# Galerkin's Spectral Method in the Analysis of Antenna Wall Operation
## 1 Introduction  
Printed antennas have been some of the most innovative areas of antenna technology for a dozen or so years. In the early fifties, radiation from strip and microstrip lines was observed. From the point of view of the use of transmission lines, however, this was an undesirable phenomenon; so, apart from a few articles suggesting its use in the construction of antennas, it did not arouse much interest. Until 1971, most published works were only in the form of presentations of experimental results. It was not until the seventies that rapid development in theoretical and experimental work on this type of antenna began. In 1979, the first international conference was held in La Cruses, devoted to a comprehensive approach to antennas on a dielectric substrate, i.e., materials, construction plans, configuration systems and theoretical foundations. Microstrip antennas are characterized by many interesting features, such as the following:  
* Precise mapping on the surface;
* Low manufacturing cost;
* High repeatability of execution;
* Small volume;
* Masking of the operating frequency;
* Simplicity of manufacturing provided that relatively advanced technologies are used;
* Flat shape and low weight, which allow the use of antennas on a dielectric substrate on fast flying objects without the risk of deterioration of their aerodynamic properties. These antennas allow for miniaturization of the antenna system and thus its greater density. This causes mutual couplings to occur, changing the field distributions on apertureantennas and the current distributions in linear antennas. This situation in turn changes the spatial radiation characteristics of the antennas and their impedance inputs. The spectral response of an antenna array placed in a homogeneous dielectric medium depends on the antenna geometry, medium parameters, angle of incidence, polarization and geometry of the exciting field. However, in many applications, the above number of parameters determining the operation of the antenna is not sufficient to model its desired directional characteristics. An increase in the number of parameters of the antenna structure can be obtained by introducing a multilayer dielectric medium with a specified number of metalized periodic surfaces placed on flat boundaries between dielectric layers [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13]. There are two complementary approaches to the analysis of such structures. In the first, the composite antenna system is analyzed by constructing supermodes of the entire structure; in the second, the system is considered as a cascaded composition of planar discrete elements, i.e., boundaries between two dielectrics, periodic metallized planes and dielectric layers. The latter approach leads to the definition of the scattering, transmission or impedance matrix of the entire structure by cascaded composition of appropriate matrices assigned to individual discrete elements of the antenna structure. It is particularly useful in modeling dielectric multilayer antenna walls, where the stored data concerning one planar antenna element can be repeatedly used in the analysis of different antenna systems with modified parameters of the remaining discrete elements of the structure. In practice, antenna arrays are most often composed of periodic metal structures placed on or immersed in a dielectric multilayer medium. Changes in the size and shape of individual antenna elements enable effective modeling of their spectral characteristics; these changes can be additionally modeled by selecting the geometrical and physical parameters of the dielectric layered substrate and coating [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13]. In the case of large transverse dimensions of the antenna of the order of 100 wavelengths and several dozen or more individual metal radiation elements, the radiation properties of the antenna correspond to the properties of an unconstrained antenna structure to a good approximation.