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Time-Varying Spiky Wave-Shape Functions for Non-Stationary Signal Decomposition

Marcelo A. Colominas, Hau-Tieng Wu

arXiv:2608.26285v1eess.SP

TL;DR

Spiky, cycle-varying biomedical waveforms are difficult to model with ANH representations because localized structures require many Fourier coefficients. The paper introduces SSAMD, which estimates waveform coefficients and regularizes them through low-dimensional PCA and singular-value entropy. Experiments on synthetic and real biomedical signals demonstrate denoising, decomposition, segmentation, and preservation of physiologically meaningful variability.

  • Problem

    Localized, impulsive biomedical waveforms with cycle-to-cycle morphological variability are difficult to capture using fixed-waveform ANH models.

  • Method

    SSAMD estimates cycle-specific waveform coefficients by linear regression and regularizes their coefficient matrix using a PCA-based representation with singular-value entropy.

  • Results

    Experiments on synthetic and real biomedical datasets demonstrated effectiveness for denoising, signal decomposition, and waveform segmentation in epileptic EEG, ECG, and single-channel fetal ECG extraction.

  • Takeaways & Limitations

    The framework preserves physiologically meaningful waveform variability and is applicable to periodic signals with localized, time-varying morphology.

  • Takeaways & Limitations

    Initial estimates can be inaccurate because phase may vary rapidly and spiky waveforms may require infinitely many Fourier coefficients under trigonometric regression.

Abstract

from arXiv · show

We propose a novel framework for decomposing nonstationary spiky signals. Unlike classical adaptive non-harmonic models, which represent signals as amplitude- and frequency-modulated (AM--FM) oscillations, the proposed model is designed for signals whose dominant structures are highly localized, impulsive, or spike-like, and contains physiological variability, which is challenging to be modeled as AM--FM representations. Representative examples include electrocardiogram (ECG) complexes, epileptic electroencephalogram (EEG) transients, and other pulse-like physiological signals. We first introduce a fixed-waveform model for repetitive spiky structures and then extend it to accommodate cycle-to-cycle waveform variability. The resulting algorithm, termed \emph{Spiky Shape-adaptive Mode Decomposition} (SSAMD), estimates waveforms in the Fourier coefficient domain. Fourier coefficients are first estimated via local regression and then regularized by exploiting their low-dimensional manifold structure. Specifically, we combine a PCA-based parametrization with an entropy regularization acting on the singular-value spectrum of the coefficient matrix, promoting morphological consistency while preserving structured waveform variability. The proposed framework is flexible and overcomes the limitations of existing models. We validate the method on synthetic and real biomedical signals, including ECG with atrial fibrillation, epileptic EEG, and trans-abdominal maternal ECG. Experimental results demonstrate effective denoising, waveform tracking, decomposition, and segmentation, while preserving physiologically meaningful morphological variations.

1. Introduction

The paper targets spiky, localized biomedical waveforms that challenge ANH models because narrow pulses require many Fourier coefficients and vary across cycles. It introduces SSAMD, which models waveform shapes in a low-dimensional coefficient space and supports denoising, tracking, decomposition, and segmentation.

  • Motivation: ANH models struggle with sharp, localized biomedical structures because narrow pulses have slowly decaying Fourier spectra and require many coefficients.Examples include ECG QRS complexes, deep-brain stimulation recordings, and cardiogenic EEG artifacts.
  • Related work: Wave-shape manifold methods provide prior foundations for denoising and decomposition, including fetal ECG extraction and stimulation-artifact removal.These methods exploit low-dimensional structure in waveform coefficient space.
  • Proposed framework: The proposed framework treats periodic spiky signals as sequences of localized waveform events whose morphology evolves across cycles, rather than primarily as AM-FM oscillations.This shifts emphasis from instantaneous-frequency trajectories to waveform geometry.
  • Method: SSAMD estimates cycle-specific waveform coefficients by linear regression and regularizes them using PCA and entropy applied to the singular-value spectrum.The optimization balances data fidelity with morphological consistency while preserving physiologically meaningful variability.
  • Method: Compared with ANH and time-varying WSF formulations, SSAMD is designed for localized, asymmetric, spike-like waveforms and uses global PCA-based optimization instead of neighborhood selection.The global formulation is described as improving robustness to nonstationary noise and artifacts while avoiding neighborhood-selection sensitivity.
  • Evaluation: Experiments on synthetic and biomedical signals demonstrate denoising, waveform tracking, decomposition, and segmentation while preserving physiologically meaningful morphology.Applications include atrial-fibrillation ECG, epileptic EEG, and single-channel fetal ECG extraction.

2. A Fixed-Waveform Model for Spiky Signals and its limitation

The fixed-waveform model represents periodic spiky signals as localized repeated waveforms and estimates the clean signal through optimization. Its initialization can be inaccurate for rapidly varying cycle timing or highly spiky waveforms, but remains useful for estimating cycles and waveform structure.

  • Model and motivation: The model targets periodic spiky signals whose localized morphology may require many Fourier coefficients under classical oscillatory representations.The proposed formulation instead approximates the waveform locally at the spike scale T, potentially reducing the required harmonic count while estimating T.
  • Model and motivation: The fixed-waveform model uses repeated scaled copies of a 1-periodic waveform h, with separated event times and a localized scale T.The waveform is normalized in L2 and modeled through h_T(t)=h(t/T); additive white noise is then included in the observation model.
  • Optimization and initialization: Recovery solves an optimization problem that approximates h using the first D harmonics, with initialization based on ANH linear regression and estimated cycle information.The initialization estimates amplitude and phase, derives the number of cycles and harmonic count, and then supplies starting values for the main optimization.
  • Optimization and initialization: ANH-based initialization can be inaccurate when inter-event intervals vary rapidly or when spiky waveforms require infinitely many harmonics, although it remains useful for cycle counts and prior waveform estimates.For atrial fibrillation, phase estimates may be smoothed and inaccurate; trigonometric regression can also underestimate D, particularly for spiky waveforms.
  • Synthetic results: At 10 dB SNR, the Gaussian-train experiment produced a narrower estimated waveform with the new model, closer to the clean waveform than the comparison result.Figure 1 compares the noisy and clean signals, LR and SWM-fixed-Alg outputs, their errors, and estimated waveforms.
  • Synthetic results: The simulations also evaluate Gaussian-derivative and synthetic-ECG trains, showing errors and estimated waveforms for LR versus SWM-fixed-Alg under 10 dB noise.The synthetic ECG experiment includes amplitude modulation, whereas the Gaussian-derivative experiment uses time-varying frequency.

3. A General Model for Time-Varying Spiky Waveforms

The time-varying model extends the fixed-waveform formulation by allowing each cycle to have its own smooth, localized waveform. These waveforms are constrained to vary on a low-dimensional manifold while retaining cycle-to-cycle morphological changes.

  • Motivation: The extension addresses biomedical spiky signals whose individual cycles exhibit morphological changes that a single template cannot capture.It replaces the fixed waveform with cycle-specific waveforms h_i(t).
  • Model formulation: Each cycle-specific waveform h_i is assumed smooth and T-periodic, with a truncated Fourier representation and boundary values fixed at zero.The time-dependent waveforms preserve the localized spiky structure while permitting variation across cycles.
  • Model formulation: The collection of waveforms is parametrized as a low-dimensional smooth wave-shape manifold embedded in L2([−T/2,T/2]).The noisy observation model adds white noise with mean zero and finite variance.
  • Model formulation: When T is sufficiently small relative to the minimum spacing between event times, the formulation is called the spiky waveform model with time-varying waveform (SWM-variable).This scale condition maintains separation between localized waveform events.

4. Proposed Algorithm: Spiky Shape-Adaptive Mode Decomposition (SSAMD)

SSAMD estimates cycle-specific spiky waveforms through local Fourier-coefficient regression followed by low-dimensional regularization. Its optimization balances fidelity to noisy coefficient estimates with morphological consistency while retaining structured cycle-to-cycle variability.

  • Algorithm overview: SSAMD uses two steps: cycle-by-cycle linear regression estimates Fourier coefficients, then SVD-entropy regularization refines them into time-varying waveforms.The first step can include artifacts and noise, motivating the second-stage coefficient refinement.
  • Low-dimensional parametrization: The noisy coefficient matrix is denoised by constraining it to a three-dimensional PCA subspace derived from the first principal components.The desired solution is parametrized as X = X0 + V A, where V contains the first three principal-component directions.
  • Entropy regularization: An entropy-based homogeneity term uses the singular-value spectrum to prevent the optimization from merely mimicking the input data.The SVD entropy is converted into an effective dimension and acts as a regularizer on the coefficient configuration.
  • Optimization objective: The regularized optimization balances data fidelity with morphological consistency across cycles while preserving physiologically meaningful waveform variability.The homogeneity parameter controls the tradeoff between configurations close to the data and spectrally concentrated configurations.
  • Optimization procedure: SSAMD solves the optimization with gradient descent over multiple regularization values and selects the solution satisfying the prescribed effective-dimension condition.The procedure starts from X0 and moves toward the noisy data reference without fully reaching it.
  • Comparison with existing algorithms: Unlike fully nonlinear manifold denoising, SSAMD imposes a three-dimensional linear-subspace constraint to reduce estimation variance at the cost of modest modeling bias.The subspace dimension can be adjusted when waveform variability is large or a particular application warrants it.

5. Numerical Results

Experiments on synthetic and real biomedical signals show that SSAMD accommodates cycle-to-cycle waveform variation and improves denoising, tracking, segmentation, and decomposition of spiky signals.

  • Real EEG signal during epileptic event.: SSAMD achieved the best EEG performance, especially near signal boundaries, by capturing time-varying waveforms.LR estimates one waveform and tends to fail near boundaries; SWM-fixed-Alg improves boundary behavior but SSAMD performs best.
  • Real EEG signal during epileptic event.: 7.251 s: SSAMD processed the 23-second EEG signal faster than MMD at 11.170 s, while producing superior residue autocorrelation.SWM-fixed-Alg took 6.876 s; SSAMD produced the best residual whiteness, whereas MMD yielded noisier, more correlated residues.
  • Real ECG signals.: SSAMD estimated cycle-to-cycle ECG waveform variation that LR and SWM-fixed-Alg, which use one repeating waveform, could not represent.The results concern two real ECG signals from the MIT-BIH Arrhythmia Database.
  • Real ECG signals.: 27.376 s: SSAMD processed the 10-second, 360-Hz ECG channel in less time than MMD at 82.74 s while capturing a distinctive fifth-cycle P-wave deflection.SWM-fixed-Alg took 26.360 s; only MMD and SSAMD adjusted to the slight cycle variation.
  • Atrial-fibrillation ECG.: SSAMD and SWM-fixed-Alg centered QRS complexes correctly in atrial-fibrillation segmentation, while LR and MMD did not.SSAMD's residual errors nearly eliminated QRS-induced spikes, whereas prominent spikes remained for LR and MMD.
  • Fetal ECG extraction.: SSAMD improved fetal ECG extraction visibility and morphology estimation, including fetal R-peaks and T-waves, across challenging simulated and real mixtures.For one noisy record, SSAMD absorbed some noise into the maternal estimate, benefiting fetal-ECG estimation; subsequent processing sometimes favored SWM-fixed-Alg.

6. Conclusions

The framework models and estimates periodic signals with spiky, time-varying wave-shape functions in the Fourier coefficient domain. Validation on synthetic and biomedical data demonstrated denoising, decomposition, and segmentation, while future work includes strengthening its theoretical and adaptive-manifold foundations.

  • Conclusions: The method represents individual wave-shape functions in the Fourier coefficient domain and refines them using SVD-entropy regularization.This encourages morphological similarity across cycles while preserving physiologically meaningful variability.
  • Conclusions: Validation on synthetic and real biomedical datasets demonstrated effectiveness for denoising, signal decomposition, and waveform segmentation.Applications included epileptic EEG, ECG, and single-channel fetal ECG extraction.
  • Conclusions: The methodology is applicable beyond biomedical processing to periodic signals with localized, time-varying waveform morphology.
  • Conclusions: Future work includes establishing a deeper theoretical connection to the wave-shape manifold model and developing adaptive manifold representations for complex noise and artifacts.

Appendix

The appendix defines an SVD of the coefficient matrix and introduces a normalized energy distribution over its singular values. It then differentiates the entropy-related expression with respect to an individual singular value and presents simplified forms.

  • Appendix: The coefficient matrix Xc is decomposed by singular value decomposition as UΣW^⊤, with Σ containing σ1 through σr.
  • Appendix: The appendix defines a normalized energy distribution before deriving the entropy-related regularization terms.
  • Appendix: The derivation computes the partial derivative of H with respect to σk and then simplifies it into expressions involving log pk and H.
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