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

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.