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

TL;DR

This work investigates spectral interference in time-frequency analysis, focusing on STFT with a Gaussian window and nonlinear refinements like SST for a two-component harmonic model. By deriving large- and small-gap regimes in terms of the frequency separation $\Delta$ and window bandwidth $\sigma$, it characterizes when STFT can resolve two nearby frequencies and how SST reassignment shapes the resulting TFR through a holomorphic/ Mobius-geometry framework. A key contribution is the explicit description of ellipse-shaped ridges (bubbles) in STFT for balanced amplitudes and small $\Delta$, plus a measure-mapping view that yields precise asymptotics for generalized SST under different weighting choices. The results show that SST can sharpen or distort energy depending on regime and parameters, and they introduce a generalized synchrosqueezing framework to mitigate interference by isolating STFT weighting, with clear implications for designing robust TF representations in applications with closely spaced components.

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.

Figures (12)

• Figure 1: All with $\sigma = \sqrt{2}, a=1.3$, and $\xi_0=1$. (Top row) Enhanced spectrogram $|V_f^{(h)}|$ with larger gap $\xi_1=2$ and cross sections in blue at constructive and destructive times and $|V_{f_0}^{(h)}(t,\eta)|$ and $|V_{f_1}^{(h)}(t,\eta)|$ in green; (Bottom row) Same as above with smaller gap $\xi_1=1.3$.
• Figure 2: All with $\sigma=\sqrt{2}, a=1$ and $\xi_0=1$. Ridges of the two component signal as defined in \ref{['eqn:ridge_def']}. (Top left) With $\xi_1=2$; (Bottom left) With $\xi_1=1.3$, green dashed curve is the ellipse from Theorem \ref{['thm:ellipse']}; (Right) Corresponding spectrograms.
• Figure 3: All with $\sigma=\sqrt{2}$ and $\xi_0=1$. Ridges of the two component signal as defined in \ref{['eqn:ridge_def']} with varying $a,\Delta$; The bottom three rows show when $\Delta\approx\Delta_{\text{critical, STFT}}(a=1)$ is near the critical point.
• Figure 4: With $\sigma=\sqrt{2}, a=1$, $\xi_0=1$, and $\xi_1$ starting at 1.2 and slowly changing to 1.3. (Top) STFT and true IFs overlayed in red dashed line; (Bottom) Ridges of the two component signal as defined in \ref{['eqn:ridge_def']}. Notice the bifurcation point becomes discontinuous as $\xi_1$ changes.
• Figure 5: The same signal as described in the bottom row of Figure \ref{['fig:diff_Delta_STFT']} (with $\sigma=\sqrt{2}, a=1.3$, $\xi_0=1$, and $\xi_1=1.3$); (Top left) The amplitude of the STFT $|V|$; (Top right) The phase $\phi(t,\eta)$ (in radians), showing a phase transition at both constructive and destructive times; (Bottom) The product $|V|\times\phi(t,\eta)$ where the phase no longer winds around zero points of $V$.

Theorems & Definitions (52)

• Definition 1: Adaptive Harmonic Model
• Proposition 1
• Definition 2: Constructive and destructive times
• Lemma 1: STFT at constructive times
• Remark 1
• Remark 2
• Lemma 2
• Lemma 3: STFT at destructive times
• Theorem 1
• proof : Proof of Theorem \ref{['thm:ellipse']}