Accurate derivative estimation from noisy data: a state-space approach

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Abstract
Numerical differentiation of discrete observations of a noisy signal is formulated as an optimal state estimation problem. A state vector is defined composed of the signal and its derivatives and a state-space representation is derived from the assumption of a band-limited signal. Under the hypothesis of additive gaussian measurement noise a fixed-lag Kalman smoother is then applied to obtain the optimal state estimate. It is shown that the main advantage of the state-space approach is that the maximum precision theoretically obtainable for the state estimate is sensitive more to the model noise than to the measurement noise, so that the inferior limit of the error covariance matrix can be made small at will provided that an adequate signal model is available. To this purpose it is shown that it is possible to obtain any prescribed accuracy on the first components of the state vector by increasing the model order. Numerical results refer to a signal of interest in ‘human motion analysis’. They are derived in a simulation context in order to obtain a precise evaluation of the smoother performance.