{"id":"2ef55368-44c5-4b1c-80f9-0929afd200cb","arxiv_id":"2605.12842","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Complex-frequency excitations increase Fisher information and improve natural frequency estimation accuracy in underdamped SDOF systems compared to harmonic excitations under Gaussian noise.","lead":"This paper examines using complex-frequency excitations (oscillating signals with exponentially decaying amplitude) to improve natural frequency estimation accuracy in noisy mechanical systems via Fisher information analysis. A smart generalist might read it for potential gains in sensor performance and nondestructive evaluation methods.","discovery_kind":"new_application","skeptic_critique":{"model":"grok-4.3","headline":"Correctness of the derived closed-form Fisher information expressions under colored noise for non-stationary complex-frequency excitations","rationale":"The reader’s weakest assumption already flags that the FI expressions must hold for the model. The concern above isolates the single most load-bearing step inside that assumption—the analytic derivation for the colored-noise, non-stationary case—and supplies a direct numerical check that would falsify it without requiring new experiments.","tokens_in":1662,"tokens_out":339,"duration_ms":36881,"concrete_test":"For a chosen set of system parameters (ω₀, ζ, noise PSD) and one complex-frequency excitation (σ, ω), compute the numerical Fisher information directly from the definition—either via the score variance or the expected Hessian of the log-likelihood under the exact Gaussian colored-noise model—and compare the value to the paper’s closed-form expression; disagreement larger than 5 % indicates the analytic result does not hold.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires that the explicit closed-form FI expressions (for both white and colored noise) are accurate when the excitation is a decaying complex exponential. The system response remains LTI, but the time-varying amplitude makes the observed signal non-stationary; the colored-noise case further requires the exact inverse or quadratic form of the noise covariance over the finite observation window. Monte Carlo simulations confirm that parameter estimates improve with chosen excitation parameters, yet this only indirectly supports the FI formula itself (via achieved variance) and does not rule out an algebraic error in the analytic expression. If the closed-form result is incorrect, the claimed enhancement relative to harmonic excitation cannot be trusted.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","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.","tokens_in":1798,"tokens_out":342,"duration_ms":8163,"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":[{"comment":"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.","section":"Derivation of Fisher information (likely §3 or §4)"}],"minor_comments":[{"comment":"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.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful review and constructive comments on our manuscript. We address the major comment below.","responses":[{"response":"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_made":"no","referee_comment":"[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."}],"tokens_in":1237,"tokens_out":295,"duration_ms":25441,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper derives explicit closed-form Fisher information expressions for natural frequency estimation on a single-degree-of-freedom damped oscillator driven by a decaying complex exponential, and reports that suitable parameter choices raise the information above what a standard harmonic drive achieves. Monte Carlo trials and lab measurements are included to support the claim.\n\nThe closed forms for both white and colored noise are the actual novelty here; earlier work on these excitations focused on quality-factor improvement, not on Fisher-information bounds for parameter recovery. The experimental section is useful because it moves past simulation to show measurable accuracy and robustness gains under real conditions.\n\nThe main soft spot is the colored-noise case. The excitation produces a non-stationary response, so the Fisher information involves the quadratic form with the inverse noise covariance over a finite window. Monte Carlo results confirm that the chosen excitations yield lower variance estimates, but that only validates the overall procedure, not the algebra inside the closed-form expression itself. An error there would undermine the direct comparison to harmonic drive. The white-noise derivation looks less exposed to this issue.\n\nThe work targets researchers in vibration sensing, nondestructive evaluation, and information-theoretic signal processing. Anyone who needs quantitative guidance on excitation design for frequency estimation will find concrete expressions and data to examine. It is worth sending for peer review because the combination of analytic results, simulations, and experiments is substantive enough to repay referee time, even if the colored-noise formula receives extra attention during review.","headline":"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.","tokens_in":2249,"tokens_out":363,"would_cite":false,"duration_ms":16428,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Complex-frequency excitations increase Fisher information and thus improve natural-frequency estimates compared with harmonic drives in noisy resonators.","keywords":["natural frequency estimation","complex-frequency excitation","Fisher information","mechanical resonator","parameter estimation","Gaussian noise","Cramér-Rao bound"],"falsifier":"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.","tokens_in":2558,"feed_emoji":"📊","tokens_out":703,"duration_ms":22956,"temperature":0.7,"pith_summary":"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.","feed_headline":"Decaying oscillations raise Fisher information for resonator frequency","feed_subtitle":"Closed-form analysis and experiments show complex-frequency drives outperform sinusoids for natural-frequency recovery under Gaussian noise.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Complex-frequency excitations raise Fisher info for natural frequency","Fisher info for natural frequency higher with complex excitations","Complex excitations raise Fisher information over harmonic ones","Natural frequency Fisher info increases under complex-frequency excitation"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Complex-frequency excitations raise Fisher info for natural frequency","Fisher info for natural frequency higher with complex excitations","Complex excitations raise Fisher information over harmonic ones","Natural frequency Fisher info increases under complex-frequency excitation"]},"model":"grok-4.3","cost_usd":0.009529,"raw_usage":{"total_tokens":4225,"prompt_tokens":612,"num_sources_used":0,"completion_tokens":56,"cost_in_usd_ticks":95287000,"prompt_tokens_details":{"text_tokens":612,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3557,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":612,"tokens_out":56,"duration_ms":28279,"temperature":1.0,"reasoning_tokens":3557,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-30T21:51:58.152456+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"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.","supporting_citations":[],"review_version":1}