From the Harmonic Oscillator to Time-Frequency Analysis of Chirp Signals

TL;DR

This paper introduces a new framework linking harmonic oscillator dynamics to time-frequency analysis of chirp signals, with implications for quantum mechanics, classical phenomena, and signal processing.

Contribution

It develops a contact transformation approach to analyze harmonic dynamics and introduces the 'Mixed Fourier Transform' for harmonizing chirp signals.

Findings

New model for anomalous and normal diffusion

Introduction of the 'Mixed Fourier Transform' concept

Implications for quantum and classical systems

Abstract

This paper presents a novel approach to understanding the role of harmonic dynamics and gaining a deeper appreciation for its impact within and outside of quantum mechanics. This includes consequences of harmonic dynamics and the uncertainty principle for anomalous diffusion and for the time-frequency analysis of chirp signals. In this approach, we consider a contact transformation to view a system of canonical variables with coordinate $x$ and momentum $p_x$ in the context of a new system of "generalized" coordinates and momentum. This new system is first studied in the context of non-relativistic quantum mechanics. The classical analog is then explored by use of the Poisson bracket equation. From this, new implications are demonstrated in classical phenomena. One is for a new model of Anomalous and Normal Diffusion. In another, we introduce the concept of the "Mixed Fourier Transform" which explores a new Gaussian Fourier Transform kernel in terms of the generalized variables. This has the ultimate objective of "harmonizing" chirp signals or producing a harmonic signal from an otherwise non-harmonic chirp.