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 a model of the signal. The second uses a discrete-time ARMA model of the signal and a discrete-time approximation of the derivative operator. Digital differentiators are presented in a shift operator polynomial form. They minimize the mean-square estimation error, and are calculated from a linear polynomial equation and a polynomial spectral factorization. The three obstacles to perfect differentiation, namely a finite smoothing lag, measurement noise, and aliasing effects due to sampling, are discussed