On causal band-limited mean square approximation

TL;DR

This paper investigates the causal approximation of non-bandlimited processes by band-limited ones using sinc series, providing a unique optimal solution that adapts dynamically as a non-time-invariant linear filter.

Contribution

It introduces a novel causal approximation method for irregular processes with a unique optimal solution expressed via sinc series, adaptable in real-time.

Findings

Unique optimal solution for causal approximation established

Approximation expressed as sinc series in time domain

Dynamic adjustment of coefficients enables real-time filtering

Abstract

We study causal dynamic approximation of non-bandlimited processes by band-limited processes such that a part of the historical path of the underlying process is approximated in $L_2$-norm by the trace of a band-limited process. This allows to cover the case of irregular non-smooth processes. We show that this problem has an unique optimal solution. The approximating band-limited process is obtained in time domain in a form of sinc series. To accommodate the current flow of observations, the coefficients of this series and the related band-limited process have to be changed dynamically. This can be interpreted as a causal and linear filter that is not time invariant.