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

TL;DR

This paper introduces a practical computational framework for multi-frequency periodograms with super resolution capabilities, applicable to various time series data, demonstrated through diverse astronomical examples and supported by open-source code.

Contribution

It presents a novel, user-friendly algorithm for multi-frequency periodograms with super resolution, surpassing the Rayleigh criterion, and extends the concept to non-sinusoidal bases and time domain applications.

Findings

Super resolution evades Rayleigh criterion

Effective for uneven and arbitrary sampling

Demonstrated on synthetic and real astronomical data

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.