Chromatic Derivatives and Approximations in Practice—Part I: A General Framework

Abstract

Chromatic derivatives are special, numerically robust differential operators that preserve spectral features of a signal; the associated chromatic approximations accurately capture local features of a signal. For this reason they allow digital processing of continuous time signals often superior to processing of discrete samples of such signals. We introduce a new concept of “matched filter” chromatic approximations, where the underlying basis functions are chosen to match the spectral profile of the signals being approximated. We then derive a collection of formulas and theorems that form a general framework for practical applications of chromatic derivatives and approximations. In the second part of this paper, we use such a general framework in several case studies of such applications that aim to illustrate how chromatic derivatives and approximations can be used in signal processing with an intention of motivating DSP engineers to find applications of these novel concepts in their own practice.