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Matrix inversion in spatial frequency analysis of electromagnetic scattering and radiation

The spatial frequency technique (SFT) was developed for solving the integral equation (IE) in electromagnetic scattering and radiation problems and shown to work faster than many existing solution methods. In this paper, the new formulation, which is based on the efficient FFT algorithm, is analyzed further for analytically solving the IE for certain types scattering and radiation problems. It is shown that the SFT description of the symmetric matrix which represents the Green's function may be decomposed into an orthogonal basis using Takagi factorization. Transforming the Green's function into this orthogonal basis has the potential to invert the Green's function analytically. The SFT is also compared with the Fast Multipole Method (FMM) and shown to have similar underlying properties which may also be utilized to analytically invert the Green's function using FMM.