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Natural frequency estimation using complex-frequency excitations

T0 review · 1 major / 1 minor · reviewed 2026-06-30 · grok-4.3

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Pith's one-line read Complex-frequency excitations increase Fisher information and thus improve natural-frequency estimates compared with harmonic drives in noisy resonators.

desk verdict Closed-form Fisher info for complex-frequency excitations in frequency estimation is the new piece, with experiments showing gains, but the colored-noise derivation needs checking. read the letter →

arxiv 2605.12842 v1 pith:VCXSMYXB submitted 2026-05-13 physics.class-ph cs.ITmath.ITphysics.app-ph

Wenbo Li , Raj Kumar Pal This is my paper · ORCID
classification physics.class-phcs.ITmath.ITphysics.app-ph
keywords naturalfrequencyestimationcomplex-frequencyexcitationFisherinformationmechanicalresonatorparameterGaussiannoiseCramér-Raobound
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper examines how driving an underdamped single-degree-of-freedom oscillator with a signal whose amplitude decays exponentially (a complex-frequency excitation) affects the precision of natural-frequency recovery when measurements contain Gaussian noise. Using the Fisher information as the metric, the authors derive closed-form expressions that relate the information content to the real and imaginary parts of the excitation frequency and to system parameters. Both analytic results and Monte Carlo simulations show that suitable choices of the imaginary part raise the information above the level achieved by a pure sinusoid of the same amplitude. Laboratory experiments on a mechanical resonator confirm that the resulting estimates are both more accurate and more robust to noise. The work therefore supplies a concrete way to lower the Cramér-Rao bound on frequency estimation simply by redesigning the drive waveform.

What carries the argument

Closed-form Fisher-information expressions obtained by differentiating the likelihood of the measured response with respect to the natural-frequency parameter under complex-frequency drive.

What would settle it

A controlled experiment or Monte Carlo trial in which the sample variance of natural-frequency estimates obtained with an optimally chosen complex-frequency drive remains larger than or equal to the variance obtained with a pure sinusoid of identical amplitude and duration.

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Extended reading notes

Core claim

For an underdamped linear time-invariant single-degree-of-freedom spring-mass-damper system subject to additive Gaussian white or colored noise, the Fisher information for the natural frequency is an explicit function of the complex excitation frequency; when the imaginary part is chosen appropriately this information exceeds the value obtained from a conventional real-frequency harmonic excitation, thereby tightening the lower bound on estimation variance.

Load-bearing premise

The physical system is exactly an underdamped linear time-invariant single-degree-of-freedom oscillator whose response to the prescribed decaying drive is observed in additive Gaussian noise.

Editorial extensions

If this is right

  • Optimal imaginary parts of the excitation frequency exist that maximize information for given noise color and system damping.
  • The same framework supplies explicit expressions for both white and colored noise, allowing direct comparison of estimation bounds.
  • Experimental demonstrations on a mechanical resonator show measurable gains in both bias and variance of the recovered frequency.
  • The approach provides a waveform-level route to higher sensor performance without hardware changes.
  • The derived information formulas can be used to select drive parameters before any measurement is taken.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same information-theoretic gain may appear in other resonant systems whose governing equations remain linear and time-invariant.
  • Parameter estimation for damping ratio or mass could be examined under the same complex-frequency drive to check for analogous improvements.
  • If the noise is non-Gaussian the closed-form expressions would no longer hold, but numerical evaluation of the Fisher information could still be performed.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 1 minor

Summary. The paper claims that complex-frequency excitations (decaying complex exponentials) applied to an underdamped LTI SDOF spring-mass-damper system yield higher Fisher information for natural-frequency estimation than conventional harmonic excitations. It derives explicit closed-form FI expressions for both white and colored Gaussian noise, verifies them via Monte Carlo simulations showing improved parameter-estimate variance with optimized excitation parameters, and reports experimental results confirming gains in accuracy and robustness.

Significance. If the closed-form FI expressions hold, the work supplies a quantitative design tool for choosing excitation parameters to maximize estimation precision in noisy mechanical systems. The combination of analytic results, Monte Carlo verification, and hardware experiments provides a concrete basis for improved sensors and NDE techniques.

major comments (1)
  1. [Derivation of Fisher information (likely §3 or §4)] The central claim rests on the correctness of the closed-form FI expressions for colored noise under non-stationary complex-frequency excitation. Because the observed signal is non-stationary and the noise covariance must be inverted over a finite window, an algebraic error in the quadratic form would invalidate the claimed enhancement relative to harmonic excitation. Monte Carlo results only indirectly support the formula via achieved variance and do not rule out such an error.
minor comments (1)
  1. [Abstract] The abstract states that the expressions relate FI to excitation and system parameters but does not list the precise assumptions on the observation interval or the form of the colored-noise covariance.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful review and constructive comments on our manuscript. We address the major comment below.

read point-by-point responses

  1. Referee: [Derivation of Fisher information (likely §3 or §4)] The central claim rests on the correctness of the closed-form FI expressions for colored noise under non-stationary complex-frequency excitation. Because the observed signal is non-stationary and the noise covariance must be inverted over a finite window, an algebraic error in the quadratic form would invalidate the claimed enhancement relative to harmonic excitation. Monte Carlo results only indirectly support the formula via achieved variance and do not rule out such an error.

    Authors: Thank you for this comment. The derivation in Section 4 explicitly constructs the non-stationary mean signal vector under complex-frequency excitation and forms the quadratic term using the exact inverse of the finite-window covariance matrix of the colored Gaussian noise. The resulting closed-form FI is therefore specific to the non-stationary case. The Monte Carlo study shows that the empirical estimator variance reaches the Cramér-Rao bound computed from this FI expression; an algebraic error in the quadratic form would produce a systematic mismatch between simulated variance and the predicted bound, which is not observed. This constitutes direct numerical verification of the formula rather than indirect support. We therefore see no need to alter the claimed enhancement. revision: no

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity in derivation of Fisher information expressions

full rationale

The paper derives explicit closed-form Fisher information expressions relating FI directly to excitation parameters, system parameters, and noise covariance for both white and colored Gaussian noise on an LTI SDOF system. These follow from standard application of the FI definition (score function or quadratic form of inverse covariance) to the known deterministic response under complex-frequency excitation. No steps reduce by construction to fitted inputs renamed as predictions, self-definitional loops, or load-bearing self-citations. Monte Carlo verification and experiments are downstream checks, not part of the analytic derivation itself. The central claim of enhanced FI relative to harmonic excitation rests on these independent closed-form results evaluated at different parameter choices.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; all assessments are therefore preliminary.

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Cite this review

Pith. "Pith review of Natural frequency estimation using complex-frequency excitations." pith.science (2026). https://pith.science/paper/VCXSMYXB

@misc{pith2026260512842,
  author       = {Pith},
  title        = {Pith review of: Natural frequency estimation using complex-frequency excitations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VCXSMYXB}},
  note         = {Machine review of arXiv:2605.12842}
}
read the original abstract

Complex frequency excitations, oscillating signals whose amplitude decreases exponentially in time, have recently been demonstrated to significantly increase the effective quality factor of mechanical resonators. In this work, we investigate the accuracy of natural frequency estimation in mechanical systems under noise using such excitations. The analysis is performed on an underdamped linear time-invariant single-degree-of-freedom spring-mass-damper system. We employ tools from information theory, namely Fisher information, to systematically quantify the sensitivity of complex-frequency excitation to measurement noise. Explicit closed-form expressions are derived relating Fisher information to excitation and system parameters under both Gaussian white and colored noise. The theoretical predictions are verified through Monte Carlo numerical simulations. The results indicate that appropriate selection of excitation parameters can significantly enhance the Fisher information, leading to improved estimation accuracy under complex-frequency excitations compared with conventional harmonic excitations. Experimental results demonstrate the advantages of complex-frequency excitation in terms of both accuracy and robustness of natural-frequency estimation. These findings establish a foundation for the development of high-performance sensors and novel nondestructive evaluation methods.

Figures

Figures reproduced from arXiv: 2605.12842 by the authors.

Figure 1
Figure 1. Basic principle of the complex-frequency excitation framework. An operator [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Three cases of transient-response effects on the Fisher information of the system under [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Dependence of the equivalent Fisher information ratio [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Dependence of the equivalent Fisher information ratio [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: Dependence of the equivalent Fisher information ratio [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: RI obtained from simulations and theory for the case Ω ≪ ωnζ. The simulation results agree closely with the theoretical predictions. With all other parameters held constant, increasing Ω leads to a more pronounced amplification of noise, thereby resulting in a decrease…
Figure 9
Figure 9. Figure 9: Experimental setup. A prescribed excitation is applied near the clamped end of an [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]

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