Resampled random processes in gravitational-wave data analysis

TL;DR

This paper proves that the non-stationary random process resulting from resampling gravitational-wave data is mathematically well-defined and extends fundamental stationary process results to this context.

Contribution

It establishes the mathematical foundation for the non-stationary processes arising in gravitational-wave data analysis due to resampling.

Findings

Proves the non-stationary process is well-defined

Extends Wiener-Khintchine theorem to non-stationary processes

Extends Cramér representation to this context

Abstract

The detection of continuous gravitational-wave signals requires to account for the motion of the detector with respect to the solar system barycenter in the data analysis. In order to search efficiently for such signals by means of the fast Fourier transform the data needs to be transformed from the topocentric time to the barycentric time by means of resampling. The resampled data form a non-stationary random process. In this communication we prove that this non-stationary random process is mathematically well defined, and show that generalizations of the fundamental results for stationary processes, like Wiener-Khintchine theorem and Cram\`{e}r representation, exist.