SigSpec User's Manual

Abstract

SigSpec computes the spectral significance levels for the DFT∗ amplitude spectrum of a time series at arbitrarily given sampling. It is based on the analytical solution for the Probability Density Function (PDF) of an amplitude level, including dependencies on frequency and phase and referring to white noise. Using a time series dataset as input, an iterative procedure including step-by-step prewhitening of the most significant signal components and MultiSine leastsquares fitting is provided to determine a whole set of signal components, which makes the program a powerful tool for multi-frequency analysis. Instead of the step-by-step prewhitening of the most significant peaks, the program is also able to take into account several steps of the prewhitening sequence simultaneously and check for the combination associated to a minimum residual scatter. This option is designed to overcome the aliasing problem caused by periodic time gaps in the dataset. SigSpec can detect non-sinusoidal periodicities in a dataset by simultaneously taking into account a fundamental frequency plus a set of harmonics. Time-resolved spectral significance analysis using a set of intervals of the time series is supported to investigate the development of eigenfrequencies over the observation time. Furthermore, an extension is available to perform the SigSpec analysis for multiple time series input files at once. In this MultiFile mode, time series may be tagged as target and comparison data. Based on this selection, SigSpec is capable of determining differential Note from the Editor: We report with sadness that Peter Reegen, the developer of SigSpec and author of this manual, unexpectedly passed away this year. SigSpec was one of the main achievements of his scientific career. During the previous year he was able to observe the adoption of his program by a large number of astronomers. We sincerely regret the loss of our dear friend and colleague “Piet”. We thank Michael Gruberbauer, who has thoroughly looked through the paper and provided several comments and supplement information. These are represented in uncounted footnotes marked with *. Note by M. Gruberbauer: Discrete Fourier Transform (DFT) 4 SigSpec User’s Manual significance spectra for the target datasets with respect to coincidences in the comparison spectra. A built-in simulator to generate and superpose a variety of sinusoids and trends as well as different types of noise completes the software package at the present stage of development. 1. What is SigSpec? SigSpec (abbreviation of ‘SIGnificance SPECtrum’) is a program that computes a significance spectrum for a time series. It evaluates the Probability Density Function (PDF) of a given DFT amplitude level analytically, making use of the theoretical concept introduced by Reegen (2005, 2007). The FalseAlarm Probability, ΦFA (A), is the probability that an amplitude in the DFT spectrum exceeds a given limit A, and is obtained through integration of the PDF (e. g. Scargle 1982). Instead of this frequently used quantity, SigSpec calculates the spectral significance (abbreviated by ‘sig’) of an amplitude A by sig (A) := − log [ΦFA (A)] . (1) E. g., a sig equal to 5 indicates that the considered amplitude level is due to noise in one out of 10 cases.∗ This value is used as the default threshold for the termination of the prewhitening sequence. SigSpec performs an iterative process consisting of four steps: 1. computation of the significance spectrum, 2. exact determination of the peak with maximum sig, 3. a MultiSine least-squares fit of the frequencies, amplitudes and phases of all significant signal components detected so far, 4. prewhitening of the sinusoidal components. The residuals are used as input for the next iteration. If SigSpec is called without any special settings, it produces four files: 1. the DFT amplitude spectrum s000000.dat of the original time series, containing also sig and phase, 2. the DFT amplitude spectrum resspec.dat of the residual time series after prewhitening all significant signal components, containing also sig and phase, Note by M. Gruberbauer: Note that this quantitiy is defined for a single frequency rather than for the whole spectrum. See Section 15.5. for a discussion. 1The AntiAlC computation (p. 47) differs slightly from this procedure.