Use of discrete Laguerre sequences to extrapolate wide-band response from early-time and low-frequency data

Extrapolation of wide-band response using early-time and low-frequency data has been accomplished by the use of the orthogonal polynomials, such as Laguerre polynomials, Hermite polynomials, and Bessel-Chebyshev functions. It is a good approach to reduce the computational loads and obtain stable results for computation intensive electromagnetic analysis. However, all the orthonormal basis functions that have been used are all continuous or analog functions, which means we have to sample the polynomials both in time and frequency domains before we can use them to carry out the extrapolation. The process of sampling will introduce some errors, especially for high degrees or small scaling factors and, hence, may destroy the orthogonality between the polynomials of various degrees in a discrete sense. In this paper, we introduce the discrete Laguerre functions, which are directly derived using the Z transform and, thus, are exactly orthonormal in a discrete sense. The discrete Laguerre polynomials are fundamentally different from its continuous counterparts, except asymptotically when the sampling interval approaches zero. The other advantage of using these discrete orthonormal functions is that they do not give rise to the Gibbs phenomenon unlike its continuous counterpart. Using it in the extrapolation, the range or convergence can be extended both for the scaling factor and order of expansion, and at the same time, the quality of performance can be improved. Since the error of extrapolation is sensitive to the scaling factor, an efficient way to estimate the error as a function of the scaling factor is explained and its feasibility for any problem is validated by numerical examples of antennas.