Discrete derivative estimation in LISA Pathfinder data reduction

TL;DR

This paper develops and analyzes five-point discrete derivative estimators, including parabolic fit and Taylor series methods, for processing LISA Pathfinder data to accurately determine test mass forces from interferometer measurements.

Contribution

It introduces a generalized class of five-point discrete derivative estimators tailored for LISA Pathfinder data analysis, including methods to reduce high-frequency noise effects.

Findings

Three five-point second derivative estimators analyzed with simulated data.

Estimators with high high-frequency noise can cause systematic errors.

Performance varies depending on noise characteristics and estimator design.

Abstract

Data analysis for the LISA Technology package (LTP) experiment to be flown aboard the LISA Pathfinder mission requires the solution of the system dynamics for the calculation of the force acting on the test masses (TMs) starting from interferometer position data. The need for a solution to this problem has prompted us to implement a discrete time domain derivative estimator suited for the LTP experiment requirements. We first report on the mathematical procedures for the definition of two methods; the first based on a parabolic fit approximation and the second based on a Taylor series expansion. These two methods are then generalized and incorporated in a more general class of five point discrete derivative estimators. The same procedure employed for the second derivative can be applied to the estimation of the first derivative and of a data smoother allowing defining a class of simple five points estimators for both. The performances of three particular realization of the five point second derivative estimator are analyzed with simulated noisy data. This analysis pointed out that those estimators introducing large amount of high frequency noise can determine systematic errors in the estimation of low frequencies noise levels.