# Galerkin's Spectral Method in the Analysis of Antenna Wall Operation
## 2 Radiation Field of Unconfined Antenna Array Placed on Multilayer Dielectric
 \tag{29}\]  
After changing the integration variables,  
\[x^{\prime}-x_{m}=x^{\prime\prime} \tag{30}\]  
\[y^{\prime}-y_{m}=y^{\prime} \tag{31}\]  
and appropriate rearrangement of the order of the elements of the integrand function:  
\[\begin{array}{c}\underline{E}_{t}^{s}(x,y,z)\ =(2\pi)^{-2}\int_{-\infty}^{+\infty}\int_{- \infty}^{+\infty}\int_{S_{00}}\underline{E}_{t00}^{s}(x^{\prime\prime},y^{ \prime\prime},0)\sum_{m,n}exp\big{(}-im\phi_{x}-in\phi_{y}\big{)}\cdot\\ \exp\big{\{}i\big{[}k_{x}(x-x^{\prime\prime})+k_{y}(y-y^{\prime}))+  
to be equal to zero. We will call this type of approximation the zero approximation of the scattered field of the limited antenna array. Within the zero approximation of the field,  
\[\underline{E}_{t}^{s}(x,y,z)\ \cong\underline{E}_{t}^{s(0)}(x,y,z)\ \equiv \underline{E}_{tMN}^{s}(x,y,z) \tag{41}\]  
For any given scattered field, it can be represented as a Fourier transform:  
\[\underline{E}_{t}^{s}(x,y,z)\ =\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}F( \underline{E}_{t}^{s})\left(k_{x},k_{y},z\right)e^{i\left[k_{x},x+k_{y}y \right]}dk_{x}dk_{y}, \tag{42}\]  
where in k-space the Fourier transform of the field is factored as follows:  
\[F(\underline{E}_{t}^{s})\left(k_{x},k_{y},z\right)=F^{\prime}(\underline{E}_{ t00}^{s})\left(k_{x},k_{y},0\right)=Af\left(k_{x},k_{y}\right). \tag{43}\]  
The first term on the left side of the equation denotes the Fourier transform of the field scattered on the central cell, multiplied by the number of antenna wall cells modified by a phase factor related to the propagation of the field in the \(z\) direction:  
\[F^{\prime}(\underline{E}_{t00}^{s})\left(k_{x},k_{y},0\right)=(2M+1)(2N+1)F( \underline{E}_{t00}^{s})\left(k_{x},k_{y},0\right)exp(ik_{z}z) \tag{44}\]  
The second factor is as follows:  
\[Af\left(k_{x},k_{y}\right)=\frac{sin[(2M+1)\phi_{x}/2]}{(2M+1)sin[\phi_{x}/2] }\frac{sin[(2N+1)\phi_{y}/2]}{(2N+1)sin[\phi_{y}/2]} \tag{45}\]  
It denotes the normalized grid factor (array factor), satisfying the condition normalizing its amplitude to unity \(Max(Af)=1\). This factor, for suitably small grid steps, can be approximated as follows:  
\[Af\left(k_{x},k_{y}\right)=\frac{sin[(2M+1)\phi_{x}/2]}{(2M+1)sin[\phi_{x}/2] }\frac{sin[(2N+1)\phi_{y}/2]}{(2N+1)sin[\phi_{y}/2]} \tag{46}\]  
Alternatively, the scattered field in physical space is expressed by the convolution of a modified field distribution on the central cell and the inverse Fourier transform of the grid coefficient:  
\[\underline{E}_{t}^{s}(x,y,z)=(2\pi)^{-2}\iint\left(\underline{E}_{t00}^{s} \right)\left(x-x^{\prime},y-y^{\prime}\right)F^{-1}(Af)\left(x^{\prime},y^{ \prime}\right)dx^{\prime}dy^{\prime} \tag{47}\]  
where  
\[\left(\underline{E}_{t00}^{s}\right)(x,y,0)-(2M+1)(2N+1)F^{-1}\left[F( \underline{E}_{t00}^{s})\left(k_{x},k_{y},0\right)exp(ik_{z}z)\right] \tag{48}\]  
The spectral factorization of the scattered field determines the basic characteristics of the antenna. The first factor--the field on the surface of the central cell--mainly affects the polarization and energy characteristics of the antenna radiation/reception. The grid factor mainly determines the directional characteristics of the antenna; dependent on the geometry of the grid, the phase and amplitude modulation of the grid, the grid jumps of the number of its elements. For example, let us assume the infinity of the periodic antenna structure in the y-axis direction and the antenna dimension in the x-axis direction equal to \((2M+1)\)dx. The directivity of the antenna radiation in the y = 0 plane is mainly determined by the maximum and minimum of the grid factor.