A new averaging-extrapolation method for quasi-periodic frequency refinement

Abstract

In a recent work, we presented an averaging-extrapolation approach for the numerical computation of frequencies and amplitudes of a discrete-time quasi-periodic signal. This approach assumes analyticity of the signal and a Diophantine frequency vector. Given an approximation to one of the frequencies and a sufficiently large number of iterates of the signal, the main outcome of the method is a refined approximation to the frequency. The crucial aspect of this method consists in building an appropriate complex signal which, by the geometrical implications of its construction, is referred to as an unfolded signal. The projection of the unfolded signal on the unit circle defines a quasi-periodic signal of the circle whose rotation frequency is the target frequency. This allows to refine the target frequency by adapting a previously developed method for computing, with great accuracy, Diophantine rotation numbers of analytic circle maps. Both the unfolding and refinement processes require computing appropriate weighted averages of the iterates and performing Richardson’s extrapolation. In the present work, we reformulate this averaging-extrapolation approach to frequency refinement. This leads to a simpler method that completely avoids the unfolding process but that is capable of producing accurate values for frequencies and amplitudes at a reasonable computational cost, which is mostly independent of the number of basic frequencies of the signal.