H∞ optimal approximation for causal spline interpolation

Abstract

In this paper, we give a causal solution to the problem of spline interpolation using H∞ optimal approximation. Generally speaking, spline interpolation requires filtering the whole sampled data, the past and the future, to reconstruct the inter-sample values. This leads to non-causality of the filter, and this becomes a critical issue for real-time applications. Our objective here is to derive a causal system which approximates spline interpolation by H∞ optimization for the filter. The advantage of H∞ optimization is that it can address uncertainty in the input signals to be interpolated in design, and hence the optimized system has robustness property against signal uncertainty. We give a closed-form solution to the H∞ optimization in the case of the cubic splines. For higher-order splines, the optimal filter can be effectively solved by a numerical computation. We also show that the optimal FIR (finite impulse response) filter can be designed by an LMI (linear matrix inequality), which can also be effectively solved numerically. A design example is presented to illustrate the result.