# Galerkin's Spectral Method in the Analysis of Antenna Wall Operation
## 2 Radiation Field of Unconfined Antenna Array Placed on Multilayer Dielectric
Thus, the carrier functions \(h_{nm}\) are as follows:  
\[h_{nm}(x,y)=1\ for\ (x,y)\in cell\ surface\ (m,n) \tag{7}\]  
\[h_{nm}(x,y)=0\ for\ (x,y)\ \
otin cell\ surface\ (m,n) \tag{8}\]  
Individual cells of the periodic antenna in the directions of the \(x\)-axis and the \(y\)-axis with steps \(d_{x}\), \(d_{y}\), respectively, with a unit increase in the indices m, n are shown:  
\[h_{m+1,\ n+1}(x,y)=h_{m,n}\big{(}x-d_{x},y-d_{y}\big{)} \tag{9}\]  
They satisfy the following condition at any point (\(x\),\(y\),\(0\)) of the antenna surface:  
\[\sum_{m,n}h_{m,n}(x,y)=1 \tag{10}\]  
From the distribution of the scattered field on the individual antenna cells, we obtain the following:  
\[ \begin{array}{c}\underline{E}_{t}^{s}(x,y,z)=(2\pi)^{-2}\times\\ \sum_{m,n}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\int_{-\infty}^{+ \infty}\int_{-\infty}^{+\infty}\underline{E}_{nm}^{s}(x^{\prime},y^{\prime},0 )exp\big{\{}i\big{[}k_{x}(x-x^{\prime})+k_{y}(y-y^{\prime})+k_{z}z\big{]}\, \big{\}}dx^{\prime}dy^{\prime}dk_{x}dk_{y}\end{array}  \tag{11}\]  
We additionally assume a uniform phase excitation of the antenna by an incident plane wave with direction angles \(\phi^{i}\), \(\phi^{i}\). The phase shifts of the scattered field on individual cells in the \(z=0\) plane at a unit shift \(d_{x}\) in the \(x\)-axis direction and \(d_{y}\) in the \(y\)-axis direction are, respectively, as follows:  
\[\phi_{x}^{s}=k_{x}d_{x}=kd_{x}sin\theta cos\phi, \tag{12}\]  
\[\phi_{y}^{s}=k_{y}d_{y}=kd_{y}sin\theta sin\phi, \tag{13}\]  
\[\phi_{x}^{i}=k_{x}^{i}d_{x}=kd_{x}sin\theta^{i}cos\phi^{i}, \tag{14}\]  
\[\phi_{y}^{i}=k_{y}^{i}d_{y}=kd_{y}sin\theta^{i}sin\theta^{i}, \tag{15}\]  
where  
\[k_{x}=ksin\theta cos\phi, \tag{16}\]  
\[k_{y}=ksin\theta sin\phi, \tag{17}\]\[k_{x}^{i}=ksin\theta^{i}cos\phi^{i}, \tag{18}\]  
\[k_{y}^{i}=ksin\theta^{i}sin\phi^{i}, \tag{19}\]  
\(\theta\) and \(\varphi\) and \(\theta^{i}\) and \(\phi^{i}\) denote the direction angles of the wave vector of the scattered field and the incident field, respectively. Hence, we obtain the global phase shifts for the individual cells as follows:  
\[\phi_{x}=\phi_{x}^{s}-\phi_{x}^{i}=\left(k_{x}-k_{x}^{i}\right)d_{x}\ =kd_{x} \Big{(}sin\theta cos\phi-sin\theta^{i}cos\phi^{i}\Big{)} \tag{20}\]  
\[\phi_{y}=\phi_{y}^{s}-\phi_{y}^{i}=\left(k_{y}-k_{y}^{i}\right)d_{y}\ =kd_{y} \Big{(}sin\theta sin\phi-sin\theta^{i}sin\phi^{i}\Big{)}. \tag{21}\]  
\[\phi_{y}^{s}=k_{y}d_{y}=kd_{y}sin\theta sin\phi, \tag{13}\]0  
\[\phi_{y}^{s}=k_{y}d_{y}=kd_{y}sin\theta sin\phi, \tag{13}\]1  
\[\phi_{y}^{s}=k_{y}d_{y}=kd_{y}sin\theta sin\phi, \tag{13}\]2  
\[\phi_{y}^{s}=k_{y}d_{y}=kd_{y}sin\theta sin\phi, \tag{13}\]3  
\[\phi_{y}^{s}=k_{y}d_{y}=kd_{y}sin\theta sin\phi, \tag{13}\]4  
\[\phi_{y}^{s}=k_{y}d_{y}=kd_{y}sin\theta sin\phi, \tag{13}\]5  
For an unconstrained antenna wall, the scattered field distribution of the cell (\(m\),\(n\)) is a replica of the field distribution on the central cell (0,0) shifted by the vector \((x_{m},y_{m})\) with an appropriate modification of the field phase:  
\[ \begin{array}{c}\underline{E}_{tmn}^{s}(x,y,0)=\underline{E}_{t00}^{s}(x-x _{m},y-y_{m},0)exp\Big{(}im\phi_{x}^{i}+in\phi_{y}^{i}\Big{)}=\\ =U\underline{E}_{t00}^{i}(x-x_{m},y-y_{m},0)exp\Big{(}im\phi_{x}^{i}+in\phi_{ y}^{i}\Big{)}\end{array}  \tag{28}\]  
The scattered field can be expressed, considering the phase shifts, by an appropriate integral convolution over the surface of a (single) central cell:  
\[ \begin{array}{c}\underline{E}_{t}^{s}(x,y,z)=(2\pi)^{-2}\sum\limits_{m,n} \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\int_{S_{00}}\underline{E}_{t 00}^{s}(x^{\prime}-x_{m},y^{\prime}-y_{n},0)exp\Big{(}im\big{(}\phi_{x}^{i}- \phi_{x}^{s}\big{)}+in\Big{(}\phi_{y}^{i}-\phi_{y}^{s}\Big{)}\Big{)}\\ \exp\big{\{}i\big{[}k_{x}(x-(x^{\prime}-x_{m})+k_{y}(y-(y^{\prime}-y_{n}))+k_ {z}z\big{]}\big{\}}d(x^{\prime}-x_{m})d(y^{\prime}-y_{n})dk_{x}dk_{y}\end{array}