Algebraic parameter estimation of a biased sinusoidal waveform signal from noisy data

Abstract

To cite this version:Rosane Ushirobira, Wilfrid Perruquetti, Mamadou Mboup, Michel Fliess. Algebraic parameter estimation of a biased sinusoidal waveform signal from noisy data. Abstract: The amplitude, frequency and phase of a biased and noisy sum of two complex exponential sinusoidal signals are estimated via new algebraic techniques providing a robust estimation within a fraction of the signal period. The methods that are popular today do not seem able to achieve such performances. The efficiency of our approach is illustrated by several computer simulations.

Towards an effective study of the algebraic parameter estimation problem

“…In the algebraic parameter estimation problem, "estimating" means, in the noiseless case, i.e., when = 0, exactly and explicitly determine the parameters Θ in terms of y and of its integrals. Ushirobira et al (2012Ushirobira et al ( , 2013 and the references therein. The use of integrals of y is made to filter the effects of the noise .…”

“…Note also that the coefficients of the annihilators Π j 's cannot depend on the parameters to be estimated till they are unknown. For more details, see Fliess et al (2003) Ushirobira et al (2012Ushirobira et al ( , 2013. Remark 3.…”

“…In this paper, we follow the algebraic approach developed by Fliess, Sira-Ramirez, Mboup, and their coauthors. See Fliess et al (2003); SiraRamírez et al (2014); Mboup et al (2009) ;Ushirobira et al (2012Ushirobira et al ( , 2013, and the references therein. In the absence of noise, i.e., = 0, the algebraic approach provides explicit and exact formulas for the constant parameters θ in terms of integrals of the observed signal y.…”

“…, polynomials (Abramowitz et al (1964); Chyzak et al (2005)). Note that the algebraic parameter estimation problem was studied for Taylor expansion series (Mboup et al (2009)) and initiated for Fourier expansions, i.e., expansions into the orthogonal basis {e i n θ } n∈Z of L 2 (T), where T = {z ∈ C | |z| = 1} is the unit torus of C (Ushirobira et al (2012(Ushirobira et al ( , 2013). …”

“…In the algebraic approach to parameter estimation, the dynamics of the signal x is defined by a time-invariant ordinary differential (OD) system (Fliess et al (2003); Sira-Ramírez et al (2014); Mboup et al (2009); Ushirobira et al (2012Ushirobira et al ( , 2013). In Fliess et al (2010), there is a remark about the possibility to extend this framework to ODEs with polynomial coefficients (see Remark 1 in page 133).…”