重新审视柯西 - 保罗小波变换：间歇性非正弦振荡的框架
Cauchy-Paul wavelet transforms revisited: A framework for intermittent non-sinusoidal oscillations
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中文总结 AI 辅助
该研究重新审视柯西 - 保罗小波变换，为间歇性非正弦振荡建立新框架。通过引入特征时间尺度确保维度一致，分析其频谱特性，并用基于相位的代数估计器等方法跟踪脑电图信号，在合成及真实记录上验证，成功分离解码睡眠纺锤波非正弦特征。
中文摘要 AI 辅助
本文通过为间歇性非正弦电生理振荡建立严格的柯西 - 保罗母小波物理和维度公式，重新审视连续小波变换框架。与传统纯数学定义不同，在母小波频域公式中引入特征时间尺度τ，确保维度一致性。对所得小波进行维度和结构分析，表明其频谱不对称决定了三个参考频率间的数学层次。还部署了基于相位的代数估计器和先进的脊提取方法来跟踪多分量脑电图睡眠模式信号，先在合成信号上验证，后应用于真实脑电图记录以分离和解码睡眠纺锤波的非正弦特征。
英文摘要
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 $L^1(\mathbb{R})$ and $L^2(\mathbb{R})$ 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 ($L^\infty$), the centroid ($L^1$), and the energy-weighted ($L^2$) 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.
