Direct extrapolation of a causal signal using low-frequency and early-time data based on integral equations

In this paper we provide two direct procedures to extrapolate the early-time and the low-frequency response of a causal signal simultaneously in the time and frequency domain. We show that the extrapolation introduced by Adve and Sarkar (1998) 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. Finally, a numerical example is presented and the performance of the two direct methods is compared.