The WDM time-frequency transform in gravitational-wave data analysis II: Evolutionary spectrum and noise covariance
Abstract
Noise cannot be treated as stationary over the timescales required for upcoming gravitational-wave detectors, minutes for ground-based and months for space-based detectors. Nonstationarity leads to a non-diagonal frequency-domain noise covariance matrix, thus motivating a shift to the time-frequency domain. We introduce \emph{evolutionary power spectra} for describing nonstationary noise in the Wilson-Daubechies-Meyer time-frequency basis. Under the framework of oscillatory processes, we show that, to leading order in the ratio of the Meyer filter and the nonstationarity timescales, the evolutionary spectrum is a power-frequency distribution that can be estimated from a single noise realization. We derive the corresponding noise covariance matrix and show that it is band tri-diagonal, like its stationary counterpart. The exact covariance to all orders is positive semi-definite by construction and dense depending on the noise. We further derive the optimal tiling of the time-frequency plane that minimizes the error of a diagonal covariance approximation and show that coarse-graining the plane suppresses correlations quadratically. Using simulated data with a LISA Galactic foreground and a drifting powerlaw spectrum, we show that the stationarity approximation leads to signal-to-noise ratio loss and severely biases inference. For realistic LISA data, approximating the covariance as diagonal in the Wilson-Daubechies-Meyer domain is accurate for the narrow-band analysis of a single resolved Galactic binary, but insufficient for a broad-band analysis. Finally, we provide mitigation strategies for the wraparound correlations induced across the data time edges whenever the spectrum differs between the start and the end of the data.
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BibTeX
@misc{johnson2026timefrequency,
title = {{The WDM time-frequency transform in gravitational-wave data analysis II: Evolutionary spectrum and noise covariance}},
author = {Aaron D. Johnson and Katerina Chatziioannou and Robert Rosati},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19002},
eprint = {2610.19002},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19002},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The WDM time-frequency transform in gravitational-wave data analysis II: Evolutionary spectrum and noise covariance
AU - Aaron D. Johnson
AU - Katerina Chatziioannou
AU - Robert Rosati
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.19002
UR - https://arxiv.org/abs/2610.19002
AB - Noise cannot be treated as stationary over the timescales required for upcoming gravitational-wave detectors, minutes for ground-based and months for space-based detectors. Nonstationarity leads to a non-diagonal frequency-domain noise covariance matrix, thus motivating a shift to the time-frequency domain. We introduce \emph{evolutionary power spectra} for describing nonstationary noise in the Wilson-Daubechies-Meyer time-frequency basis. Under the framework of oscillatory processes, we show that, to leading order in the ratio of the Meyer filter and the nonstationarity timescales, the evolutionary spectrum is a power-frequency distribution that can be estimated from a single noise realization. We derive the corresponding noise covariance matrix and show that it is band tri-diagonal, like its stationary counterpart. The exact covariance to all orders is positive semi-definite by construction and dense depending on the noise. We further derive the optimal tiling of the time-frequency plane that minimizes the error of a diagonal covariance approximation and show that coarse-graining the plane suppresses correlations quadratically. Using simulated data with a LISA Galactic foreground and a drifting powerlaw spectrum, we show that the stationarity approximation leads to signal-to-noise ratio loss and severely biases inference. For realistic LISA data, approximating the covariance as diagonal in the Wilson-Daubechies-Meyer domain is accurate for the narrow-band analysis of a single resolved Galactic binary, but insufficient for a broad-band analysis. Finally, we provide mitigation strategies for the wraparound correlations induced across the data time edges whenever the spectrum differs between the start and the end of the data.
ER -