Cubic-spline interpolation. 1

The need to interpolate is widespread, and the approaches to interpolation are just as widely varied. For example, sampling a signal via a sample and-hold circuit at uniform, T-second intervals produces an output signal that is a piecewise-constant (or zero-order) interpolation of the signal samples. Similarly, a digital-to-analog (D/A) converter that incorporates no further post-filtering produces an output signal that is (ideally) piecewise-constant. One very effective, well-behaved, computationally efficient interpolator is the cubic spline. The approach is to fit cubic polynomials to adjacent pairs of points and choose the values of the two remaining parameters associated with each polynomial such that the polynomials covering adjacent intervals agree with one another in both slope and curvature at their common endpoint. The piecewise-cubic interpolating function g(x) that results is twice continuously differentiable. We develop the basic algorithm for cubic-spline interpolation.