Digital/analog hybrid implementation of cardinal spline interpolation
A digital/analog hybrid system is presented which implements the cardinal polynomial spline interpolation of arbitrary degree. Based on the fact that the (m - 1)st derivative of a spline of degree m - 1 is a staircase function, this system generates a cardinal spline of degree m - 1 by m - 1 cascaded integrators with a staircase function input. A given sequence of sampled values are transformed by a digital filter into coefficients for the B-spline representation of the spline interpolating the sampled values. The values of its (m - 1)st derivative with respect to time are computed by the recurrence formula interpreting differentiation of the spline as difference of the coefficients. Then a digital-to-analog converter generates a staircase function representing the (m - 1)st derivative, which is integrated by a cascade of m - 1 analog integrators to make the expected spline. In order to cope with the offset errors involved in the integrators, a dynamical sampled-data control is attached. An analog-to-digital converter is employed to sample the output of the cascaded integrators. Target state of the cascaded integrators at each sampling instance is computed from the coefficients for the B-spline representation. The state error between the target and the estimated is compensated by feeding back a weighted sum of the state error to the staircase input.
spline interpolation digital-to-analog conversion sampled-data feedback control