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High-Order Sinusoidal Modeling for Precise Parameter Refinement

Shaowen Chen, Tomoki Toda

arXiv:2610.21643v1eess.SPeess.AS

TL;DR

Accurate sinusoidal modeling remains difficult when existing methods produce frequency mismatches, and QHM-family methods have limited extraction and correction capabilities. The paper analyzes QHM’s model-order constraint, proposes a higher-order sinusoidal model, and uses Levenberg–Marquardt estimation. The reported results show improved parameter estimation and signal modeling in simulations and speech experiments, including under additive noise.

  • Problem

    QHM-family methods can refine frequency estimates while extracting amplitudes, but their extraction and correction capabilities remain limited, especially under frequency deviations and noise.

    Source: §I · p. 2, B. Limitations of QHM methods · p. 3

  • Method

    The paper revisits QHM’s mismatch correction and proposes a higher-order sinusoidal model, using Levenberg–Marquardt for nonlinear least-squares parameter estimation.

    Source: §I · p. 2, A. Optimization method · p. 7

  • Results

    4th-SM consistently outperforms QHM and eaQHM on frequency trajectories, RMSEA, and RMSEf, and the conclusion reports more accurate amplitude and frequency estimates than eaQHM in simulations and speech modeling.

    Source: A. Case 1: Noisy Frequency Scenario · p. 9, §VII · p. 12

  • Takeaways & Limitations

    The high-order model more accurately estimates sinusoidal parameters and improves signal reconstruction in the reported simulation and speech experiments.

    Source: §VII · p. 12

    The paper states that QHM methods retain notable limitations, without specifying their scope in this passage.

    Source: B. Limitations of QHM methods · p. 3

Abstract

from arXiv · show

Although the amplitude estimation of sinusoids is a linear and tractable problem when their frequencies are known, the accuracy of signal modeling is fundamentally limited by the inevitable frequency mismatches produced by existing methods. Quasi-harmonic modeling (QHM) and its extensions (aQHM and eaQHM) offer the ability to refine frequency estimates while extracting amplitudes; however, their extraction and correction capabilities remain limited. In this paper, we first revisit QHM methods from the perspective of frequency mismatches, and reveal that their performances are essentially constrained by the order of the model. Specifically, QHM methods are linear models and thus incapable of capturing higher-order frequency variations, leading to suboptimal modeling accuracy. Second, to address this limitation, we propose a higher-order modeling framework that enables more precise approximation of signal components. Eventually, we qualitatively characterize the optimization problem and employ the Levenberg--Marquardt (LM) algorithm for parameter estimation. As a result, the proposed method allows for significantly improved estimation of amplitudes, frequencies, and phases of sinusoidal components, enabling more accurate signal modeling. Experiments on both simulation study and speech signals demonstrate the superiority of the high-order model in frequency refinement and parameter extraction and its robustness under additive noise conditions.

I. INTRODUCTION

Accurate extraction of sinusoidal amplitude, frequency, and phase matters because synthesis quality depends on these parameters, yet existing methods remain limited under frequency deviations and noise. The paper revisits QHM’s frequency-correction mechanism and proposes a high-order sinusoidal model with Levenberg–Marquardt parameter estimation.

I. INTRODUCTION

  • Precise amplitude, frequency, and phase extraction is important because signal synthesis quality depends heavily on parameter accuracy, while QHM-family methods face frequency-error and noise limitations.Large initial frequency errors can degrade iterative correction, and noise makes least-squares complex-amplitude estimation less reliable.Source: §I · pp. 1–2
  • QHM’s complex time-varying amplitude model approximates frequency mismatch to first order, limiting its ability to model larger mismatches.The paper proposes a high-order sinusoidal model and uses Levenberg–Marquardt estimation for its nonlinear least-squares problem.Source: §I · p. 2

II. REVIEW OF QHM METHODS

  • The analysis focuses on well-separated sinusoidal components, whose instantaneous frequencies are separated by at least 2d under Hypothesis 1.The paper relates this structure to voiced speech segments and periodic mechanical-vibration signals.Source: §II · p. 2

A. Quasi-Harmonic Modeling

  • QHM’s complex slope term provides a first-order approximation for subtle frequency mismatch, but iterative correction retains errors because it assumes intra-frame stationarity.The orthogonal slope coefficient is used to estimate frequency deviation; large initial mismatch calls for iteration.Source: A. Quasi-Harmonic Modeling · p. 3
  • aQHM uses a non-stationary phase function and iterative updates, while eaQHM adds an adaptive amplitude amplifier to track rapid within-frame amplitude modulation.The amplifier is based on the ratio of instantaneous amplitude over the frame to amplitude at its center.Source: A. Quasi-Harmonic Modeling · p. 3

B. Limitations of QHM methods

The paper identifies limitations in QHM methods.

B. Limitations of QHM methods

1) Insufficient Parameter Estimation:

QHM and its extensions lose parameter-estimation accuracy when initial frequency mismatches are large, and iterative correction can worsen errors in closely spaced components. The paper attributes the limits of QHM to its first-order model and motivates higher-order approximation.

1) Insufficient Parameter Estimation:

  • With modest initial frequency deviations ηk ∈[-10, 10], QHM and eaQHM estimate accurate trajectories and amplitudes, but substantial mismatches cause oscillating trajectories, amplitude errors, and eaQHM swaps of closely spaced components.The swap occurs around t = 0.4s, when eaQHM assigns the second component’s frequency to the third and vice versa.Source: 1) Insufficient Parameter Estimation · pp. 3–4

2) Heavy computation:

  • Iterative aQHM and eaQHM incur substantial computational cost through repeated signal reconstruction, spline and linear interpolation, and least-squares optimization.More iterations may be needed for large initial deviations, creating a tradeoff between accuracy and computational efficiency.Source: 2) Heavy computation · p. 4

III. HIGH ORDER APPROXIMATION

  • The proposed high-order sinusoidal model increases model order to improve parameter extraction accuracy and address QHM methods’ mismatch-correction limitations.Source: §III · p. 4

A. On the Insufficiency of Low-Order Models in QHM

B. Towards High-Order Model

The section develops a structurally aligned high-order sinusoidal model to address the limitations of higher-order polynomial fitting, with more reliable frequency estimates and improved amplitude estimation reported for 2nd-SM.

B. Towards High-Order Model

  • The higher-order polynomial model represents local amplitude structure and implicit frequency mismatch using a polynomial amplitude fitted to the framewise signal.It uses N+1 independent coefficients to capture local amplitude, phase, and frequency mismatch.Source: B. Towards High-Order Model · p. 5
  • The general higher-order polynomial model uses N+1 free coefficients but lacks structural alignment with the exponential mismatch term, risking non-robust estimates in noise.Its coefficient relations to the frequency mismatch generally fail in practical estimation.Source: B. Towards High-Order Model · p. 5
  • The model construction derives higher-order coefficients from the mismatch-term Taylor expansion, then simplifies them to avoid parameters in the polynomial denominator.The denominator is avoided because it may cause numerical instability in practical computation.Source: B. Towards High-Order Model · p. 5
  • The proposed high-order SM retains Taylor-expansion alignment as model order increases and reduces estimation to two key coefficients, a_k and b_k.The model is extended to multicomponent signals.Source: B. Towards High-Order Model · p. 5
  • In Fig. 3, 2nd-SM nearly overlaps the second-order Taylor expansion, while QHM and 2nd-PM deviate from their corresponding expansions.The figure compares mismatch-term trajectories and their Euclidean distances from the corresponding Taylor expansions.Source: B. Towards High-Order Model · p. 5, Figure 3 · p. 6
  • 2nd-SM yields more reliable frequency estimates under substantial initial mismatch and significantly improves amplitude estimation compared with QHM and eaQHM.The frequency estimates are shown in Fig. 4 and the amplitude estimates in Fig. 5.Source: B. Towards High-Order Model · pp. 5–6, Figure 4 · p. 6, Figure 5 · p. 6

C. Quantitative Performance Analysis of High-Order SM in Parameter Estimation

The analysis derives least-squares estimates and error bounds for high-order SM parameters. It finds that increasing model order reduces amplitude and frequency-mismatch estimation errors, which converge to zero for sufficiently large order.

C. Quantitative Performance Analysis of High-Order SM in Parameter Estimation

  • A least-squares analysis within a frame derives estimates and biases for amplitude A and frequency mismatch η from the high-order sinusoidal model.The theorem considers a monocomponent signal and an initially estimated frequency.Source: C. Quantitative Performance Analysis… · p. 6
  • For sufficiently large N, amplitude and frequency-mismatch estimation errors converge to zero, and increasing N substantially reduces both errors.The analysis illustrates the errors and their upper bounds as functions of initial mismatch and model order, using a Hamming window and T = 10 ms.Source: C. Quantitative Performance Analysis… · p. 6
  • The estimation errors depend on both initial frequency mismatch η and model order N, with larger initial mismatch generally producing larger errors.Proposition 1 gives bounds for the amplitude and frequency-mismatch errors.Source: C. Quantitative Performance Analysis… · p. 6

IV. IMPLEMENTATION OF HIGH-ORDER SM

High-order SM requires nonlinear optimization because it is linear in a but nonlinear in b. The implementation uses frame-parallel LM updates to estimate parameters and reconstruct the signal.

A. Optimization method

  • Because high-order SM is nonlinear in b, direct least squares is inapplicable, unlike the linear QHM parameter estimation.The model remains linear in a, but nonlinear optimization is needed to estimate b.Source: A. Optimization method · p. 7
  • LM jointly estimates a and b, using a damping factor to improve numerical stability while retaining fast convergence.A large damping factor makes LM more like gradient descent early on; a small factor makes it more like Gauss-Newton near the optimum.Source: A. Optimization method · p. 7
  • Initial frequencies and hyperparameters are initialized per frame, and frames are processed in parallel for computational efficiency.The initial frequencies are roughly detected from a time-frequency representation before LM optimization.Source: A. Optimization method · p. 7
  • At each iteration, parameters are selectively updated by comparing residual norms, while the damping factor is adapted to support stable convergence.The algorithm recomputes Jacobians and residuals and estimates amplitude, phase, and final frequency mismatch after the iterations.Source: A. Optimization method · p. 8

B. Signal Reconstruction

Signal reconstruction follows frequency correction and interpolation of framewise amplitude and frequency estimates. The resulting instantaneous quantities define phase and component reconstruction.

  • After correcting frequencies and interpolating framewise amplitude and frequency, the method depicts instantaneous versions and obtains instantaneous phase.The phase construction uses the framewise phase estimate and a quantity computed from adjacent frames.Source: B. Signal Reconstruction · p. 8
  • An arbitrary signal component can then be reconstructed from the resulting instantaneous parameter estimates.The reconstruction step follows the phase-construction procedure.Source: B. Signal Reconstruction · p. 8

V. NUMERICAL PERFORMANCE EVALUATION

The evaluation compares high-order SM with QHM and eaQHM on a simulated multicomponent signal under frequency and waveform noise. It assesses parameter estimation and reconstruction quality.

  • The experiments compare 4th-SM with QHM and eaQHM, and test 2nd-, 4th-, and 6th-order SMs on a signal with complex amplitude and frequency modulations.Two conditions are considered: noisy frequency, and noisy frequency combined with noisy waveform.Source: §V · p. 8
  • Frequency and amplitude estimation are evaluated with RMSE, while reconstruction quality is measured by SRER.A larger SRER indicates a result closer to the reference.Source: §V · p. 8
  • The evaluation includes estimated frequency trajectories and a table of per-component amplitude and frequency RMSE scores.The table reports RMSEA and RMSEf.Source: Figure 8 · p. 9, Table I · p. 9

A. Case 1: Noisy Frequency Scenario

Across noisy-frequency and noisy-waveform tests, high-order SM estimates parameters and reconstructs signals more accurately than QHM and eaQHM. Higher model orders improve robustness as mismatch or noise increases.

A. Case 1: Noisy Frequency Scenario

  • For the noisy-frequency case, 4th-SM consistently outperforms QHM and eaQHM in frequency trajectories, RMSEA, and RMSEf.QHM has the highest RMSEA for all components; eaQHM improves amplitude estimation but may fail to correct frequency mismatch.Source: A. Case 1: Noisy Frequency Scenario · p. 9
  • In the reconstruction comparison, 4th-SM achieves the highest SRER value, 38.8 dB, and produces a waveform closer to ground truth than the synthesized waveforms from other methods.The compared waveform segments are around t = 0.613 s.Source: A. Case 1: Noisy Frequency Scenario · p. 9
  • With increasing initial frequency mismatch, QHM and eaQHM degrade, while 6th-SM remains stable and 4th-SM outperforms 2nd-SM.The comparison averages RMSEA, RMSEf, and SRER over 20 Monte-Carlo simulations.Source: Figure 10 · p. 9, Figure 11 · p. 9, A. Case 1: Noisy Frequency Scenario · p. 9

B. Case 2: Noisy Frequency and Noisy Waveform Scenario

VI. APPLICATION IN SPEECH MODELING

In speech modeling, 4th-SM estimates frequencies and amplitudes more accurately than the QHM methods and improves reconstruction, while requiring more computation than QHM. The conclusion attributes this performance to a higher-order model estimated with Levenberg–Marquardt.

VI. APPLICATION IN SPEECH MODELING

  • Because individual speech amplitudes and frequencies lack ground truth, estimation accuracy is assessed through a spectrogram distance, with lower Lspec indicating more accurate amplitude and frequency estimates.The speech experiments use utterances from LJspeech at 22.05 kHz, with f0 initialized by YAAPT.Source: §VI · p. 11
  • 4th-SM estimates frequencies consistently across all harmonics; QHM has serrations around the 4th–6th harmonics, while eaQHM fails to reliably correct the 1st–3rd.The comparison uses speech frequency trajectories estimated by all methods.Source: Figure 16 · p. 11, §VI · p. 11
  • 4th-SM achieves both accurate amplitude and frequency estimates and the smallest Lspec; QHM has larger amplitude errors, while eaQHM has noticeable frequency deviations.The speech spectrogram comparison evaluates the extracted parameters against the ground truth around t = 1.16s.Source: Figure 17 · p. 11, §VI · p. 11
  • 12.2 dB SRER: 4th-SM provides the most reliable speech reconstruction, and consistently achieves the best parameter-estimation accuracy and reconstruction fidelity.4th-SM and eaQHM have higher RTFs than QHM; eaQHM’s RTF approaches 26 when initial frequency mismatch is large, while 4th-SM typically uses 8 LM iterations.Source: Figure 18 · p. 11, §VI · p. 11

VII. CONCLUSION

  • The conclusion reports that high-order SM more accurately estimates amplitudes and frequencies, improves reconstruction, and is robust to waveform noise and frequency perturbance.The authors also report limited time consumption and faster estimation than eaQHM.Source: §VII · pp. 11–12

APPENDIX A THE PROOF OF THEOREM 1

The proof first eliminates the complex coefficient through an orthogonal projection, then minimizes a nonlinear cost in the frequency-mismatch parameter to estimate amplitude and mismatch.

APPENDIX A THE PROOF OF THEOREM 1

  • The complex coefficient a is the orthogonal projection of x onto the subspace spanned by yN, and setting the derivative with respect to a* to zero gives its estimate.The proof then substitutes this estimate into E(a,b).Source: Appendix A · p. 12
  • After substituting the estimate of a, the cost E(a,b) is nonlinear in b because yN has an exponential-like polynomial structure.The proof equivalently rewrites the solution for b under the stated positive lower bound on ∥x∥2.Source: Appendix A · p. 12
  • The resulting estimates of a and b are used to estimate amplitude and frequency mismatch.Source: Appendix A · p. 12

APPENDIX B THE PROOF OF PROPOSITION 1

The proposition proof analyzes parameter-estimation errors by linearizing the cost near the true parameters. It concludes that estimation error decreases as model order increases, with stability requiring a positive derivative-related quantity.

APPENDIX B THE PROOF OF PROPOSITION 1

  • The analysis linearizes E(a,b) around the true parameters using small estimation errors εa and εb, relying on optimality conditions for a and b.Source: Appendix B · p. 12
  • The amplitude estimation error decreases as model order N increases and approaches zero in the limit.The proof separately analyzes amplitude and frequency-mismatch estimation errors.Source: Appendix B · p. 12
  • Stability requires D⋆N to remain positive; empirical results with symmetrically tapered windows show its minimum stays strictly positive for a wide range of η and N.The example uses a Blackman window.Source: Appendix B · p. 13, Figure 19 · p. 13
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