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.
Deep Dive into Discrete derivative estimation in LISA Pathfinder data reduction.
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 s
Among the main goals of the LTP experiment, to be flown aboard the LISA PATHFINDER mission, is the estimation of the residual force noise acting on the test masses, perturbing them from their geodesic motion. The measurement is performed in a differential configuration, by looking at the relative acceleration of two test masses hosted on the same spacecraft (SC) [1 -4]. In the framework of the first LTP Mock Data Challenge (MDC) [1], the differential acceleration of the test masses was reconstructed by the output of the interferometric readout according to the equation:
where:
• o 1 and o ∆ are the output of the interferometer channels reading the position of test mass 1(TM1) and the differential position between TMs respectively. • H df (s) and h lfs (s) are the gains of the drag-free loop and low-frequency-suspension loop respectively [3,4].
• ω p1 2 and ω p2 2 are the residual coupling of test mass 1 and test mass 2 (TM2) to the SC.
• δ is the cross-talk coefficient between the two interferometer (IFO) channels.
In writing Equation (1) we neglected small terms associated with the gravitational gradient and the mass ratio between TMs and S/C. Acceleration of the two test masses is calculated by differentiating the interferometer outputs o 1 and o ∆ taking into account the effects from residual coupling of the TMs to the SC and the presence of digital control loops applied to the spacecraft (following test mass 1 by means of low noise microthrusters) and to test mass 2 (forced electrostatically to follow TM1). Discrete representation of the controllers was obtained by impulse-invariant discretization of their Laplace domain representations H df (s) and h lfs (s). Available IFO signals are discrete therefore the evaluation of the derivatives involves a discretized differentiation of the signals, which is obtained by suitably designed filters that mimic the frequency response of the derivative operators. The subsequent step in estimating the residual noise is evaluating the Power Spectral Density (PSD) of the relative acceleration of the TMs, which is performed employing the modified Welch periodogram, associated with windowing and detrending of the time series to suppress long-term drifts. We discuss here the implications on these low frequency estimates as long as different methods are employed for signal differentiation. The most sensitive frequency range for LISA will be 1x10 -3 -3x10 -2 Hz. Therefore the need to extract as much information as possible from available data demand for an accurate estimation of the components of the acceleration PSD in the frequency range mentioned above.
Given a discretized series of data points one of the possible ways to numerically estimate the derivative is to making use of a five point equation. The advantage of such estimators is to ensure high accuracy with reduced calculation efforts [5]. The five point method for the derivative approximation, at a given time t = kT (k being an integer and T being the sampling time), is calculated by means of finite differences between the element at t with its four neighbors. In other words, considering the discrete series of data:
First and second derivative at a certain time can be approximated by a five point difference equation:
where T is the sampling time. The problem of the identification of the derivative estimator is reduced to the definition of the five coefficients [a’, b’, c’, d’, g’] and[a’’, b’’, c’’, d’’, g’’] for the first and second derivative respectively. In the framework of the first LTP MDC [1], we have used two different identification methods. The first is based on a parabolic fit approximation, and the second is based on a Taylor series expansion. In the following, it will be demonstrated that these two methods can be considered as two particular cases of a general method for the definition of a five point stencil for the numerical derivative estimation.
This method estimates the coefficients of the difference equation by means of a least square fit with a second order equation on a generic data series:
Given the equation ( 4), it is easy to see the connection between fit coefficients and the derivative estimator:
The least square procedure is equivalent to solve the system of equations:
The vector of coefficients can be found from the solution of 1 ( )
It is worth to note that K is independent from the particular series of data. Comparing (7) with equation ( 5), we see that the first row of K matrix is the vector of coefficients for the five point estimator of y[k] acting like a smoother for the data series, the second row of K matrix is the vector of coefficients for the five point estimator of the first derivative of y[k] and the third row of K matrix is the vector of coefficients for the five point estimator of one half the second derivative of y[k]. This method was successfully employed to reconstruct the external force acting on a harmonic oscillator as described in [6].
The Taylor serie
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