General method for sinusoidal frequencies estimation using ARMA algorithms with nonlinear prediction error transformation

Abstract

A new general approach to estimating the frequencies of sinusoidal signals corrupted by an additive non-Gaussian noise is presented. The mixture of sinusoids and noise is modeled by an ARMA model with non-Gaussian model noise. A class of ARMA recursive algorithms with nonlinear prediction error transformation is proposed for frequencies estimation. For a given probability density function of the model noise, known except of the scale parameter, the presented method enables the derivation of the algorithms ensuring the fastest convergence of the covariance error matrix to the asymptotic one. The robust version of the algorithms is also discussed. The performance of the ARMA nonlinear algorithms is illustrated by simulation results.

Real time prediction of irregular periodic time series data

“…with 𝜃 1 = 𝜙 1 and 𝜃 2 = 𝜙 2 . See Zhang et al (2018) and Platonov et al (1992) for a more detailed discussion on the frequency estimation problem. It is natural to expect that r should be close to one if the data exhibit a sine or cosine function pattern.…”

“…The frequency estimation problem based on the time-constant ARMA model is discussed in Zhang, Ng, and Na (2018), B. Chen and Gel (2010), Stoica, Friedlander, andSoderstrom (1987), andPlatonov, Gajo, andSzabatin (1992).…”

Computational explosion in the frequency estimation of sinusoidal data

“…Can the ARMA model be used to handle the sinusoidal pattern? The answer is positive, see Platonod et al (1992). The crucial point is that sine/cosine function is the solution to a recursive relationship.…”

“…The feasibility of analyzing sinusoidal data via ARMA model is studied in Platonod et al (1992). It is illustrated that sine/cosine model perturbed with error terms can be rewritten as an ARMA model with unit complex characteristic roots.…”

“…In addition, the combined linear trend and sinusoidal pattern can be described via the multiplicity of complex characteristic roots. In the simulation study, Platonod et al (1992) only consider examples with small sample size n = 100. As noted in p.292 of Ozaki (2012), most existing numerical nonlinear optimization procedure of ARMA model estimation face the risk of "computational explosion" due to the violation of invertibility condition of MA parameters during the iteration.…”