A Fast Template Periodogram for Detecting Non-sinusoidal Fixed-shape Signals in Irregularly Sampled Time Series

Astrophysical time series often contain periodic signals. The large and growing volume of time series data from photometric surveys demands computationally efficient methods for detecting and characterizing such signals. The most efficient algorithms available for this purpose are those that exploit the $\mathcal{O}(N\log N)$ scaling of the Fast Fourier Transform (FFT). However, these methods are not optimal for non-sinusoidal signal shapes. Template fits (or periodic matched filters) optimize sensitivity for a priori known signal shapes but at a significant computational cost. Current implementations of template periodograms scale as $\mathcal{O}(N_f N_{obs})$, where $N_f$ is the number of trial frequencies and $N_{obs}$ is the number of lightcurve observations, and due to non-convexity, they do not guarantee the best fit at each trial frequency, which can lead to spurious results. In this work, we present a non-linear extension of the Lomb-Scargle periodogram to obtain a template-fitting algorithm that is both accurate (globally optimal solutions are obtained except in pathological cases) and computationally efficient (scaling as $\mathcal{O}(N_f\log N_f)$ for a given template). The non-linear optimization of the template fit at each frequency is recast as a polynomial zero-finding problem, where the coefficients of the polynomial can be computed efficiently with the non-equispaced fast Fourier transform. We show that our method, which uses truncated Fourier series to approximate templates, is an order of magnitude faster than existing algorithms for small problems ($N\lesssim 10$ observations) and 2 orders of magnitude faster for long base-line time series with $N_{obs} \gtrsim 10^4$ observations. An open-source implementation of the fast template periodogram is available at https://www.github.com/PrincetonUniversity/FastTemplatePeriodogram.