Error analysis of Jacobi derivative estimators for noisy signals

Recent algebraic parametric estimation techniques (see Fliess and Sira-Ramirez, 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.