[Paper Review] Extracting Phases from Aperiodic Signals

Alexander Kraskov, Thomas Kreuz+4 more|arXiv (Cornell University)|Sep 15, 2004

Magnetic Properties and Applications1 references3 citations

TL;DR

This paper argues that phase extraction from aperiodic signals is fundamentally ambiguous due to non-monotonic phase evolution and topological pathologies in phase space. It proposes decomposing signals into components with positive, non-constant phase velocities—generalizing Fourier decomposition—to achieve robust, physically meaningful phase definitions, illustrated via a biased random walk phase model.

ABSTRACT

We demonstrate by means of a simple example that the arbitrariness of defining a phase from an aperiodic signal is not just an academic problem, but is more serious and fundamental. Decomposition of the signal into components with positive phase velocities is proposed as an old solution to this new problem.

Motivation & Objective

• To demonstrate that phase definition for aperiodic signals is fundamentally ambiguous, not just practically difficult.
• To show that standard methods—geometric, Hilbert transform, and cosine-based—fail to provide unique, robust phase definitions in the presence of phase reversals or self-intersecting trajectories.
• To argue that the lack of a unique phase definition undermines the reliability of phase-based analysis in nonlinear and non-stationary systems.
• To propose a new framework for phase extraction based on decomposition into components with positive phase velocities, generalizing Fourier analysis.
• To illustrate the feasibility and utility of this approach using a model of a cosine signal driven by a biased random walk in phase.

Proposed method

• The authors use a model signal defined as $ x(t) = \cos(\phi(t)) $, where $ \phi(t) $ undergoes a biased random walk with $ d\phi/dt = \omega + \eta(t) $, $ \eta(t) $ being white noise.
• They analyze the signal using standard phase estimation techniques: delay embedding (phase portrait), Hilbert transform, and spectral analysis (Welch window).
• They identify that self-intersecting trajectories and zero-crossings in the Hilbert transform lead to non-unique and unstable phase definitions.
• They propose decomposing signals into components with positive phase velocities, inspired by Kepler’s replacement of epicycles with elliptic orbits.
• The method avoids reliance on geometric centers or analytic signal properties, instead focusing on ensuring monotonic phase evolution in each component.
• The approach is not algorithmic in a general sense but is conceptually grounded in selecting physically meaningful, non-reversing phase components.

Experimental results

Research questions

• RQ1 Can a unique and robust phase be defined for aperiodic signals when standard methods fail due to phase reversals or self-intersecting trajectories?
• RQ2 Why do geometric, Hilbert transform, and cosine-based phase definitions fail to provide consistent results in the presence of non-monotonic phase evolution?
• RQ3 Is there a general framework for phase extraction that avoids the ambiguities inherent in traditional methods?
• RQ4 Can signal decomposition into components with positive phase velocities yield more physically meaningful phase representations than existing approaches?
• RQ5 What are the implications of phase ambiguity for signal analysis in nonlinear dynamics and time series processing?

Key findings

• The geometric phase definition fails when the trajectory self-intersects, as different central points yield non-equivalent phases with different mean angular velocities.
• The Hilbert transform method becomes unstable when the signal passes near the origin, leading to extreme sensitivity to noise even without exact zero-crossings.
• Spectral analysis shows a broad peak, indicating no well-defined frequency, which invalidates phase estimation based on dominant frequency.
• Despite the lack of a unique phase via standard methods, the underlying model signal has a clear, physically meaningful phase defined by the biased random walk.
• The authors demonstrate that phase ambiguity is not a numerical artifact but a fundamental limitation of phase definitions in non-periodic, non-monotonic systems.
• Decomposing signals into components with positive phase velocities offers a viable, conceptually sound alternative to traditional phase extraction, generalizing Fourier decomposition.