ML Estimation and Detection of Multiple Frequencies Through Periodogram Estimate Refinement

This letter presents a method to detect and estimate multiple frequencies based on the maximum-likelihood principle. The method addresses the three main difficulties in this kind of computation, which are the detection of the number of frequencies, the coarse localization of the cost function's global maximum, and the iterative refinement of an initial estimate. Fundamentally, it consists of first detecting and estimating single frequencies or frequency clusters using the periodogram, and then refining this last estimate through a Newton-type method. This second step is fast because its complexity is independent of the number of samples, once a single fast Fourier transform (FFT) has been computed. These two steps are iteratively repeated until no mode frequency is above a fixed detection threshold. The main advantage of the proposed method is its low complexity, given that its computational burden is just that of a few FFTs in typical scenarios. The method is assessed in a numerical example.