# Galerkin's Spectral Method in the Analysis of Antenna Wall Operation
## 4 Dissipation Radiation Field Analysis of Antenna Array of Limited Dimensions  
### Antenna Area Limited Features  
We consider the case of a limited periodic antenna array in plane \(z=0\) in the directions of the \(x\)- and \(y\)-axes. The antenna array is immersed in any unconfined dielectric medium, layered in planes perpendicular to the \(z\)-axis and placed on the boundary of two layers or dielectric media; in the special case, in free space. The antenna cells with the carrier function \(h_{mn}\) cover the part of the plane \(z=0\) corresponding to the carrier function \(H_{MN}(x\),\(y)\):  
\[\sum_{m,n}h_{mn}(x,y)=H_{MN}(x,y), \tag{151}\]  
where  
\[H_{MN}(x,y)=1\quad for\quad x\in[-Md_{x},Md_{x}]\ and\ y\in[-Nd_{y}Nd_{y}] \tag{152}\]  
\[H_{MN}(x,y)=o\quad for\quad x\
otin[-Md_{x},Md_{x}]\ or\ y\
otin[-Nd_{y}Nd_{y}] \tag{153}\]  
where integers \(m\), \(n\) number the cells in the \(x\)-axis direction \(m=-M\), \(-M+1\), \(\ldots\), \(0\), \(\ldots\), \(M-1\), \(M\) and in the y-axis direction \(n=-N\), \(-N+1\), \(\ldots\), \(0\), \(\ldots\), \(N-1\), \(N\). The metallized elements of the antenna are distributed periodically in the same limited area of the \(z=0\) plane in individual cells of the grid (\(m\),\(n\)) and cover the area with the carrier function \(A_{MN}(x\),\(y)\):  
\[\sum_{m,n}a_{mn}(x,y)=A(x,y)H_{MN}(x,y)=A_{MN}(x,y) \tag{154}\]  
Note that the functions \(H_{MN}(x\),\(y)\), \(A_{MN}(x\),\(y)\) factorize into components depending on \(x\) and \(y\).  
\[H_{MN}(x,y)=H_{MNx}(x)H_{MNy}(y), \tag{155}\]  
\[A_{MN}(x,y)=A_{MNx}(x)A_{MNy}(y). \tag{156}\]  
Due to the antenna limitation in the plane \(z=0\), the antenna cells are distinguished, e.g., in the solution of the scattering problem, by their position relative to the origin of the coordinate system (\(x\),\(y\)) = (0,0), assumed as the center point of the central cell (0,0). However, in this case the solution (current distribution) changes from cell to cell. In an unconstrained periodic system, each cell (\(m\),\(n\)) can be a central cell. In a constrained periodic system, the central cell is uniquely determined by its position in the center of the antenna grid area; each cell other than the central one (\(m\),\(n\)) is uniquely determined by the position of its center relative to the center of the grid area (\(x\),\(y\)) = (0,0). For further considerations, it is therefore necessary to define the carrier (window) function \(H_{MNm}(x,y)\) of the antenna grid with the origin of the coordinate system at points (\(x_{m}\),\(y_{m}\)) of subsequent cells (\(m\),\(n\)). The \(H_{MNmn}\)(\(x\),\(y\)) functions change depending on the hypothetical cell in which the origin of the coordinate system is located and are defined by translations by the vector (\(x_{m}\),\(y_{n}\)) = (\(md_{x}\),\(nd_{y}\)) of the \(H_{MN}\): function:  
\[H_{mn}(x,y)=H_{00}(x-x_{m},y-y_{m})=H_{MN}(x-x_{m},y-y_{m}). \tag{157}\]  
Their Fourier transforms are, respectively, as follows:  
\[F(H_{00})(k_{x},k_{y})=\frac{MNd_{x}d_{y}}{\pi^{2}}\frac{sinMk_{x}d_{x}}{Md_{x }}\frac{sinNk_{y}d_{y}}{Nd_{y}}, \tag{158}\]  
\[\overline{H}_{\text{ook}}\equiv\overline{H}_{MN00}(k_{xk},k_{yk})=(d_{x}d_{y })^{-1}F(H_{00})(k_{xk},k_{yk})=\frac{MN}{\pi^{2}}\frac{sinMk_{x}d_{x}}{Md_{x }}\frac{sinNk_{yk}d_{y}}{Nd_{y}}, \tag{159}\]  
\[\begin{split}\overline{H}_{mnk}\equiv\overline{H}_{mn}& \left(k_{xk},k_{yk}\right)=\overline{H}_{00}\left(k_{xk},k_{yk} \right)\textit{exp}\left[-i\left(mk_{xk}d_{x}+nk_{yk}d_{y}\right)\right]\\ &=\overline{H}_{00}\left(k_{xk},k_{yk}\right)\textit{exp}\left[ -i\left(m\phi_{xk}+n\phi_{ky}\right)\right].\end{split} \tag{160}\]  
Therefore, the Fourier transforms of the cell carrier function experience a phase modulation equal to the phase modulation of the corresponding Floquet harmonics of the incident field.