(\underline{r})\) and mathematically expressed in the observation domain \(\Omega\) as [21]  
\[\underline{E}_{scatt}(\underline{r}_{m})=j2\pi r/\mu_{0}\int_{S}\underline{G} \left(\underline{r}_{m}|\underline{r}^{\prime}\right)\cdot\underline{I}_{eq} \left(\underline{r}^{\prime}\right)d\underline{r}^{\prime} \tag{2}\]  
where \(\underline{J}_{eq}(\underline{r})=\tau(\underline{r})\underline{E}_{tot}( \underline{r}),\underline{r}\in S\), and  
\[\tau(\underline{r})=[\varepsilon(\underline{r})-1]-j\frac{\sigma(\underline {r})}{2\pi f\varepsilon_{0}} \tag{3}\]  
is the so-called object function, \(\underline{G}(\underline{r}_{m}|\underline{r}^{\prime})\) is the Green function, \(f\) is the operation frequency, \(j=\sqrt{-1}\), and \(\varepsilon_{0}\), \(\mu_{0}\) are the dielectric permittivity and the permeability constants of the free space, respectively. In order to guarantee the desired coverage within \(\Omega\), the scattering environment is opportunistically exploited and the design problem is formulated as the synthesis of the set of complex weights \(\underline{w}\) of the BTS antenna array such that a desired target field distribution \(\underline{E}_{tar}(\underline{r}_{m})\), \(\underline{r}_{m}\in\Omega\), \(m=1,\ldots,M\) (\(M\) being the number of probing locations in \(\Omega\)), is obtained. Towards this aim, due to the nonlinearity and ill-posedness of the arising  
Figure 1: Smart EM environment.