On the Practical Use of Blaschke Decomposition in Nonstationary Signal Analysis

TL;DR

This work analyzes the practical limitations of Phase Dynamics Unwinding (PDU), a Blaschke-decomposition-based method for nonstationary signals, notably poor amplitude modulation capture and winding-induced mode-mixing in biomedical time series. It introduces windowed PDU with divide-and-conquer tapering and a cumsum-based anti-derivative technique to enhance local AM/trend modeling and suppress high-frequency winding effects, enabling more robust decomposition and instantaneous frequency estimation. The approach is validated on both simulated adaptive harmonic model signals and real-world photoplethysmography data, showing superior AM recovery, phase accuracy, and clearer separation of components (e.g., cardiac vs. respiratory) compared with vanilla PDU; code is provided for reproducibility. By connecting Blaschke factorization with multiscale/TF analysis, the paper offers a practical, efficient framework for biomedical signal decomposition that improves interpretability and potential downstream analyses.

Abstract

The Blaschke decomposition-based algorithm, {\em Phase Dynamics Unwinding} (PDU), possesses several attractive theoretical properties, including fast convergence, effective decomposition, and multiscale analysis. However, its application to real-world signal decomposition tasks encounters notable challenges. In this work, we propose two techniques, divide-and-conquer via tapering and cumulative summation (cumsum), to handle complex trends and amplitude modulations and the mode-mixing caused by winding. The resulting method, termed {\em windowed PDU}, enhances PDU's performance in practical decomposition tasks. We validate our approach through both simulated and real-world signals, demonstrating its effectiveness across diverse scenarios.

Figures (10)

• Figure 1: Examples of the $f_\alpha(t):=\frac{e^{i2\pi t}-\alpha}{1-\overline{\alpha}e^{i2\pi t}}=e^{i\phi_\alpha(t)}$, where $t\in [0, 1)$, $\alpha=r_\alpha e^{i2\pi\theta_\alpha}\in \mathbb{D}$, $r_\alpha\in [0,1)$ and $\theta_\alpha\in [0,1)$. In each subplot, the red curve represents the real part of the signal, while the blue curve depicts the corresponding instantaneous frequency. The blue cross marks $\theta_\alpha\in [0,1)$, the phase of the associated root. In the right subpanel, the blue dot denotes the root located inside the unit disk, and the multiplicity of this root is indicated next to the blue cross. (a) $f_{\alpha_1}$, where $\alpha_1=\frac{1+i}{5\sqrt{2}}$, (b) $f_{\alpha_1}^4$, (c) $f_{\alpha_2}$, where $\alpha_2=\frac{-14+7i}{10\sqrt{5}}$, (d) $f_{\alpha_2}^4$, (e) $f_{\alpha_3}$, where $\alpha_3=\frac{9-18i}{10\sqrt{5}}$, (f) $f_{\alpha_3}^4$, (g) $f_{\alpha_2}^4f_{\alpha_3}^4$, and (h) $f_{\alpha_2}^6f_{\alpha_3}^5$.
• Figure 2: The left panel shows $g(e^{i\theta})$ and the right panel shows $h(e^{i\theta})$, where $\theta\in [0,2\pi)$. The colorbar indicates $\theta$.
• Figure 3: Illustration of the window effect in the proposed windowed PDU algorithm. Consider $f(t)=f_\alpha^5(t)$, where $f_\alpha(t):=\frac{e^{i2\pi t}-\alpha}{1-\overline{\alpha}e^{i2\pi t}}$, $t\in [0, 1)$ and $\alpha=\frac{9-18i}{10\sqrt{5}}$. The top panel shows the real part of $f$, with the gray vertical lines indicates the truncation range. The window is defined by $B=0.045$ and $T=B/4$. The resulting truncated signal, denoted as $f_w$, is shown in the middle panel. In the bottom left panel, the magnitude of the holomorphic function $F_w=P_+f_w$ over $\mathbb{D}$ evaluated by Poisson integration, is displayed. In the bottom middle panel, the phase of the holomorphic function $F_w$ over $\mathbb{D}$ is displayed. In the bottom right panel, the region where $|F_w|<0.005$ is shown in white, while all other regions are shown in red.
• Figure 4: A continuous of Figure \ref{['fig windowed PDU example']} with the same signal and meaning of each panel, while from top to bottom rows the window described in \ref{['window definition w']} are centered at 0.82 with $B=0.055$ and $T=B/8$, $B=0.045$ and $T=B/6$ and $B=0.035$ and $T=B/9$ respectively.
• Figure 5: Left panel: results of PDU; right panel: results of windowed PDU. In each panel, the gray curve shown in the top is the input signal $f\in \mathbb{R}^{\lfloor f_sT\rfloor}$, which is the summation of two IMT functions shown as gray curves below. The decomposed components by PDU and windowed PDU are superimposed as red and blue curved, and the summation of the decomposed components, $\Re(\tilde{f}_1+\tilde{f}_2)$, is superimposed on the top black curve. The difference between $f-\Re(\tilde{f}_1+\tilde{f}_2)$ is shown in the bottom magenta curve. To enhance the visualization, only the middle 2.5 seconds are shown.