Optimal Differentiation based on Stochastic Signal Models
IEEE Transactions on Signal Processing,
vol SP-39, pp 341-353, February 1991. © 1991 IEEE.
Also: Internal Report UPTEC 89031R, Dept. of Technology,
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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
properties of signals and noise.
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.
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:
in IEEE Trans. SP 1992.
in IEEE Trans. ASSP 1989, on the design of deconvolution
estimators and differentiators.
in Automatica 1993, which in Section 5 outlines a robustification
against model errors.
(Academic Press 1994) on the polynomial approach to Wiener
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