Refreshing idea on Fourier analysis

TL;DR

The paper argues that the conventional time–frequency resolution limit in Fourier analysis arises from numerical methods and boundary conditions rather than fundamental physics, citing the classical bound $|\Delta f \Delta t| \ge h/2\pi$ as a misattributed origin. It proposes a generalized complex-mode expansion with $S(t)=\sum_{m=1}^M e^{H_m(t)}$ and $H_m'(t)=2\pi i f_m(t)+\lambda_m(t)$, in which Fourier is recovered as a special case, together with a locally linearized solution around $t_k$ and a non-standard LPC-based estimation of $H_m'(t_k)$. The instantaneous spectrum is derived via a Laplace–Fourier hybrid integral, yielding $F_{\rm disc}(f,t_k)=\sum_m|\frac{c_m(t_k)}{\lambda_m(t_k)+2\pi i(f_m(t_k)-f)}|$, which provides per-term spectra and a power spectrum $F_{\rm disc}^2$ while addressing boundary-condition artifacts. Demonstrations on frequency-modulated signals show that the method enables less-than-a-cycle time–frequency analysis and spectrograms that reveal instantaneous frequency dynamics, with resolution tied to data granularity rather than the classical windowing constraint. Overall, the work offers a new mathematical perspective on boundary conditions in Fourier analysis and presents a practical framework for high-resolution time–frequency analysis with potential benefits for signal separation and nonlinear-system 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.

Figures (5)

• Figure 1: Periodic boundary condition.
• Figure 2: Proposed linear extrapolation condition.
• Figure 3: Time series for analysis.
• Figure 4: Spectrum of each obtained mode.
• Figure 5: (a) Time series for analysis Eq. (\ref{['eq-fm-ts']}), and (b) obtained spectrogram Eq. (\ref{['eq-fm-ts-f']}). Area of interest is magnified.