Conditions for generation of stable and accurate hybrid TD-FD MoM solutions

Broadband characterization of any electromagnetic (EM) data (e.g., surface currents, radiation pattern, and network parameters) can be carried out using partial information in the time domain (TD) and the frequency domain (FD). In this hybrid TD-FD approach, one generates the early time response using a TD code at a spatial location and uses a FD code to generate the low-frequency response at the same place. Then, the partial complementary information in both the TD and FD is fit by a set of orthogonal functions and its Fourier transform having the same expansion coefficients. Three different types of functions, namely, Hermite, Bessel-Chebyshev, and Laguerre, have been used for extrapolation. Once the expansion coefficients for these functions are known, the response can be extrapolated either for late times or high frequencies using the initial partial information. The objective of this paper is to explore the conditions under which this hybrid approach yields a stable and accurate solution. We investigate bounds for both the number of orthogonal functions needed to carry out the extrapolation and the scale factors needed to accurately fit the data in time and in frequency. Numerical examples have been presented to illustrate the efficacy of these bounds. It is important to point out that, in this hybrid approach of extrapolation, we are not creating new information but processing the available information in an intelligent fashion.