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REVIEW 4 major objections 6 minor 23 references

A high-sensitivity frequency counter for free-induction-decay signals

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read An FPGA frequency counter extracts pulsed-signal frequencies with better than 100 µHz/√Hz sensitivity, matching offline nonlinear fitting.

desk verdict A solid FPGA frequency counter for FID signals that matches offline LM fitting; the robustness claims are the main caveat. read the letter →

arxiv 2506.04780 v1 pith:RVQFHIZI submitted 2025-06-05 physics.ins-det physics.atom-ph

Tong Gong , Ming-Rui Shu , Jiang He , Kai Liu , Yi-Ren Li , Xin-Jun Hao , Dong Sheng , Yu-Ming Wang
show 1 more author
Yu-Kun Feng
This is my paper
classification physics.ins-detphysics.atom-ph
keywords frequencycounterHilberttransformfree-induction-decayatomicmagnetometerFPGAreal-timeextractionsensitivitylinearregression
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 aims to show that a frequency counter built around a Hilbert-transform linear-regression (HT-LR) algorithm can read out the oscillation frequency of pulsed free-induction-decay (FID) signals in real time, at a sensitivity that matches offline nonlinear fitting. Tested on synthetic damped oscillations between 10 kHz and 500 kHz, the counter reports better than 100 µHz/√Hz sensitivity at 10 Hz when outputting at 200 Hz, and stays below 400 µHz/√Hz at a 1000 Hz output rate. The motivation is that FID atomic magnetometers convert magnetic-field measurements into frequency measurements, so a fast, precise, and algorithm-agnostic frequency readout would make these magnetometers practical for field use. The scheme requires no pre-knowledge of the signal's analytic form, which distinguishes it from fitting-based approaches.

What carries the argument

The engine of the work is the Hilbert-Transform Linear-Regression (HT-LR) algorithm: a discrete Hilbert transform truncated at $K=20$ converts the real signal $x[n]$ into its quadrature $y[n]$, forming the analytic signal $z[n]=x[n]+iy[n]$; the instantaneous phase $\varphi[n]=\arctan(y[n]/x[n])$ is unwrapped into a monotone cumulative phase $\Phi[n]$, and a weighted linear fit of $\Phi[n]$ versus time yields the frequency as its slope. The weights are the instantaneous amplitudes $|z[n]|$, which suppress low-SNR late-time samples. The hardware implementation uses an FPGA with a dual-core ARM processor and an 18-bit ADC at 1.53846 MSa/s, referenced to a 50 ppb oven-controlled crystal oscillator; one core acquires data while the other processes it. The truncation parameter $K$ trades computation time against precision, and $K=20$ keeps processing faster than an offline Levenberg-Marquardt fit of the same data.

What would settle it

Feed the counter a signal with two close-frequency damped tones, an amplitude drift, or a different relaxation time, or a real FID magnetometer output, and measure the 10 Hz noise spectral density; if the sensitivity rises above 100 µHz/√Hz or departs from Levenberg-Marquardt fitting of the same data, the 'no analytic-form pre-knowledge' claim fails for realistic envelopes.

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Extended reading notes

Core claim

The central claim is that the HT-LR algorithm, implemented in an FPGA-based counter, turns nonlinear frequency fitting into a linear phase regression without losing accuracy. For each pulse, the device computes the discrete Hilbert transform of the acquired samples to build an analytic signal, extracts the unwrapped instantaneous phase, and performs a weighted linear regression of that phase versus time; the slope is the oscillation frequency, with sample amplitudes as weights to downweight the noisy tail of the decay. On 2.5 ms damped sinusoidal test pulses between 10 kHz and 500 kHz, the extracted frequency's noise spectral density at 10 Hz is better than 100 µHz/√Hz at a 200 Hz output rate, comparable to Levenberg-Marquardt fitting of the same data offline. The paper further shows that the truncation parameter $K$ of the Hilbert transform introduces a systematic frequency offset one order of magnitude smaller than typical heading errors of geomagnetic FID magnetometers, and that the weighted fit improves sensitivity by 15–40% in simulation depending on gate time.

Load-bearing premise

The counter's tested signals are a single synthetic damped sinusoid with fixed amplitude and fixed 2.5 ms decay time, so the claimed sensitivity assumes real FID signals resemble that ideal pulsed tone.

Editorial extensions

If this is right

  • The counter's 100 µHz/√Hz floor at 200 Hz output corresponds to about 10 fT/√Hz for 87Rb magnetometers, bringing real-time field sensitivity to that level.
  • The chosen truncation K=20 keeps per-pulse processing faster than an offline LM fit of the same data on a high-end CPU, while adding a systematic frequency offset one order of magnitude below typical heading errors.
  • Increasing the output rate to 1000 Hz sacrifices less than a factor of four in sensitivity, which remains below 400 µHz/√Hz.
  • Within the 10–500 kHz band, HT-LR reaches the same sensitivity as an ideal 20 ps TDC Omega-counter, but with simpler hardware.

Reading between the lines

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

  • Because the tests use a single damped tone from a signal generator, a natural next check is a real atomic magnetometer's FID output, where amplitude drift, detuning, and multi-component spectra would test the 'no analytic-form pre-knowledge' claim in the field.
  • The weighted fit was evaluated only in simulation, showing 15–40% improvement depending on gate time; a field implementation could verify whether real envelope noise follows the same weighting benefit.
  • The truncation-induced systematic frequency offset is nearly constant across pulses, so gradiometric or differential measurements that subtract two channels might cancel it; this is a testable consequence not explored in the paper.
  • The same phase-linear-regression idea could be extended to multi-frequency FID signals by first filtering into bands, since a weighted linear fit of unwrapped phase is model-free per band.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The manuscript reports an FPGA-based frequency counter that implements a Hilbert-transform linear-regression (HT-LR) algorithm for extracting the frequency of pulsed exponential-decay oscillations. The authors measure the noise spectral density (NSD) of the extracted frequency using function-generator signals with fixed amplitude 2.5 V and relaxation time 2.5 ms over 10 kHz to 500 kHz, reporting better than 100 µHz/√Hz at 10 Hz for a 200 Hz output rate and better than 400 µHz/√Hz at 1000 Hz. They compare the results with offline Levenberg-Marquardt fitting on the same data, study the effect of the Hilbert-transform truncation order K and of clock stability, and propose a weighted linear-regression variant tested in simulation. The conclusion is that the counter is suitable for FID atomic magnetometers and other pulsed-frequency applications.

Significance. If the reported sensitivity numbers hold, the HT-LR counter is a useful real-time alternative to offline nonlinear fitting for FID-type signals, with modest hardware requirements. The direct experimental comparison with the LM algorithm on identical recorded data is a strength, and the clock-stability check adds practical value. The main limitation is that the experimental evidence covers only one synthetic signal class with fixed amplitude and relaxation time; the weighted-fit robustness improvement is simulation-only, and the comparison with Ω-counters is dimensionally unclear. These gaps do not invalidate the central measurement, but they do limit the breadth of the conclusions as currently stated.

major comments (4)
  1. [Sec. III, Fig. 2 and text around "We choose K=20"] The truncation order K is selected using the same data set that is then used to report the headline sensitivity values. Because Fig. 2(b) shows that the NSD is still varying with K in the vicinity of K=20 for some of the tested frequencies, the reported <100 µHz/√Hz value may depend on this in-sample choice. The authors should either provide an out-of-sample rule for selecting K or present a sensitivity analysis showing that the headline number is stable over a range of K values.
  2. [Sec. III, Fig. 6 and accompanying text] The claimed 15-40% improvement from weighted fitting is demonstrated only on simulated white-noise data and is not verified experimentally. Because this improvement is presented as a robustness result with a quantitative range, the authors should confirm it on measured signals (for example, by adding controlled noise to the function-generator output) or explicitly label it as a simulation-based prediction that has not yet been experimentally validated.
  3. [Sec. III, Eq. (5) and the following comparison] The comparison with the Ω-counter is dimensionally inconsistent as written. Equation (5) evaluates to a frequency error in hertz (about 277 µHz for f=250 kHz, T=2.5 ms, δT=20 ps), not to a noise spectral density in Hz/√Hz. To obtain the quoted 20 µHz/√Hz one must apply an additional conversion factor, approximately √T, and this step is omitted. The comparison should be re-derived with consistent units or the claims should be restated accordingly.
  4. [Abstract and Secs. I and IV] The claim that the method "does not require the pre-knowledge of the analytic expression of the input signals" is broader than the evidence supports. All experimental tests use the single signal class A e^{-t/τ} sin(2π f t) with τ=2.5 ms and A=2.5 V; no tests cover DC offsets, non-exponential envelopes, detuning, multiple frequency components, or correlated noise. The abstract and conclusion should be restricted to the tested signal class, or additional experiments covering these cases should be reported.
minor comments (6)
  1. [Sec. II] The text contains the typo "Field-Prgrammable Gate Array"; it should read "Field-Programmable Gate Array."
  2. [Sec. III] The word "determins" should be "determines" in the sentence about crystal oscillator stability.
  3. [Fig. 2 caption] The caption refers to a CPU model "i9-139000"; this should be "i9-13900" to match the text.
  4. [Fig. 2(b) axis] The axis label "NSD (10^-5 Hz/Hz^1/2 @ 10 Hz)" would be clearer as "NSD (10^-5 Hz/√Hz @ 10 Hz)" to avoid confusion between Hz^1/2 and √Hz.
  5. [Abstract and Sec. III] The phrase "frequency range of 10 to 500 kHz" should be written as "10 kHz to 500 kHz" for consistency and clarity.
  6. [Sec. II, Eqs. (1) and (3)] The truncation in Eq. (3) should state explicitly that the sum runs over odd values of k, as the surrounding text implies, so that the reader can reproduce the finite-sum implementation.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: HT-LR is an independent phase-based estimator benchmarked against LM on identical recorded data, with no load-bearing self-citation or fitted-input-as-prediction step.

full rationale

The derivation is self-contained. The HT-LR algorithm estimates frequency as the slope of the unwrapped instantaneous phase obtained from a discrete Hilbert transform, with a weighted linear regression whose weights are the instantaneous amplitudes; none of these quantities is defined in terms of the LM benchmark or the Omega-counter limit. The truncation parameter K is selected by a convergence/efficiency trade-off shown in Fig. 2(a)-(c), not by matching the LM output, and the same recorded data are then independently processed offline by LM for comparison (Sec. III). The Omega-counter comparison is an analytic theoretical reference (Eq. 5) with stated assumptions, not a fitted prediction. No load-bearing self-citation appears: the cited prior Hilbert-transform work [16], [17], [20] is external and does not supply the FPGA implementation or the weighted-fit extension. The fact that robustness is demonstrated only on simulated signals of the same exponential form is a limitation on external validity, but it is not circular because the weighted fit is evaluated against LM on fresh simulated noise realizations and is not constructed to reproduce LM outputs. Therefore no circular step is present.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard signal-processing identities and on the adequacy of the synthetic test signals. No free parameters other than K are fitted; the algorithm does not use the target frequency as an input. No new physical entities are introduced.

free parameters (1)
  • K (Hilbert transform truncation order) = 20
    Chosen in Sec. III as a trade-off between computation time and precision based on Fig. 2; the systematic frequency offset from LM varies by up to about 0.8 Hz with K.
assumptions (5)
  • standard math Discrete Hilbert transform (Eq. 1) is a valid approximation for the band-limited FID signals
    From Oppenheim [19]; the paper truncates the sum at K=20 (Eq. 3).
  • domain assumption FID signal is a single-frequency damped sinusoid with slowly varying amplitude
    Stated in Sec. III test model; the algorithm targets this signal class.
  • domain assumption Weighting the phase linear fit by instantaneous amplitude improves frequency estimation
    Validated in simulation (Fig. 6); no theoretical derivation given.
  • domain assumption OCXO frequency reference (50 ppb) does not limit the counter sensitivity
    Supported by Fig. 3, which shows no change from 50 ppb to 5 ppb.
  • domain assumption The theoretical Ω-counter sensitivity (Eq. 5) with δT=20 ps is a fair benchmark
    From Ref [22-23]; used for the comparison in Sec. III.

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Cite this review

Pith. "Pith review of A high-sensitivity frequency counter for free-induction-decay signals." pith.science (2026). https://pith.science/paper/RVQFHIZI

@misc{pith2026250604780,
  author       = {Pith},
  title        = {Pith review of: A high-sensitivity frequency counter for free-induction-decay signals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RVQFHIZI}},
  note         = {Machine review of arXiv:2506.04780}
}
read the original abstract

Real-time frequency readout of time-dependent pulsed signals with a high sensitivity are key elements in many applications using atomic devices, such as FID atomic magnetometers. In this paper, we propose a frequency measurement algorithm based on the Hilbert transform and implement such a scheme in a FPGA-based frequency counter. By testing pulsed exponential-decay oscillation signals in the frequency range of 10 to 500 kHz, this frequency counter shows a frequency sensitivity better than 0.1 mHz/Hz^(1/2) at 10 Hz, with an output rate of 200 Hz. When the output rate is increased to 1000 Hz, the sensitivity remains better than 0.4 mHz/Hz^(1/2) at 10 Hz. The performance on frequency sensitivity is comparable with results obtained by off-line nonlinear fitting processes. In addition, this frequency counter does not require the pre-knowledge of the analytic expression of the input signals. The realization of such a device paves the way for practical applications of highly-sensitive FID atomic magnetometers.

Figures

Figures reproduced from arXiv: 2506.04780 by the authors.

Figure 1
Figure 1. (a) Schematic diagram of the frequency counter. (b) A picture of the [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Comparison of frequency counter results at different signal frequencies [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 2
Figure 2. (a) Computation time of the HT-LR algorithm for different [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: (a) Comparison between the HT-LR algorithm and Levenberg [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 6
Figure 6. Figure 6: NSD of 250 kHz simulated signal processed by three algorithms (gate [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]
Figure 5
Figure 5. Figure 5: NSD of the frequency counter at different output rates. [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

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