Envelope-Tracking Adaptive Integral Method for Band-Pass Transient Scattering Analysis

Abstract

An envelope-tracking (ET) algorithm is presented for efficiently solving integral equations pertinent to the analysis of transient scattering from perfect electrically conducting surfaces. The algorithm solves space-time samples of the complex envelope of the surface current density by implicit time marching; thus, the time-step size is inversely proportional to the bandwidth—not the maximum frequency content—of the fields of interest. At each time step, anterpolation, propagation, interpolation, and near-zone correction stages are performed to compute the right-hand side of a matrix equation, which involves 4-D space-time FFTs, and to accelerate its iterative solution, which involves 3-D space FFTs. These FFTs dominate the computational costs of the method: they require $O({N_{\mathrm{C}}} [\log^2 {N_{\mathrm{g}}} + {\bar{N}_{\mathrm{I}}} \log {N_{\mathrm{C}}}])$ operations per time step; here, $N_{\mathrm{C}}$ is the number of nodes on an auxiliary regular grid, $N_{\mathrm{g}}$ is determined by the number of time steps that fields take to travel the maximum distance between two points on the scattering surface, and $\bar{N}_{\mathrm{I}}$ is the average number of iterations needed for the iterative solution to converge at each time step. Moreover, the algorithm is systematically compared to its time- and frequency-domain counterparts. To this end, radar profiles of benchmark targets are computed, a time-domain dB-error norm is proposed, the algorithms’ parameters are optimized subject to at most 2% error in the radar profile according to this norm, and the computational costs are measured, as target size, shape, and excitation pulse are varied. The results show that the algorithm outperforms its time-domain counterpart and is competitive with its frequency-domain counterpart.