Optimal design of interpolation methods for time-delay interferometry

Abstract

Time-delay interferometry (TDI) suppresses laser frequency noise by forming linear combinations of time-shifted interferometric measurements. The time-shift operation is implemented by interpolating discretely sampled data. To enable in-band laser noise reduction by eight to nine orders of magnitude, interpolation has to be performed with high accuracy. Interpolation can be understood as the convolution of an interpolation kernel with the data to be shifted. Optimizing the design of this interpolation kernel is the focus of this work. Previous research that studied constant time-shifts suggested Lagrange interpolation as the interpolation method for TDI. Its transfer function is maximally flat at DC and therefore performs well at low frequency. However, to be accurate at high frequencies, Lagrange interpolation requires a high number of coefficients. Furthermore, when applied in TDI we observed prominent time-domain features when a time-varying shift scanned over a pure integer sample shift. To limit this effect we identify an additional requirement for the interpolation kernel: when considering time-varying shifts the interpolation kernel must be sufficiently smooth to avoid unwanted time-domain transitions that produce glitch-like features in power spectral density estimates. The Lagrange interpolation kernel exhibits a discontinuous first derivative by construction, which is insufficient for the application to LISA or other space-based gravitational-wave observatories. As a solution we propose a novel design method for interpolation kernels that respect a predefined requirement on in-band interpolation residuals and that possess continuous derivatives up to a prescribed order. Using this method we show that an interpolation kernel with 22 coefficients is sufficient to respect LISA’s picometre-requirement and to allow for a continuous first derivative which suppresses the magnitude of the time-domain transition adequately. The reduction from 42 (Lagrange interpolation) to 22 coefficients enables us to increase robustness against artifacts in the data and, as a side effect, save computational cost.