FFT-based matrix compression for layered media

We consider an extension of the Fast Fourier Transforms based impedance matrix compression method to problems involving complex conducting structures embedded in layered media of infinite extent in the transverse directions. The method regains generalization of the compression technique to the multilayered medium Green's function. The method is applicable to structures which may be electromagnetically large and, at the same time, discretized with highly sub-wavelength resolution. We analyze two approaches of compressing the far field part of the impedance matrix: through an approximation to the Fourier transforms of the basis functions <(<phi>)over tilde>(q): (A) based on the Taylor expansion about q = 0, and (B) based on the least-squares approximation at the q values giving dominant contribution to the impedance matrix element. While compression (A) is applicable already to distances much smaller than the wavelength, compression (B) is based ox the field behavior in the asymptotic wave region, and thus applies only to distances comparable to and larger than the wavelength. Therefore. approach (A) is better suited to structures discretized with highly sub-wavelength resolution. Its other advantages are simplicity and independence of the structure of the Green's function. For densely packed structures, the method is characterized by O(N log N) computational complexity and O(N) memory requirements with a small, when compared to other approaches. proportionality coefficient in front of the estimates. Error estimates for the proposed algorithm an discussed.