Estimating the power spectra of unevenly sampled X-ray time series: Unresolved Gaussian fitting to the autocorrelation function

Abstract

We present a new method for estimating the power spectral density function (PSD) of a stochastic light curve where the available observations are not uniformly sampled in time. This analysis is particularly appropriate to X-ray data from satellites in low Earth orbit, as the temporal sampling of such data can be extremely heterogeneous. The method uses the fact that the PSD of a stochastic light curve is the Fourier transform of its autocorrelation function (ACF). We therefore estimate the ACF from the available data, and fit a very general model to this function. The Fourier transform of this model is the best estimate of the underlying PSD, subject only to the physical constraints that the PSD is positive everywhere, and that it is a smoothly varying function of frequency (with smoothing scale specified by the user). The analysis also returns a measure of the uncertainty in the PSD at each frequency, and the covariances between the estimates at different frequencies. Tests of this method show that it is capable of recovering the power spectra from a wide range of models, even when the observations are temporally very non-uniform. Since the method avoids the smoothing imposed by many methods of PSD estimation, it is even capable of recovering the shapes of PSDs when the power spectrum varies very rapidly with frequency. Application to Ginga data from the active galaxy MCG-6-30-15 shows how this method can be used to derive power spectra from astrophysical data, and fitting of a power law to the derived spectrum returns a power-law index of - 1.5+0.30.4, in agreement with the results obtained by other less general methods.