OPTIMAL DIFFERENTIATION BASED ON STOCHASTIC SIGNAL MODELS

The problem of estimating the time derivative of a signal from sampled measurements is addressed. The measurements may be corrupted by colored 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.