Fourier Analysis of Time-Stamped Signals Using a Small Number of Samples

Abstract

This paper concerns the Fourier analysis of measurements taken at times that are not at all uniformly spaced yet are accurately known. Such can be the case when measurements are obtained by networked sensors with accurate or synchronized clocks using common Internet Protocol networks or other networks that do not guarantee timely data delivery. Such can also be the case when the object being measured is only sporadically available for measurement. A method for computing a complex spectrum from a minimal number of samples taken at nonuniform time intervals is described. The novelty of this paper lies in the method's ability to minimize the number of samples required to ensure a desired accuracy. A practical application of the method lies in doing spectral analysis of signals obtained from networked sensors where acquiring each sample is expensive in time, power, or money, and so, the number of samples taken is to be minimized. The method has been tested on two rather different sets of actual measurements: one taken via smart sensors connected to the Internet and the other taken by a number of observational astronomers using a variety of apparatus over decades.