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Astrophysical time series often contain periodic signals. The large andgrowing volume of time series data from photometric surveys demandscomputationally efficient methods for detecting and characterizing suchsignals. The most efficient algorithms available for this purpose are thosethat 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 prioriknown signal shapes but at a significant computational cost. Currentimplementations 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 oflightcurve observations, and due to non-convexity, they do not guarantee thebest fit at each trial frequency, which can lead to spurious results. In thiswork, we present a non-linear extension of the Lomb-Scargle periodogram toobtain a template-fitting algorithm that is both accurate (globally optimalsolutions are obtained except in pathological cases) and computationallyefficient (scaling as $\mathcal{O}(N_f\log N_f)$ for a given template). Thenon-linear optimization of the template fit at each frequency is recast as apolynomial zero-finding problem, where the coefficients of the polynomial canbe computed efficiently with the non-equispaced fast Fourier transform. We showthat 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 longbase-line time series with $N_{obs} \gtrsim 10^4$ observations. An open-sourceimplementation of the fast template periodogram is available athttps://www.github.com/PrincetonUniversity/FastTemplatePeriodogram.

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