Studies in Astronomical Time Series Analysis: The Double Lomb-Scargle Periodogram and Super Resolution

TL;DR

This work develops practical multi-frequency time series tools by introducing double periodograms that model data as a sum of two independent sinusoids and extend to an omnigram framework with arbitrary basis functions. It presents four variants based on maximizing or marginalizing the likelihood, with or without Lomb phase shifts, enabling super resolution that can separate closely spaced frequencies beyond the Rayleigh limit. Demonstrations on sunspot numbers and the pulsating white dwarf WD J0135+5722 show the method can resolve frequencies with separations well below the classical Rayleigh threshold and extract amplitudes and phases. The approach generalizes to arbitrary bases, offers a computationally straightforward implementation, and provides open-source code to promote broad use across astronomy and related fields.

Abstract

Multiple-frequency periodograms -- based on time series models consisting of two or more independent sinusoids -- have long been discussed. What is new here is the presentation of a practical, simple-to-use computational framework implementing this concept. Our algorithms have super resolution that evades the Rayleigh criterion, as well as provision for statistical weighting and tapering. They can be used for essentially any time series (e.g. time-tagged events or point measurements) with arbitrary sampling -- even or uneven. Examples of super resolution of synthetic data, sunspot numbers, and the rich pulsations of white dwarf J0135+5722, demonstrate practical applications. Appendices derive generalized periodograms using an arbitrary number of arbitrary basis functions (following Bretthorst, 1988)and define several examples of non-sinusoidal bases for these ``omnigrams.'' Application beyond the frequency domain is demonstrated with an autoregressive model exhibiting super resolution in the time domain. A GitHub repository containing omnigram code, and symbolic algebra scripts for generating it, will soon be available.

Figures (9)

• Figure 1: Resolution of two synthetic sinusoids: Upper Left: the double Lomb-Scargle periodogram $P(\omega_{1}, \omega_{2})$ for two sinusoids of length 10000 seconds ($\omega_{0} =$ 6.2886e-04 radians per second) with frequencies of .0235 and .0237 radians per second (i.e. ${\bf \mathcal{R}}_{ayleigh}$ = .3), amplitudes $1$ and $4$, and with additive Gaussian noise of amplitude $0.1$. The circle is centered on the true values $\omega_{1} = 0.0235$ and $\omega_{2} = 0.0237$; the black dot is the local maximum of $P(\omega_{1}, \omega_{2})$, precisely recovering the true values. Values above the diagonal, redundant due to symmetry, are not shown. Upper-right and lower-left: slices of $P(\omega_{1}, \omega_{2})$ passing through this maximum; their maxima yield frequency estimates indicated by red dots. Lower-right: the 1D LSP, with vertical lines indicating the true input frequencies.
• Figure 2: DLSP Super Resolution. The estimated frequency separation of two sinusoids, with four different levels of additive Gaussian noise, is plotted as a function of their true separation, in units of the fundamental. Indicated S/N values are the ratio of the sinusoid amplitude to the noise standard deviation. Abscissas are slightly offset for clarity. The points are averages, and $1\sigma$ error bars are standard deviations, of the estimated component separations over 128 realizations of the noise and time-sampling.
• Figure 3: Recovery of Amplitudes. The estimated values of the four parameters $a_{1}, b_{1}, a_{2}$ and $b_{2}$ in Eq. \ref{['two_sine_model']} (from which the amplitudes and phases of the two component sinusoids can be computed) are plotted as a function of the Rayleigh parameter R. The arbitrarily chosen input values (1, 3.25, 2.1 and 4) are open circles at the ends of the range. These plots are based on the same simulation as in Fig. \ref{['super_resolution_scan_0']}, and use the same colors to indicate the four signal-to-noise cases.
• Figure 4: One dimensional LS periodograms of the sunspot time series, oversampled in frequency by a factor of 16: untapered and with a Slepian taper (thick blue and red lines, respectively). Window functions computed from the sample times as described in the footnote in Section \ref{['criticisms']} are thin lines in corresponding colors, shifted to coincide with the periodogram peaks.
• Figure 5: Top: DLSP for the sunspot number time series. The "+" symbol indicates the peak periodogram value at frequency coordinates (.0917 .0971), periods of 10.905 and 10.297 years, respectively. Bottom: Slices of the double periodogram passing through this peak: blue = vertical; red = horizontal. The two peak frequencies are marked with vertical lines.