The analysis of indexed astronomical time series – XII. The statistics of oversampled Fourier spectra of noise plus a single sinusoid
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Date
2015
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Publisher
Oxford University Press
Abstract
With few exceptions, theoretical studies of periodogram properties focus on pure noise time
series. This paper considers the case in which the time series consists of noise together with a
single sinusoid, observed at regularly spaced time points. The distribution of the periodogram
ordinates in this case is shown to be of exponentially modified Gaussian form. Simulations
are used to demonstrate that if the periodogram is substantially oversampled (i.e. calculated
in a dense grid of frequencies), then the distribution of the periodogram maxima can be
accurately approximated by a simple form (at least at moderate signal-to-noise ratios). This
result can be used to derive a calculation formula for the probability of correct signal frequency
identification at given values of the time series length and (true) signal-to-noise ratio. A set of
curves is presented which can be used to apply the theory to, for example, asteroseismic data.
An illustrative application to Kepler data is given.
Description
Keywords
Astronomy, Periodogram, Asteroseismic data, Kepler, Sinusoidal signal
Citation
Koen, C. (2015). The analysis of indexed astronomical time series – XII. The statistics of oversampled Fourier spectra of noise plus a single sinusoid. Monthly Notices of the Royal Astronomical Society, 453(2): 1793-1798.