Error analysis of a class of derivative estimators for noisy signals and applications

Da-Yan Liu 1, 2
1 NON-A - Non-Asymptotic estimation for online systems
Inria Lille - Nord Europe, CRIStAL - Centre de Recherche en Informatique, Signal et Automatique de Lille (CRIStAL) - UMR 9189
2 SyNeR - Systèmes Non Linéaires et à Retards
CRIStAL - Centre de Recherche en Informatique, Signal et Automatique de Lille (CRIStAL) - UMR 9189
Abstract : This thesis concerns the construction and analysis of robust estimators for the numerical calculations of the derivatives of noisy signals and the parameters of noisy sinusoidal signals. These estimators, originally introduced by Fliess, Sira Ramirez and Mboup, are currently being studied in the Non-A Project-Team of INRIA Lille-Nord Europe. For a class of estimators, we get them by rewriting the Laplace transform of the linear differential equations of analyzed signals in the operational domain. By some simple algebraic manipulations in the ring IR(s)[d/ds] of differential polynomials where the variable is d/ds and their rational coefficients contain the operational variable s, we show that these estimators are non-asymptotic and the obtained numerical estimates are robust for a small number of samples of signals, even in the presence of noise. Moreover, we show that these properties are checked for a large class of type of noises. These estimators expressed in the time domain are written in general through fractions of iterated integrals of the analyzed signals. In the first part of this thesis, we study some families of derivative estimators obtained by these algebraic methods. We show that a class of them can be directly obtained by truncating the Jacobi orthogonal series. This consideration allows us to extend the set of the parameters defining these estimators to IR. Then, we analyze the influence of these extended parameters on the truncation error which produces a time-delay estimation in causal case, on the error due to noises considered as stochastic processes, and finally on the error due to numerical integration methods. Thus, we show how to reduce the time-delay and the noise effect. A validation of this approach is achieved by constructing a non-asymptotic observer of the state variables of a nonlinear system. In the second part of this thesis, by using the algebraic method we construct estimators of the parameters of a noisy sinusoidal signal the amplitude of which varies with time. Moreover, we show that the modulating functions method has a link to the algebraic method. We then study the influence of parameters defining the estimators on the noise error contribution and the numerical integration error. In particular, some error bounds of these errors are given for a class of parameter estimators. Finally, a comparison between these estimators and the classical synchronous detection method is performed so as to demonstrate the effectiveness of our approach on such signals.
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Da-Yan Liu. Error analysis of a class of derivative estimators for noisy signals and applications. Numerical Analysis [math.NA]. Université des Sciences et Technologie de Lille - Lille I, 2011. English. ⟨tel-00634652v2⟩

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