Chromatic Derivatives and Approximations in Practice—Part II: Nonuniform Sampling, Zero-Crossings Reconstruction, and Denoising

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. In the first part of this paper, entitled “Chromatic Derivatives and Approximations in Practice–Part I: A General Framework,” we have derived a collection of formulas and theorems which we use in this paper to demonstrate practical applications of chromatic derivatives and approximations. We present four case studies: a highly accurate (better than 170 dB) reconstruction of a signal from its nonuniformly spaced samples; a highly accurate method for obtaining timing information, such as timing of the zeros of a signal as well as timing of the zeros of its first derivative; a method of reconstruction of a sampled speech signal of 64 000 samples using such timing information only, with a reconstruction error of only about 1% of the rms of the original signal; and finally, a denoising algorithm that significantly outperforms the well-known Cadzow's denoising algorithm. The main purpose of these case studies is to illustrate the potential of chromatic derivatives and expansions and motivate DSP engineers to find applications of these novel concepts in their own practice.