Direct extrapolation of a causal signal using low-frequency and early-time data

In this paper, we provide three direct procedures to extrapolate the early-time and the low-frequency response of a causal signal simultaneously in the time-and frequency domain. Compared with the extrapolation by orthonormal basis functions, direct extrapolation is straightforward and we do not need to evaluate the basis functions and search for the optimal scaling factor and the optimal number of basis functions. We show that the extrapolation introduced by Adve and Sarkar is equivalent to a Neumann-series solution of an integral equation of the second kind. It is further shown that this iterative Neumann expansion is an error-reducing method. We propose to solve this integral equation efficiently by employing a conjugate gradient iterative scheme. The convergence of this scheme is also demonstrated. We provide the matrix equations and show the equivalence to the integral equations, and demonstrate that the method of singular value decomposition (SVD) of solving the matrix equation provides accurate and stable results. Finally, a number of illustrative numerical examples are presented and the performances of the three direct methods are compared.