# 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
For any \(z\), the scattered field and consequently also the total field are expressed by the Fourier transform of the appropriate integrand function:  
\[\begin{array}{l}\underline{E}^{s}(x,y,0)=\sum_{k}\!\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty}F\big{(}\underline{E}^{s}_{kk00}\big{)}\big{(}k_{x},k_ {y},0\big{)}\!\sum_{m,n}\!\varepsilon^{im\phi_{x}+in\phi_{y}}e^{i[k_{x}x+k_{y}y +k_{z}z]}dk_{x}dk_{y}\\ =(2M+1)(2N+1)\!\sum_{k}\!\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}F\big{(} \underline{E}^{s}_{kk00}\big{)}\big{(}k_{x},k_{y}0\big{)}\,Af_{k}\big{(}k_{x}, k_{y}\big{)}e^{i[k_{x}x+k_{y}y+k_{z}z]}dk_{x}dk_{y}\\ +4\!\sum_{k}\!\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}F\big{(} \underline{E}^{s}_{kk00}  
The unknown to be determined here is the distribution of currents \(\underline{J}(x,y)\) or alternatively the values of the coefficients \(\overline{\underline{J}}_{k}\) of the spectral distribution of currents on the harmonics of the periodic structure.  
The solution of the system of Equations (115)-(118) will be presented below using the [PERSON] method and the decomposition into vector basis functions. The construction of the solution is based on the decomposition of the basic functions into Floquet harmonics:  
\[\underline{\phi}_{q}(x,y)=\sum_{k}\overline{\underline{\phi}}_{qk}e_{k}(x,y) \tag{119}\]  
\[\overline{\underline{\phi}}_{qk}=\iint\limits_{S_{00}}e_{k}^{*}(x,y)\underline {\phi}_{q}(x,y)dxdy \tag{120}\]  
with distribution coefficients \(\overline{\underline{\phi}}_{qk}=\overline{\phi}_{qkj}\), where the indices \(j=1\), \(2\) correspond to the vector indices \(x\) (\(j=1\)) and \(y\) (\(j=2\)). The distribution of currents induced on the metal elements of the antenna into basis functions:  
\[\underline{J}(x,y)=\sum_{q}\underline{j}_{q}\underline{\phi}_{q}(x,y)=\sum_{ k}\overline{\underline{J}}_{k}e_{k}(x,y) \tag{121}\]  
where  
\[\overline{\underline{J}}_{k}=\sum_{q}\underline{j}_{q}\overline{\underline{ \phi}}_{qk} \tag{122}\]  
In the above expressions, there are two equivalent representations of the decomposition of currents into scalar Floquet harmonics and basic functions. Knowledge of the relationship between basis functions and Floquet harmonics is essential for formulating and solving the considered scattering problem. The basic (test) functions are related to Floquet harmonics by means of equations:  
\[\underline{\underline{\varphi}}_{q}=\sum_{k}\overline{\underline{\varphi}}_{ qk}e_{k}(x,y) \tag{123}\]  
\[\overline{\underline{\varphi}}_{qk}=\iint\limits_{S_{00}}e_{k}^{*}(x,y) \underline{\phi}\varphi_{q}dxdy \tag{124}\]  
This is assuming orthonormality of basic functions (separately for each component \(\underline{\underline{\phi}}_{p}\equiv\ \ \left(\phi_{p1},\phi_{p2}\right)\); \(\ \phi_{p1}\equiv\ \phi_{px}\), \(\phi_{p2}\equiv\ \phi_{py}\)). \[\int\limits_{S_{00}}\underline{\phi}_{p}^{*}(x,y)\underline{\underline{\phi} }_{q}(x,y)dxdy=\delta_{pq} \tag{125}\]  
We obtain, from the orthonormality of scalar harmonics, the condition for the orthonormality of the coefficients of the distribution of the basis (testing) functions:  
\[\sum_{k}\overline{\underline{\varphi}}_{pk}^{*}\overline{\underline{\varphi} }_{qk}=\delta_{pq} \tag{126}\]  
The above dependencies also result in the relationships between the current distribution coefficients in the physical and spectral spaces.