# Galerkin's Spectral Method in the Analysis of Antenna Wall Operation
## 5 Spectral Representation of the Current Distribution on a Limited Antenna Array  
The solution to the scattering problem analyzed is based on the spectral decomposition of the window function on the Floquet harmonics \(e_{k}\)(\(x\),\(y\)). In general, the analysis of the problem also allows for the possibility of decomposing the window function on the harmonics of the grid with other values of the grid steps, for example, equal to \(Md_{x}\) and \(Nd_{y}\) in the \(x\)- and \(y\)-axes. In this case, this corresponds to the spectral decompositions on the harmonics of a planar rectangular unconstrained periodic structure with a single cell carrier equal to the carrier of the \(H_{MN}\) function. The basic relations specifying this type of spectral decompositions used in this work will be presented below. The effectiveness of the solution to the problem is conditioned by the formulation of an appropriate spectral representation of the current distribution function of the constrained periodic structure:  
\[\underline{J}_{mn}(x,y)=H_{nm}(x,y)\underline{J}(x,y) \tag{201}\]  
in a form, analogous to the case of an unconstrained structure. The functions \(J_{mn}(x\),\(y)\) and \(\underline{J}(x\),\(y)\) denote the distribution of currents on, respectively, a constrained and unconstrained periodic grid, \(H_{mn}\) is the window function of the constrained structure and the pair of integers \(m\), \(n\) determines the position of the origin of the coordinate system relative to the central point (geometric center) of the constrained antenna structure. Let us introduce the decomposition of the function \(J(x\),\(y)\) into the Floquet harmonics of the periodic array with the array jumps equal to \(d_{x}\), \(d_{y}\), respectively:  
\[\underline{J}(x,y)=\sum_{k}\underline{J}_{x}e_{k}(x,y), \tag{202}\]  
where the distribution function is expressed by the Fourier transform of the function \(\underline{J}(x,y)\):  
\[F(\underline{J})\left(k_{x},k_{y}\right)=\sum_{k}(d_{x}d_{y})^{-1/2}\underline {J}_{k}\delta(k_{x}-k_{xk})\delta\left(k_{y}-k_{yk}\right). \tag{203}\]  
Figure 4: The radiation pattern of an antenna array. Figure 3: Eight-element antenna array. and in the spectral decomposition, we use the normalized Floquet harmonics of the periodic antenna array. Then, we introduce the decomposition of the function \(H_{mn}(x,y)\) into new Floquet harmonics of the periodic grid with grid steps equal to \(Md_{x}\), \(Nd_{y}\), respectively:  
\[H_{mn}(x,y)=\sum_{\kappa}\overline{H}_{mnx}f_{\kappa}(x,y) \tag{204}\]  
with a distribution function defined by the Fourier transform of the window function:  
\[F(H_{mn})\big{(}k_{x},k_{y}\big{)}=\sum_{k}(Md_{x}Nd_{y})^{-1/2}\overline{H}_{ mnk}\delta(k_{x}-k_{xx})\delta\big{(}k_{y}-k_{yx}\big{)} \tag{205}\]  
and using Floquet harmonics with a higher spectrum sampling rate and zero phase shift in the harmonic exponent:  
\[f_{\kappa}(x,y)=\Big{(}Md_{x}Nd_{y})^{-1/2}exp[i\big{(}k_{xx}x+k_{yx}y\big{)} \Big{]}, \tag{206}\]  
\[k_{xx}=2\pi m^{\prime\prime}/Md_{x}, \tag{207}\]  
\[k_{yx}=2\pi n^{\prime\prime}/Md_{y}, \tag{208}\]  
\[\underline{J}(x,y)=\sum_{k}\underline{J}_{x}e_{k}(x,y), \tag{202}\]0  
Ultimately, the current distribution in the limited antenna array is expressed by the double sum of the Floquet harmonic distribution of both types:  
\[\underline{J}(x,y)=\sum_{k}\underline{J}_{x}e_{k}(x,y), \tag{202}\]1  
This is the starting formula for modifying the solution of the scattering problem on an unconstrained antenna grid, where the relationship holds the following:  
\[\underline{J}(x,y)=\sum_{k}\underline{J}_{x}e_{k}(x,y), \tag{202}\]2  
\[\underline{J}(x,y)=\sum_{k}\underline{J}_{x}e_{k}(x,y), \tag{202}\]3  
From the form of the above expressions, several parallel specifications of the numerical analysis of the solution follow.