Unevenly-sampled signals: a general formalism for the Lomb-Scargle periodogram

¹ Chip Computers Consulting srl, Viale Don L. Sturzo 82, S. Liberale di Marcon, 30020 Venice, Italy e-mail: robertovio@tin.it,
² ESO, Karl Schwarzschild strasse 2, 85748 Garching, Germany
³ INAF - Osservatorio Astronomico di Trieste, via Tiepolo 11, 34143 Trieste, Italy e-mail: pandrean@eso.org
⁴ ESO, Karl Schwarzschild strasse 2, 85748 Garching, Germany e-mail: abiggs@eso.org

Received: 15 January 2010
Accepted: 12 May 2010

Abstract

The periodogram is a popular tool that tests whether a signal consists only of noise or if it also includes other components. The main issue of this method is to define a critical detection threshold that allows identification of a component other than noise, when a peak in the periodogram exceeds it. In the case of signals sampled on a regular time grid, determination of such a threshold is relatively simple. When the sampling is uneven, however, things are more complicated. The most popular solution in this case is to use the Lomb-Scargle periodogram, but this method can be used only when the noise is the realization of a zero-mean, white (i.e. flat-spectrum) random process. In this paper, we present a general formalism based on matrix algebra, which permits analysis of the statistical properties of a periodogram independently of the characteristics of noise (e.g. colored and/or non-stationary), as well as the characteristics of sampling.