Digital signal processing
Finite-part integrals and the Euler-Maclaurin expansion
Proceedings of the conference on Approximation and computation : a fetschrift in honor of Walter Gautschi: a fetschrift in honor of Walter Gautschi
New finite difference formulas for numerical differentiation
Journal of Computational and Applied Mathematics
On stable numerical differentiation
Mathematics of Computation
Signals and Systems
Lanczos' generalized derivative for higher orders
Journal of Computational and Applied Mathematics
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Numerical differentiation for the second order derivatives of functions of two variables
Journal of Computational and Applied Mathematics
Letter to the editor: Numerical differentiation for high orders by an integration method
Journal of Computational and Applied Mathematics
Differentiation by integration with Jacobi polynomials
Journal of Computational and Applied Mathematics
Linear time-derivative trackers
Automatica (Journal of IFAC)
A new approach to numerical differentiation and integration
Mathematical and Computer Modelling: An International Journal
Convergence rate of the causal jacobi derivative estimator
Proceedings of the 7th international conference on Curves and Surfaces
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Recent algebraic parametric estimation techniques (see Fliess and Sira-Ramírez, ESAIM Control Optim Calc Variat 9:151---168, 2003, 2008) led to point-wise derivative estimates by using only the iterated integral of a noisy observation signal (see Mboup et al. 2007, Numer Algorithms 50(4):439---467, 2009). In this paper, we extend such differentiation methods by providing a larger choice of parameters in these integrals: they can be reals. For this, the extension is done via a truncated Jacobi orthogonal series expansion. Then, the noise error contribution of these derivative estimations is investigated: after proving the existence of such integral with a stochastic process noise, their statistical properties (mean value, variance and covariance) are analyzed. In particular, the following important results are obtained: the bias error term, due to the truncation, can be reduced by tuning the parameters, such estimators can cope with a large class of noises for which the mean and covariance are polynomials in time (with degree smaller than the order of derivative to be estimated), the variance of the noise error is shown to be smaller in the case of negative real parameters than it was in Mboup et al. (2007, Numer Algorithms 50(4):439---467, 2009) for integer values. Consequently, these derivative estimations can be improved by tuning the parameters according to the here obtained knowledge of the parameters' influence on the error bounds.