Abstract
Time series analysis is ubiquitous in many fields of science including gravitational-wave astronomy, where strain time series are analyzed to infer the nature of gravitational-wave sources, e.g., black holes and neutron stars. It is common in gravitational-wave transient studies to apply a tapered window function to reduce the effects of spectral artifacts from the sharp edges of data segments. We show that the conventional analysis of tapered data fails to take into account covariance between frequency bins, which arises for all finite time series---no matter the choice of window function. We discuss the origin of this covariance and derive a framework that models the correlation induced by the window function. We demonstrate this solution using both simulated Gaussian noise and real Advanced LIGO/Advanced Virgo data. We show that the effect of these correlations is similar in scale to widely studied systematic errors, e.g., uncertainty in detector calibration and power spectral density estimation.
Highlights
Time-series analysis underpins recent advances in gravitational-wave astronomy
We show that the conventional analysis of tapered data fails to take into account covariance between frequency bins, which arises for all finite time series—no matter the choice of window function
As we probe increasingly higher signal-to-noise ratio (SNR), and as we combine larger ensembles of data segments, our analyses are increasingly susceptible to systematic error from approximations in our models and data analysis
Summary
Time-series analysis underpins recent advances in gravitational-wave astronomy. The vast majority of gravitational-wave data analysis relies on windowing, a procedure that multiplies the time-domain data segment by a window function that tapers off at the beginning and end of the segment. Analysts apply tapered windows to mitigate two effects: (1) spectral artifacts arising from the Fourier transform of the data segment edges (Gibbs phenomena) and (2) correlations between neighboring frequency bins. We derive from first principles a formalism which accounts for the correlations between neighboring frequencies introduced by the window function applied to obtain finite time series. We show how correlations between frequency bins arise from the fact that quasistationary Gaussian noise processes are fundamentally described in the frequency domain by continuous functions, which imply infinite-duration time series. We derive a simple expression for the “finite-duration” covariance matrix, which encodes the correlations naturally present in all finite time series, and identify our result as a specific basis for a Karhunen-Loève transformation (KLT) (see, e.g., [11]).
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