Frequency Estimation And Generalized Lomb-Scargle Periodograms

Using Bayesian probability theory we demonstrate that the Lomb-Scargle periodogram may be generalized in a straightforward manner to nonuniformly nonsimultaneously sampled quadrature data when the sinusoid has arbitrary amplitude modulation. This generalized Lomb-Scargle periodogram is the sufficient statistic for single frequency estimation in a wide class of problems ranging from stationary frequency estimation in real uniformly sampled data, to frequency estimation for a single sinusoid having exponential, Gaussian, or arbitrary amplitude modulation. In addition we define the bandwidth of a nonuniformly sampled data set and show that the phenomenon of aliases exists in both uniformly and nonuniformly sampled data and that the phenomenon has the same cause in both types of data. Finally, we show that nonuniform sampling does not affect the accuracy of the frequency estimates; although it may affect the accuracy of the amplitude estimates.