Causal Cubic Splines: Formulations, Interpolation Properties and Implementations

The paper presents two formulations of causal cubic splines with equidistant knots. Both are based on a causal direct B-spline filter with parallel or cascade implementation. In either implementation, the causal part of the impulse response is realized with an efficient infinite-impulse-response (IIR) structure, while only the anticausal part is approximated with a finite-order finite-impulse-response (FIR) filter. Resulting cubic coefficients are computed from the causal B-spline coefficients by using a third-order output FIR filter with either single-input multiple-output (SIMO) or multiple-input multiple-output (MIMO) structure, depending on the chosen formulation of the cubic spline. The paper demonstrates and proves that the properties of the resulting causal splines are quite different, whether they are based on a more popular B-spline formulation, or a bit neglected tridiagonal matrix formulation. It is shown that the proposed low-complexity but accurate causal interpolators can be realized for many practical applications with the delay of only a few samples.