Number-theoretic approach to optimum velocity decoding given quantized position information

The problem of optimum velocity estimation, given a sequence of outputs from a digital position sensor, is important in many process applications and motion control systems. The sequence obtained when the output of a constant rate system is quantized, and then uniformly sampled, can be represented as a nonhomogeneous spectrum. An analogous sequence finds application in computer graphics in problems such as the digitization of straight lines. In this paper, the number-theoretic framework developed to represent graphic systems is modified for application to velocity sensing. The assumption of close-to-constant velocity often holds true in high-inertia and regulator-type applications. To cater for periods when this assumption is invalid, the velocity estimation algorithms described include an optimum, linear finite-impulse response (FIR) differentiator. Both on-line and off-line (look-up table based) versions of the algorithm are presented. The mean-squared error (MSE) associated with the number-theoretic algorithm is shown to be superior to that of linear filters. This is experimentally demonstrated through using digital signal processor (DSP)-based instrumentation applied to a motion-control-based test-rig. The new filter is also applicable to other areas in which quantized signals are differentiated.