Numerical differentiation with annihilators in noisy environment

Abstract

Numerical differentiation in noisy environment is revised through an algebraic approach. For each given order, an explicit formula yielding a pointwise derivative estimation is derived, using elementary differential algebraic operations. These expressions are composed of iterated integrals of the noisy observation signal. We show in particular that the introduction of delayed estimates affords significant improvement. An implementation in terms of a classical finite impulse response (FIR) digital filter is given. Several simulation results are presented.

Synthesis on a class of algebraic differentiators and application to nonlinear observation

“…The objective is to estimate the n th order derivative of x using x ̟ . For this purpose, we apply a class of algebraic differentiators involving Jacobi polynomials, which were introduced in [7,8] using a recent algebraic parametric method (see [13] for other algebraic differentiators).…”

“…where q = N − n ∈ N. It is shown in [8,11] that this differentiator can also be obtained by taking the q + 1 first terms in the Jacobi series expansion of x (n) , i.e. we locally approximate x (n) by a q th order polynomial on [t 0 − T, t 0 ]:…”

“…[3][4][5][6][7][8][9][10][11][12][13]). The obtained so-called algebraic differentiators are divided into two classes: model-based differentiators and model-free differentiators.…”

“…The first model-free differentiator was introduced in [6] by applying the algebraic method to the truncated Taylor series expansion of the signal to differentiate. Then, two model-free differentiators were studied in [7,8], where the so-called Jacobi differentiator is the most used. Moreover, it was shown that the Jacobi differentiator can also be obtained by taking the truncated Jacobi orthogonal series expansion of the signal to differentiate.…”

“…Moreover, it was shown that the Jacobi differentiator can also be obtained by taking the truncated Jacobi orthogonal series expansion of the signal to differentiate. Then, it was significantly improved by admitting a time-delay [7,8].…”