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

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.