Discrete Scaling Based on Operator Theory

TL;DR

This paper introduces a novel discrete signal scaling method based on hyperdifferential operator theory, providing a self-consistent and elegant approach aligned with the discrete Fourier transform.

Contribution

It proposes a new discrete scaling approach grounded in operator theory, improving upon previous interpolation-based methods.

Findings

The new method is fully consistent with the discrete Fourier transform.

It offers a self-consistent and elegant definition of discrete scaling.

The approach enhances the theoretical understanding of signal magnification.

Abstract

Signal scaling is a fundamental operation of practical importance in which a signal is enlarged or shrunk in the coordinate direction(s). Scaling or magnification is not trivial for signals of a discrete variable since the signal values may not fall onto the discrete coordinate points. One approach is to consider the discretely-spaced values as the samples of a signal of a real variable, find that signal by interpolation, scale it, and then re-sample. However, this approach comes with complications of interpretation. We review a previously proposed alternative and more elegant approach, and then propose a new approach based on hyperdifferential operator theory that we find most satisfactory in terms of obtaining a self-consistent, pure, and elegant definition of discrete scaling that is fully consistent with the theory of the discrete Fourier transform.