Refreshing idea on Fourier analysis

TL;DR

This paper challenges the common belief that Fourier analysis limitations are due to fundamental physical or mathematical constraints, showing instead they stem from numerical methods and proposing a hybrid approach for improved analysis.

Contribution

It reveals the true origin of Fourier analysis resolution limits as numerical rather than theoretical, and introduces a hybrid Laplace-Fourier transform method for better time-frequency analysis.

Findings

Numerical methods cause the perceived limits in Fourier analysis.

Replacing Fourier integrals with complex integrals offers new analysis perspectives.

Hybrid Laplace-Fourier transform approach enhances time-frequency analysis.

Abstract

The "theoretical limit of time-frequency resolution in Fourier analysis" is thought to originate in certain mathematical and/or physical limitations. This, however, is not true. The actual origin arises from the numerical (technical) method deployed to reduce computation time. In addition, there is a gap between the theoretical equation for Fourier analysis and its numerical implementation. Knowing the facts brings us practical benefits. In this case, these related to boundary conditions, and complex integrals. For example, replacing a Fourier integral with a complex integral brings a hybrid method for the Laplace and Fourier transforms, and reveals another perspective on time-frequency analysis. We present such a perspective here with a simple demonstrative analysis.