On spectral interference of the short-time Fourier transform and its nonlinear variations

TL;DR

This paper analyzes spectral interference in the short-time Fourier transform and its nonlinear variants, revealing conditions for resolution limits, phase behavior, and how reassignment methods can sharpen or distort time-frequency representations.

Contribution

It provides a detailed mathematical analysis of spectral interference phenomena and introduces a generalized synchrosqueezing framework to mitigate interference effects.

Findings

Identifies a critical frequency gap for resolving two components.

Describes bifurcation and bubble formation in spectrogram ridges.

Explains how reassignment can both sharpen and distort TFRs depending on parameters.

Abstract

Spectral interference, the frequency counterpart of the beating phenomenon in the time domain, can severely distort time-frequency representations (TFRs) in physical applications. We study this phenomenon for the short-time Fourier transform (STFT) with a Gaussian window and for nonlinear refinements based on the reassignment method, with an emphasis on the synchrosqueezing transform (SST). Working with a two-component harmonic model, we quantify when STFT can (and cannot) resolve two nearby frequencies: a sharp transition occurs at a critical gap that scales inversely to kernel bandwidth and depends explicitly on the amplitude ratio. Below this threshold, the spectrogram ridges undergo bifurcation and form repeating time-frequency bubbles, which we describe asymptotically and, in the balanced-amplitude case, approximate closely by ellipses. We then analyze the STFT phase, showing a canonical winding behavior, and relate the complex-valued SST reassignment map to a holomorphic structure via the Bargmann transform. In the two-component setting the reassignment rule admits an explicit Mobius-geometry description, sending frequency lines to circular arcs in the complex plane. Finally, viewing SST and reassignment through a measure mapping perspective, we derive small-kernel asymptotics that explain when reassignment sharpens energy and when it produces distorted or misleading TFRs; we also introduce a generalized synchrosqueezing framework that isolates the role of STFT weighting and clarifies how alternative choices can mitigate interference in certain regimes.