# Galerkin's Spectral Method in the Analysis of Antenna Wall Operation
## 3 Radiation Field Analysis of Unrestricted Antenna Array
### Spectral Representation of the Field--The Case of Arbitrary Excitation
\[v_{k}=v_{k}^{e}+v_{k}^{s}=(1+\Gamma_{k})v_{k}^{i}+v_{k}^{s} \tag{106}\]0  
\[v_{k}=v_{k}^{e}+v_{k}^{s}=(1+\Gamma_{k})v_{k}^{i}+v_{k}^{s} \tag{106}\]1Having at our disposal two sets of functions spanning the solution space--harmonics for the electromagnetic field and basic functions for the currents--we obtain, by virtue of the orthonormality property of the harmonics, two alternative expressions for the scattered field:  
\[v_{k}=v_{k}^{e}+v_{k}^{s}=(1+\Gamma_{k})v_{k}^{i}+v_{k}^{s} \tag{106}\]2  
\[v_{k}=v_{k}^{e}+v_{k}^{s}=(1+\Gamma_{k})v_{k}^{i}+v_{k}^{s} \tag{106}\]3  
which can be reduced in spectral space to a system of algebraic equations relating the \(v^{s}_{k}\) coefficients of the scattered field distribution to the coefficients of the current distribution \(\underline{j}_{q}\):  
\[\underline{W}\,\underline{j}\equiv\left[ \begin{array}{cc}\underline{W}_{1} &0\\ 0&\underline{W}_{2}\end{array} \right]\,\left[ \begin{array}{c}\underline{j}_{1 }\\ \underline{j}_{2}\end{array} \right]=\underline{V}^{s}, \tag{131}\]  
\[[W]_{kqj}\equiv W_{kqj}=\sum_{k}\psi^{s}_{ki}\overline{G}_{kij}\overline{ \varphi}_{qkj} \tag{132}\]  
\[[j]_{qj}\equiv j_{qj}, \tag{133}\]  
\[[V^{s}]_{k}\equiv v^{s}_{k} \tag{134}\]  
where the indices \(j=1\); \(2\) correspond to the tensor components \(x\) and \(y\), respectively. The equation describing the condition of zeroing of the tangential component of the total electric field on the antenna's conductive elements and the expression for the scattered field are as follows:  
\[\sum_{k}\sum_{m,n}a_{mn}(\underline{E}^{s}_{kbmn}(x,y,0))+\underline{E}^{s}_{ tbmn}(x,y,0) \tag{135}\]  
\[\underline{E}^{s}(x,y,0)=\sum_{k}\underline{\overline{G}}_{k}\underline{ \underline{J}}_{k}e_{k}[x,y] \tag{136}\]  
This is reduced by testing these equations with basic functions \(\underline{\overline{\varphi}}_{qk}=\overline{\varphi}_{qkj}\), to a system of algebraic equations relating the coefficients \(v^{i}_{k}\) of the incident field distribution to the coefficients of the current distribution \(\underline{j}_{q}\):  
\[\underline{\underline{N}}\,\underline{V}^{i}=-\underline{U}\underline{j} \equiv-\left[ \begin{array}{cc}\underline{U}_{11}&\underline{U}_{12}\\ \underline{U}_{21}&\underline{U}_{22}\end{array} \right]\left[ \begin{array}{c} \underline{j}_{1}\\ \underline{j}_{2}\end{array} \right] \tag{137}\]  
\[\left[ \begin{array}{c}\underline{N}\\ \underline{J}\end{array} \right]_{qkj}\equiv N_{qkj}=(1+\bar{I}_{k})\overline{ \varphi}^{s}_{qkj}\psi_{kj} \tag{138}\]  
\[\left[ \begin{array}{c}\underline{U}\\ \underline{J}\end{array} \right]_{pqij}\equiv U_{pqij}=\sum_{k}\overline{\varphi} ^{s}_{pki}\overline{G}_{kij}\overline{\varphi}_{qkj} \tag{139}\]  
\[\left[ \begin{array}{c}\underline{j}\\ \underline{j}\end{array} \right]_{q}\equiv j_{qj} \tag{140}\]  
\[\left[ \begin{array}{c}\underline{V}^{i}\\ \underline{J}\end{array} \right]_{k}\equiv v^{i}_{k} \tag{141}\]  
The equivalent representation of the \(\underline{N}\) matrix is also used below:  
\[\underline{\underline{N}}=\underline{\underline{M}}\left(\underline{I}+ \underline{\underline{I}}\right) \tag{142}\]  
where the following notations were introduced:  
The elements of the scattering matrix depend exclusively, through the matrix of reflection coefficients \(\underline{L}\), [PERSON]'s function \(\underline{G}\) and the form of Floquet harmonics \(\underline{y}_{k}\), on the geometry and material parameters of the antenna structure as well as on the system of basic (testing) functions \(\underline{\varphi}_{q}\) adopted in the analysis. By the very definition of the scattering matrix, its elements do not depend on the specifically analyzed case of excitation of the antenna structure and on the currents induced on its elements. Thus, the scattering matrix completely characterizes the antenna structure in the general case for any of its excitations.