Time-Varying Amplitude Harmonic Model

• Time-Varying Amplitude Harmonic Model is a framework that adaptively represents oscillatory signals with smoothly varying amplitudes, frequencies, and waveforms.
• It employs estimation techniques such as polynomial and spline regression, trigonometric fitting, and alternating minimization to recover dynamic signal characteristics.
• The model finds widespread application in biomedical signal analysis, voice processing, nonlinear optics, and energy systems for robust nonstationary signal decomposition.

The time-varying amplitude harmonic model is a general mathematical and algorithmic framework for the representation and analysis of oscillatory signals whose harmonic amplitudes, frequencies, and, in some extensions, wave-shapes can vary smoothly or adaptively in time. This paradigm underpins contemporary approaches to nonstationary signal modeling in diverse domains such as biomedical engineering, voice analysis, nonlinear optics, astrophysics, and electromagnetic energy accumulation, enabling the adaptive recovery, decomposition, and interpretation of complex oscillatory patterns under realistic (noisy, nonstationary) conditions. Models in this class generalize classical Fourier and harmonic representations by encoding non-sinusoidal, amplitude/frequency-modulated, and even shape-adaptive morphologies, thus accommodating phenomena ranging from pathophysiological signal morphodynamics to pump-induced nonlinearities in metamaterials.

1. Mathematical Formulation of Time-Varying Amplitude Harmonic Models

The canonical time-varying amplitude harmonic model expresses a signal $x(t)$ as a sum of harmonics whose amplitudes and phases (or frequencies) are allowed to vary with time:

$$x(t) = A_0(t) + \sum_{k=1}^K A_k(t)\,\cos\big[2\pi\,\phi_k(t) + \varphi_k(t)\big] + v(t),$$

where $A_k(t)$ are the harmonic amplitudes, $\phi_k(t)$ are the instantaneous phases (with $\phi_k'(t)$ as the instantaneous frequency), $\varphi_k(t)$ residual phases, and $v(t)$ denotes an aperiodic or noise component. In fixed “wave-shape function” (WSF) harmonic models, the oscillatory $\cos[\cdot]$ term is replaced by a more general $2\pi$-periodic $s_k(\cdot)$, typically with a bandlimited or sparse Fourier expansion (“adaptive non-harmonic” or ANH model). The time-varying amplitude extension allows $A_k(t)$ and potentially higher harmonic parameters (e.g., $\alpha_\ell(t)$ for the $\ell$th harmonic) to be slowly or smoothly modulated.

This general structure encapsulates:

• Adaptive Harmonic (AH) model, where both amplitude and frequency may vary but the waveform is locally sinusoidal.
• Fixed-WSF ANH models, with fixed but non-sinusoidal shape per period, still with amplitude modulation.
• Fully adaptive, time-varying wave-shape models where both the harmonic content and their coefficients are explicit functions of time (Ruiz et al., 2023).
• Polynomial and spline-basis representations for the time-dependence of $A_k(t)$ and phase, especially for voice and astronomical signals (Ikuma et al., 2022, Motta et al., 2021).
• Special cases for physical systems, e.g., time-varying permittivity and nonlinear susceptibility in SHG at modulated optical interfaces, or engineered circuits with prescribed amplitude envelopes (Tirole et al., 2024, Mirmoosa et al., 2018).

2. Model Parameterization and Estimation Techniques

Estimation methods exploit the underlying smoothness and low-rank structure of amplitude and phase evolution:

• Polynomial or spline expansions: e.g., $$A_k(t) \approx \sum_{\ell=0}^L a_{k,\ell} h_\ell(t-t_0)$$, with $h_\ell$ polynomial bases or B-splines. This is the preferred framework when slow intra-frame variation is anticipated (Ikuma et al., 2022, Motta et al., 2021).
• Trigonometric regression: Fitting harmonics with time-varying coefficients via ordinary (possibly penalized) least squares, leveraging STFT-derived instantaneous amplitude and phase for constructing the design matrix (Ruiz et al., 2023).
• Nonlinear regression with interpolated amplitudes: Using free-node cubic Hermite interpolation for the amplitude trajectories (as in $\alpha_\ell(t)$), and jointly estimating harmonic amplitudes and non-integer phase distortions by nonlinear least squares, initialized via linear regression and refined by Levenberg–Marquardt (Ruiz et al., 2023).
• Alternating minimization: For non-convex cost functions, an alternating scheme is used, updating amplitude and phase sets separately in each iteration, with the amplitude update often admitting closed-form or pseudo-inverse solutions (Ikuma et al., 2022).
• Information-theoretic or penalized selection of harmonic order: Model complexity (number of harmonics $r$) is determined adaptively using AIC-like, MDL-like, GCV, or unbiased risk criteria, with experiments confirming robust order recovery in the presence of noise (Ruiz et al., 2023).
• Spectral estimation for physical models: In SHG at a time-varying interface, the time-dependent response is derived via direct physical modeling and matched to experimental spectra, with key analytical relationships calibrated to measured modulation depths (Tirole et al., 2024).

3. Representative Applications Across Domains

Time-varying amplitude harmonic models have demonstrated efficacy in multiple domains:

Domain	Model Structure & Focus	Reference
Biomedical signals	Time-varying wave-shape, denoising, decomposition, segmentation	(Ruiz et al., 2023)
Voice analysis	Slow-varying amplitude & frequency, robust HNR, tremor/diplophonia	(Ikuma et al., 2022)
Astrophysical lightcurves	Spline-based time-varying amplitude harmonics, missing data, forecasting	(Motta et al., 2021)
Nonlinear optics	Time-modulated permittivity and $\chi^{(2)}(t)$, enhanced SHG	(Tirole et al., 2024)
Energy accumulation	Arbitrary amplitude shaping via time-varying reactance	(Mirmoosa et al., 2018)
Frequency estimation	Robust adaptation to slowly varying amplitudes and noisy data	(Ruderman, 2021)

For example, in biomedical signal processing, adaptive ANH models enable high-resolution, noise-robust recovery of time-varying morphologies for applications such as EEG denoising, ECG segmentation (notably detecting ventricular fibrillation via harmonic amplitude jumps), and decomposition of multicomponent physiological signals (Ruiz et al., 2023). In voice processing, such models permit the extraction of physically relevant features (e.g., HNR) that are robust to pathological modulations (Ikuma et al., 2022). In time-resolved nonlinear optics, the time-varying amplitude harmonic approach precisely captures modulation-enhanced SHG effects by relating the temporal evolution of pump-induced susceptibility and reflection coefficients to observed spectral features (Tirole et al., 2024).

4. Robustness, Performance, and Algorithmic Considerations

Robustness to noise, nonstationarity, and modeling error is a central theme:

• Denoising and decomposition: Adaptive models outperform fixed-shape or static-harmonic approaches, especially at low SNR or when waveforms exhibit abrupt transitions. For example, on synthetic signals with SNR_in ≤ 15 dB, the adaptive procedure yields higher output SNR than fixed-WSF LR or polynomial-phase-based SAMD baselines (Ruiz et al., 2023).
• Order estimation and overfitting: Automatic order selection is critical, as fixed harmonic order models may under- or overfit depending on noise and waveform complexity. Information-theoretic and predictive risk criteria show close-to-optimal harmonic count recovery across varying SNR and waveform variability (Ruiz et al., 2023).
• Boundary artifacts and signal extension: Techniques such as ARIMA-based extensions and signal flipping correct for edge effects introduced by finite-window STFT and interpolation processes (Ruiz et al., 2023).
• Physical constraints and domain specialization: In energy accumulation and nonlinear optics, the engineered time profile of reactive elements or susceptibilities (e.g., mixing networks, parametrically modulated thin films) is constrained by device physics and is synthesized or modeled accordingly (Mirmoosa et al., 2018, Tirole et al., 2024).
• Computational complexity: Joint nonlinear optimization (e.g., Levenberg–Marquardt in (Ruiz et al., 2023)) is tractable for moderate numbers of harmonics and nodes, with initialization schemes exploiting closed-form regression to avoid poor local minima.

5. Physical Realization and Theoretical Implications

In certain physical systems, the time-varying amplitude harmonic model admits experimental realization or provides direct physical insight:

• Time-varying reactances and energy accumulation: By engineering a time-dependent reactance $X(t)$, it is possible to synthesize arbitrary current envelopes under a time-harmonic drive, as established analytically in (Mirmoosa et al., 2018). This enables, for example, unlimited accumulation of energy or its release as a time-compressed pulse.
• Parametric synthesis via mixers and filters: The required reactance profile for non-reflecting, energy-accumulating circuits can be constructed from a series of harmonically modulated circuit elements (mixers), each operating at a subharmonic of the base frequency, and summed with filtering to isolate the desired response (Mirmoosa et al., 2018).
• Enhanced nonlinear optical response: In optically pumped ITO interfaces, a direct relationship between time-varying carrier density (plasma frequency) and enhanced second-harmonic susceptibility $\chi^{(2)}(t)$ yields a modulation contrast in SHG that is up to $\sim4\times$ higher than for the fundamental, with frequency broadening and shifts scaling with the speed of time modulation (Tirole et al., 2024).

6. Extensions, Limitations, and Future Directions

While the time-varying amplitude harmonic model is flexible, several challenges and possibilities are recognized:

• Highly non-perturbative and nonstationary regimes: In strongly modulated or extremely noise-dominated settings, complexity may outpace available measurements or computational tractability, requiring further model reduction or new regularization strategies (Tirole et al., 2024).
• Model structure selection and adaptation: Dynamic adjustment of shape classes, number of harmonics, and interpolation nodes remains an active direction, particularly for applications with varying morphology (e.g., arrhythmic ECG, changing voice pathologies, transient astrophysical phenomena) (Ruiz et al., 2023, Ruiz et al., 2023).
• Generalization beyond harmonic content: Some phenomena involve more general non-harmonic or even chaotic oscillatory patterns, suggesting the need for further model generalization.
• Physical limitations and assumptions: In systems engineering, practical realizability is bounded by device constraints (e.g., maximal modulation speed, losses, non-idealities), and theoretical results often assume lossless or perfectly modulated elements (Mirmoosa et al., 2018).
• Domain-specific impacts: For nonlinear optics and electromagnetic metamaterials, exploitation of time-varying amplitude harmonic effects continues to evolve, particularly for applications in frequency conversion, sensing, and energy harvesting (Tirole et al., 2024).

The time-varying amplitude harmonic model thus provides a unifying formalism for adaptive, high-resolution, and robust oscillatory signal analysis and synthesis in a wide range of scientific and engineering applications, with continued innovations in both mathematical methodology and physical realization (Ruiz et al., 2023, Ikuma et al., 2022, Motta et al., 2021, Tirole et al., 2024, Mirmoosa et al., 2018).