引力波数据分析中的WDM时频变换 II：演化谱与噪声协方差
The WDM time-frequency transform in gravitational-wave data analysis II: Evolutionary spectrum and noise covariance
AI总结：
该研究针对引力波探测器的非平稳噪声，提出WDM时频域的演化功率谱，推导其噪声协方差矩阵，通过模拟与实际LISA数据验证平稳性近似的缺陷，并给出环绕相关性的缓解策略。
摘要：
在下一代引力波探测器所需的时间尺度上，噪声不能被视为平稳的——地面探测器的时间尺度为分钟级，空间探测器则为月级。非平稳性会导致频域噪声协方差矩阵非对角，因此推动了向时频域的转变。我们引入演化功率谱，用于描述Wilson-Daubechies-Meyer（WDM）时频基下的非平稳噪声。在振荡过程框架下，我们证明，在Meyer滤波器与非平稳时间尺度的比值的主导阶下，演化谱是一种功率频率分布，可从单个噪声实现中估计得到。我们推导了对应的噪声协方差矩阵，表明其与平稳情况类似，为带状三对角矩阵；而所有阶的精确协方差在构造上为半正定，具体形式则取决于噪声是否稠密。我们进一步推导了时频平面的最优划分方式，该划分可最小化对角协方差近似的误差，且对平面进行粗粒化处理可使相关性呈二次方级抑制。通过包含LISA银河系前景与漂移幂律谱的模拟数据，我们证明，平稳性近似会导致信噪比损失并严重偏倚推断结果。对于实际LISA数据，将Wilson-Daubechies-Meyer域中的协方差近似为对角形式，对单个已分辨银河系双星的窄带分析是准确的，但对宽带分析则不足。最后，我们针对数据起止时刻频谱存在差异时，数据时间边缘引发的环绕相关性，提供了缓解策略。
英文摘要：
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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