Fourier-Based Spectral Analysis with Adaptive Resolution

TL;DR

This paper introduces a generalized Fourier Transform that allows for adaptive resolution, overcoming the traditional static resolution limitation while maintaining simplicity and computational efficiency.

Contribution

It presents a novel generalization of Fourier Transform enabling adaptive resolution, which retains backward compatibility and minimal computational overhead.

Findings

Demonstrates Fourier Transform can be adapted for variable resolution

Maintains simplicity and computational efficiency of classical Fourier analysis

Overcomes the static resolution limitation of traditional Fourier methods

Abstract

Despite being the most popular methods of data analysis, Fourier-based techniques suffer from the problem of static resolution that is currently believed to be a fundamental limitation of the Fourier Transform. Although alternative solutions overcome this limitation, none provide the simplicity, versatility, and convenience of the Fourier analysis. The lack of convenience often prevents these alternatives from replacing classical spectral methods - even in applications that suffer from the limitation of static resolution. This work demonstrates that, contrary to the generally accepted belief, the Fourier Transform can be generalized to the case of adaptive resolution. The generalized transform provides backward compatibility with classical spectral techniques and introduces minimal computational overhead.