2026/07/17 by P. Argoul, J. Taillard, F. Argoul
Physics and Astronomy · #physics.bio-ph
This paper revisits the continuous wavelet transform framework by establishing a rigorous physical and dimensional formulation of the Cauchy-Paul mother wavelet, tailored specifically for intermittent, non-sinusoidal electrophysiological oscillations. Departing from conventional, purely mathematical definitions, we introduce a characteristic time scale τ into the frequency-domain formulation of the mother wavelet. This parameter ensures strict dimensional consistency by maintaining dimensionless functional arguments, thereby confining the physical dimension solely to the multiplicative normalization constant under both L1(ℝ) and L2(ℝ) norms. A sharp dimensional and structural analysis of the resulting wavelet is conducted. We demonstrate that the spectral asymmetry inherent to the Cauchy-Paul wavelet dictates a strict mathematical hierarchy between three alternative reference frequencies: the peak (L^∞), the centroid (L1), and the energy-weighted (L2) frequencies. Each frequency definition yields distinct quality factors (Q) and time-bandwidth characteristics that govern the time-frequency localization trade-off. To track multi-component EEG sleep pattern signals, a phase-based algebraic estimator is deployed alongside an advanced ridge-extraction method. The robust tracking performance and morphological adaptability of the proposed Cauchy-Paul framework are first numerically validated on synthetic transients, harmonics, and chirps, and subsequently applied to real, non-stationary EEG recordings to successfully isolate and decipher the non-sinusoidal signatures of sleep spindles.