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arxiv: 2603.20614 · v1 · submitted 2026-03-21 · 📡 eess.SP · cs.NA· math.NA· physics.app-ph

Sparse stability diagrams of LSCF method via strategic pole destabilization using orthogonal matching pursuit

Shogo Shimada , Akira Saito This is my paper

Pith reviewed 2026-05-15 07:42 UTC · model grok-4.3

classification 📡 eess.SP cs.NAmath.NAphysics.app-ph
keywords LSCF methodmodal parameter identificationstability diagramsorthogonal matching pursuitspurious polesfrequency response functionssparse polynomialsvibration analysis
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The pith

Making LSCF polynomial coefficients sparse with orthogonal matching pursuit destabilizes spurious poles to produce cleaner stability diagrams.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The LSCF method fits high-order polynomials to frequency response functions for extracting modal parameters like natural frequencies and damping ratios, but high orders introduce non-physical spurious poles that clutter stability diagrams. The paper shows that orthogonal matching pursuit can enforce sparsity on those polynomial coefficients to strategically destabilize only the spurious poles. This yields sparse diagrams from which unstable poles can be removed while physical modes remain accurate and stable. Tests on finite-element data for a rectangular plate, experimental FRFs from low- and high-damping plates, and FRFs from an electric-vehicle stator core confirm the approach works without accuracy loss. The result simplifies identification of vibration characteristics in mechanical, aerospace, and civil structures.

Core claim

The proposed method strategically destabilizes the stable yet spurious poles of the characteristic polynomials by making their coefficients as sparse as possible via orthogonal matching pursuit. This results in sparse stability diagrams because unstable poles can be eliminated. In this paper, the proposed method is first applied to a numerically-obtained FRFs of a rectangular plate using finite element model, and its validity is discussed. Then, the method is applied to experimentally-obtained FRFs of rectangular plates with low-damping and with high-damping. Furthermore, to confirm its applicability to industrial applications with realistic complexity, it has also been applied to the FRFs,

What carries the argument

Orthogonal matching pursuit applied to the coefficients of the LSCF characteristic polynomial to enforce sparsity and selectively destabilize spurious poles.

If this is right

  • Stability diagrams become sparse, allowing direct removal of unstable poles without manual inspection of each candidate mode.
  • The technique preserves accuracy of natural frequencies, damping ratios, and mode shapes across numerical and experimental cases.
  • It applies to both low-damping and high-damping structures as well as complex industrial components such as stator cores.
  • High-order polynomials can be used without the usual proliferation of spurious roots in the diagrams.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The sparsity step might reduce reliance on post-processing heuristics for pole selection in routine vibration testing workflows.
  • If the same OMP step transfers to other polynomial fitting methods, it could streamline system identification beyond LSCF.
  • Testing on structures with known nonlinear behavior would show whether the selective destabilization remains reliable outside linear modal assumptions.

Load-bearing premise

Enforcing sparsity via OMP on the characteristic polynomial coefficients will destabilize only the non-physical spurious poles while leaving the physical poles stable and accurate.

What would settle it

Applying the OMP procedure to a dataset where physical poles lose stability or extracted modal parameters show large errors would falsify the central claim.

Figures

Figures reproduced from arXiv: 2603.20614 by Akira Saito, Shogo Shimada.

Figure 1
Figure 1. Figure 1: Roots of polynomials of order 100 for various numbers of nonzero coefficients randomly chosen for [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Relationship between the percentage of the roots of the polynomial with [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Numerical model of a plate and its frequency response function [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Vibration modes of the model corresponding to the two resonant peaks. Results are obtained by [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of stability diagram obtained from the conventional LSCF method and the proposed [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Result of Curve fit when both methods are applied to the FRF obtained from the numerical [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of the stability diagrams when both methods are applied to the FRF obtained from [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Result of curve fit when both methods are applied to the FRF obtained from the numerical [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of the stability diagrams when both methods are applied to the FRF obtained from [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Result of curve fit when both methods are applied to the FRF obtained from the numerical [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Experiment model and its frequency response function [PITH_FULL_IMAGE:figures/full_fig_p015_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Comparison of the stability diagrams obtained from the conventional LSCF method and the [PITH_FULL_IMAGE:figures/full_fig_p016_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Comparison of the poles in complex plane. [PITH_FULL_IMAGE:figures/full_fig_p016_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Result of curve fit when both methods are applied to the FRF obtained from case study 1 [PITH_FULL_IMAGE:figures/full_fig_p017_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Comparison of MAC between the mode shapes obtained from the conventional LSCF method and [PITH_FULL_IMAGE:figures/full_fig_p018_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Experiment model and its frequency response function [PITH_FULL_IMAGE:figures/full_fig_p019_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Comparison of stability diagram when both methods are applied to the FRF obtained from case [PITH_FULL_IMAGE:figures/full_fig_p020_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Comparison of the poles in complex plane. [PITH_FULL_IMAGE:figures/full_fig_p021_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Result of Curve fit when both methods are applied to the FRF obtained from case study 2 [PITH_FULL_IMAGE:figures/full_fig_p021_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Comparison of MAC when both methods are applied to the FRF obtained from case study 2 [PITH_FULL_IMAGE:figures/full_fig_p022_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Experimental model and the averaged measured FRF [PITH_FULL_IMAGE:figures/full_fig_p023_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Comparison of stability diagram when both methods are applied to the FRF obtained from case [PITH_FULL_IMAGE:figures/full_fig_p024_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Comparison of the poles in complex plane. [PITH_FULL_IMAGE:figures/full_fig_p025_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Result of curve fit when both methods are applied to the FRF obtained from case study 3 [PITH_FULL_IMAGE:figures/full_fig_p025_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Mode shapes for the mode 1. MAC=0.998. measured FRF and the approximated FRF of the conventional method shows a value of 0.5213(m/s 2/N)2 , while the proposed method shows a value of 1.7747(m/s 2/N)2 , which is a slight increase from that of the conventional method. Table. 6: Comparison of natural frequencies and damping ratios for the case study 3 Natural Frequency [Hz] Damping Ratio Mode number LSCF Pro… view at source ↗
Figure 26
Figure 26. Figure 26: Mode shapes for the mode 2. MAC=0.999. Normalized displacement 1.0 0 (a) Conventional LSCF method Normalized displacement 1.0 0 (b) Proposed method [PITH_FULL_IMAGE:figures/full_fig_p027_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: Mode shapes for the mode 3. MAC=0.986. Normalized displacement 1.0 0 (a) Conventional LSCF method Normalized displacement 1.0 0 (b) Proposed method [PITH_FULL_IMAGE:figures/full_fig_p027_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: Mode shapes for the mode 4. MAC=0.999. Normalized displacement 1.0 0 (a) Conventional LSCF method Normalized displacement 1.0 0 (b) Proposed method [PITH_FULL_IMAGE:figures/full_fig_p027_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: Mode shapes for the mode 6. MAC=0.990. Normalized displacement 1.0 0 (a) Conventional LSCF method Normalized displacement 1.0 0 (b) Proposed method [PITH_FULL_IMAGE:figures/full_fig_p027_29.png] view at source ↗
Figure 30
Figure 30. Figure 30: Mode shapes for the mode 8. MAC=0.950. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_30.png] view at source ↗
Figure 31
Figure 31. Figure 31: Comparison of MAC values for case study 3 with top and bottom layers in axial direction of the core being out-of-phase, where the spatial order in circumferential direction is two whereas the one in axial direction is one. The mode 2 shown in [PITH_FULL_IMAGE:figures/full_fig_p028_31.png] view at source ↗
read the original abstract

In various engineering fields including mechanical, aerospace, and civil engineering, the identification of modal parameters, including natural frequencies, damping ratios, and mode shapes, is crucial for determining the vibration characteristics of engineered structures. A common method for identifying the modal parameters of structures involves experimental modal analysis using frequency response functions (FRFs) obtained from forced vibration tests. The least squares complex frequency (LSCF) domain method is a widely-used frequency-domain curve-fitting method for the FRFs using the polynomials of high order, which can extract modal parameters with high accuracy. However, increasing the polynomial order tends to result in the generation of non-physical spurious poles that need to be eliminated from the stability diagrams. To overcome this issue, we propose a method that strategically destabilize the stable yet spurious poles of the characteristic polynomials by making their coefficients as sparse as possible, via orthogonal matching pursuit (OMP). This results in sparse stability diagrams because unstable poles can be eliminated from the diagrams. In this paper, the proposed method is first applied to a numerically-obtained FRFs of a rectangular plate using finite element model, and its validity is discussed. Then, the method is applied to experimentally-obtained FRFs of rectangular plates with low-damping and with high-damping. Furthermore, to confirm its applicability to industrial applications with realistic complexity, it has also been applied to the FRFs of the electric machine's stator core used for electric vehicles. Based on the results, we have confirmed that the spurious roots can be eliminated from the stability diagrams without compromising accuracy for the cases considered.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper proposes applying orthogonal matching pursuit (OMP) to sparsify the coefficients of the characteristic polynomials produced by the least-squares complex frequency (LSCF) method. The goal is to strategically destabilize spurious (non-physical) poles while leaving physical modal parameters intact, thereby producing sparse stability diagrams that require less manual interpretation. The approach is demonstrated on numerically generated FRFs of a rectangular plate, experimental FRFs of low- and high-damping plates, and FRFs from an EV stator core, with the claim that spurious roots are eliminated without loss of accuracy.

Significance. If the selective destabilization property holds beyond the presented cases, the method would offer a practical post-processing step that simplifies modal parameter extraction in experimental modal analysis. The empirical validation across numerical, laboratory, and industrial datasets is a positive feature. However, the absence of theoretical justification for why OMP preferentially affects spurious poles and the lack of quantitative error metrics limit the immediate significance of the contribution.

major comments (3)
  1. [Method description (proposed OMP application to LSCF polynomials)] The central claim—that OMP-driven sparsity on the characteristic polynomial coefficients destabilizes only spurious poles while preserving physical poles—lacks supporting analysis. No examination of the geometry of the coefficient space or conditions guaranteeing selective destabilization is provided; if physical and spurious poles exhibit overlapping sensitivity to coefficient perturbations, the greedy OMP selection may alter physical frequency or damping estimates. This issue is load-bearing for the method's validity.
  2. [Numerical and experimental results sections] The results claim that accuracy is not compromised, yet no quantitative metrics (e.g., percentage errors in natural frequencies or damping ratios relative to reference values, comparison tables against standard LSCF, or statistical measures across multiple realizations) are reported. The abstract and validation sections rely on qualitative statements about the cases considered, which is insufficient to substantiate the no-loss-of-accuracy assertion.
  3. [Implementation and parameter selection] The sparsity level in OMP is treated as a free parameter alongside polynomial order. No systematic procedure, sensitivity study, or criterion for choosing the sparsity level is given, leaving open the possibility that results depend on case-specific tuning rather than a robust, general rule.
minor comments (2)
  1. [Method] Clarify the precise formulation of the OMP objective (e.g., whether it operates on the full polynomial coefficient vector or a normalized version) and include pseudocode or a step-by-step algorithmic outline for reproducibility.
  2. [Results] Add error bars or repeated-run statistics to the stability diagrams and modal parameter tables to quantify variability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments on our manuscript. We address each major comment point by point below, indicating where revisions will be made to strengthen the paper.

read point-by-point responses

  1. Referee: The central claim—that OMP-driven sparsity on the characteristic polynomial coefficients destabilizes only spurious poles while preserving physical poles—lacks supporting analysis. No examination of the geometry of the coefficient space or conditions guaranteeing selective destabilization is provided; if physical and spurious poles exhibit overlapping sensitivity to coefficient perturbations, the greedy OMP selection may alter physical frequency or damping estimates. This issue is load-bearing for the method's validity.

    Authors: We acknowledge that the original manuscript relies primarily on empirical demonstration rather than a formal geometric analysis of the coefficient space. The selectivity arises because spurious poles in over-parameterized LSCF fits are associated with smaller-magnitude higher-order coefficients that OMP prioritizes for zeroing under the sparsity constraint, while physical poles are anchored by dominant lower-order terms. In the revision we will expand the method description with this explanation drawn from the observed coefficient behavior across the tested cases and add an explicit statement that a general proof of selective destabilization remains an open theoretical question. We believe this provides the strongest honest defense without overstating the current analysis. revision: partial

  2. Referee: The results claim that accuracy is not compromised, yet no quantitative metrics (e.g., percentage errors in natural frequencies or damping ratios relative to reference values, comparison tables against standard LSCF, or statistical measures across multiple realizations) are reported. The abstract and validation sections rely on qualitative statements about the cases considered, which is insufficient to substantiate the no-loss-of-accuracy assertion.

    Authors: We agree that quantitative support is required. The revised manuscript will include new tables in the numerical and experimental results sections that report natural frequencies and damping ratios obtained with the proposed method, together with percentage errors relative to reference values (finite-element model for the plate) and direct comparisons against standard LSCF. Where multiple experimental realizations are available, we will also report standard deviations or consistency metrics. These additions will directly substantiate the accuracy claim. revision: yes

  3. Referee: The sparsity level in OMP is treated as a free parameter alongside polynomial order. No systematic procedure, sensitivity study, or criterion for choosing the sparsity level is given, leaving open the possibility that results depend on case-specific tuning rather than a robust, general rule.

    Authors: We will add a new subsection on implementation and parameter selection. It will describe a practical criterion: increase sparsity until the physical poles in the stability diagram remain unchanged while spurious poles are removed; further sparsity that begins to shift known physical poles defines the upper limit. A sensitivity study showing modal-parameter variation over a range of sparsity levels for the numerical plate and both experimental plates will be included to demonstrate robustness within the recommended operating window. revision: yes

Circularity Check

0 steps flagged

No circularity: post-processing step on standard LSCF polynomials

full rationale

The paper treats LSCF as an established frequency-domain fitting technique and introduces OMP-based coefficient sparsification as an independent algorithmic post-processing step to destabilize spurious poles. No equation or claim reduces by construction to a fitted parameter defined by the target result, nor does any load-bearing premise rest on a self-citation chain. Validation proceeds via direct application to FEM-generated FRFs and experimental data sets, with accuracy assessed against known physical modes rather than by re-deriving the input polynomials. The central assumption (selective destabilization of non-physical roots) is presented as an empirical observation, not a mathematical identity or uniqueness theorem imported from prior author work.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The method rests on the domain assumption that physical poles remain stable under coefficient sparsification while spurious ones do not; no free parameters are explicitly fitted in the abstract description beyond standard LSCF order selection.

free parameters (2)
  • polynomial order
    High-order polynomials are chosen to capture all modes; order selection is a standard but tunable parameter in LSCF.
  • OMP sparsity level
    Number of retained non-zero coefficients is selected to achieve desired destabilization; this controls the outcome.
axioms (1)
  • domain assumption Sparsity in polynomial coefficients selectively destabilizes non-physical poles without affecting physical ones
    Invoked as the core mechanism enabling elimination of spurious roots from stability diagrams.

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