Discrete splines and spline filters

An equation is derived for the Z transform of discrete polynomial splines for the general case of nonuniform knots. Two filter structures are provided for the computation and analysis of discrete splines, one for the one-sided factorial function representation and one for the B-spline representation. The filter inputs are the coefficient sequence and the corresponding knot set and the outputs are the discrete spline and its differences. The filter structures supply the input-output relations that can be used to analyze the effects of different patterns of knot nonuniformities given the coefficients, or vice versa. Digital filters with discrete spline unit-sample responses are analyzed. It is shown that filtering with a discrete spline filter can be implemented in two stages: the first stage is an MA filter with as many nodes as there are knots. and the second stage is an AR filter which performs successive summations. Because only the first stage involves multiplications, filters with large ratios of (length of unit-sample response) to (number of knots) can be implemented very efficiently