Numerical causal derivative estimators from noisy data are essential for real time applications especially for control applications or fluid simulation so as to address the new paradigms in solid modeling and video compression. By using an analytical
point of view due to Lanczos cite{C. Lanczos} to this causal case, we revisit $n^{th}$ order derivative estimators originally introduced within an algebraic framework by Mboup, Fliess and Join in cite{num,num0}. Thanks to a given noise level $delta$ and a well-suitable integration length window, we show that the derivative estimator error can be $mathcal{O}(delta ^{frac{q+1}{n+1+q}})$ where $q$ is the order of truncation of the Jacobi polynomial series expansion used. This so obtained bound helps us to choose the values of our parameter estimators. We show the efficiency of our method on some examples.
In this paper, we give estimators of the frequency, amplitude and phase of a noisy sinusoidal signal with time-varying amplitude by using the algebraic parametric techniques introduced by Fliess and Sira-Ramirez. We apply a similar strategy to estima
te these parameters by using modulating functions method. The convergence of the noise error part due to a large class of noises is studied to show the robustness and the stability of these methods. We also show that the estimators obtained by modulating functions method are robust to large sampling period and to non zero-mean noises.
