Optimal Differentiation based on Stochastic Signal Models

B Carlsson, Anders Ahlén and Mikael Sternad

IEEE Transactions on Signal Processing, vol SP-39, pp 341-353, February 1991. © 1991 IEEE.

Also: Internal Report UPTEC 89031R, Dept. of Technology, Uppsala University.

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Outline:

A filter which estimates the derivative of a signal is designed as a compromise between good differentiation and a low noise sensitivity. The paper discusses the systematic model-bases design of realizable differentiating filters, based on continuous-time or discrete-time models describing the spectral properties of signals and noise.

Abstract:

The problem of estimating the time derivative of a signal from sampled measurements is addressed. The measurements may be corrupted by coloured noise. A key idea is to use stochastic models of the signal to be differentiated and of the measurement noise. Two approaches are suggested. The first is based on a continuous-time stochastic process as model of the signal. The second approach uses a discrete-time ARMA model of the signal and a discrete-time approximation of the derivative operator. The introduction of this approximation normally causes a small performance degradation, compared to the first approach. There exists an optimal (signal dependent) derivative approximation, for which the performance degradation vanishes.

Digital differentiators are presented in a shift operator polynomial form. They minimize the mean square estimation error. In both approaches, they are calculated from a linear polynomial equation and a polynomial spectral factorization. (The first approach also requires sampling of the continuous-time model.) Estimators can be designed for prediction, filtering and smoothing problems. Unstable signal and noise models can be handled. The three obstacles to perfect differentiation, namely a finite smoothing lag, measurement noise and aliasing effects due to sampling, are discussed.

Related publications:

Related paper in IEEE Trans. SP 1992.
Paper in IEEE Trans. ASSP 1989, on the design of deconvolution estimators and differentiators.
Paper in Automatica 1993, which in Section 5 outlines a robustification against model errors.
Book chapter (Academic Press 1994) on the polynomial approach to Wiener filter design.