Simultaneous time- and frequency-domain extrapolation

In this paper, given the early-time response and the low-frequency response of a causal system, we simultaneously extrapolate them in the time and frequency domains. The approach is iterative and is based on a simple discrete Fourier transform. Simultaneous extrapolation in time and frequency domains is further enhanced by using the matrix pencil technique in the time domain and the Cauchy method in the frequency domain. The results are further enhanced through the Hilbert transform, hence enforcing the physical constraints of the system and thereby guaranteeing a causal extrapolation in time. It is, therefore, possible to generate information over a larger domain from limited data. It is important to note that through this extrapolation, no new information is created. The early-time and low-frequency data are complementary and contain all the desired information. The key is to extract this information in an efficient and accurate manner. The electric current on a scatterer is used as an example for the method.