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REVIEW 4 major objections 4 minor 31 references

Influence Mechanism of Truncation on Low-Frequency Phase Measurement

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

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Pith's one-line read Dither added before phase truncation suppresses low-frequency phase noise in digital phasemeters, yielding a 9.5 dB improvement at 10 mHz.

desk verdict The 9.5 dB dither improvement is real, but the paper's own equations make phase-truncation spurs vanish at exactly 10 MHz with 80 MHz sampling, so the central attribution is wrong. read the letter →

arxiv 2506.06788 v1 pith:MV3XP25W submitted 2025-06-07 physics.ins-det

Yujie Feng , Yuanze Jiang , Liuyang Chen , Haifeng Chen , Yurong Liang This is my paper
classification physics.ins-det
keywords phasetruncationquantizationnoisedigitalphasemeterlow-frequencyditheringlinearfeedbackshiftregisterDDSspursspacegravitationalwavedetection
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

Digital phasemeters measure tiny phase shifts by comparing an incoming carrier with a locally generated reference, and the reference generator drops low-order phase bits to save memory. The paper tries to establish that this truncation step is not benign when the carrier frequency sits near an integer multiple of the sampling clock: the truncated bits repeat periodically, turning the quantization error from white noise into concentrated spurs that elevate low-frequency phase noise. That noise floor violates the $1.3\,\mathrm{\mu rad/Hz^{1/2}}$ noise-shape requirement for space gravitational wave readout between 2 mHz and 0.1 Hz. The proposed remedy is to add Gaussian dither synthesized from linear feedback shift registers before quantization, which decorrelates the error from the signal. With dither the phase noise improves by 9.5 dB at 10 mHz and meets the 0.1 mHz–1 Hz requirement.

What carries the argument

The load-bearing object is the truncated phase word in the numerically controlled oscillator, quantified by the equivalent tuning word $\mathrm{ETW}=\mathrm{PIR}\bmod 2^B$ and the grand repetition rate $\mathrm{GRR}=2^A/\mathrm{GCD}(\mathrm{PIR},2^A)$. The truncation error behaves as a sawtooth wave with period $T_t=2^B/\mathrm{ETW}$ (or its mirror for $\mathrm{ETW}\ge 2^{B-1}$), which is why the spurs are periodic and alias into the Nyquist band. The suppression machinery is Gaussian dither synthesized from linear feedback shift registers, added to the phase before truncation so the deterministic error no longer correlates with the signal; the dither period must exceed 10,000 seconds to cover the 0.1 mHz band.

What would settle it

With the same test setup, measure a 10 MHz carrier while the numerically controlled oscillator keeps all phase bits (no truncation); if the low-frequency noise floor does not drop to the 10.3 MHz level, phase truncation is not the dominant cause of the excess noise.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is that phase truncation in the numerically controlled oscillator produces a periodic sawtooth error whose spurs alias into the measurement band exactly when the signal frequency and sampling frequency approach an integer ratio. For a 10 MHz signal this concentrates remapped spectral lines at 10 MHz and 30 MHz; the 30 MHz component acts as a $3\omega$ artifact that, after mixing and aliasing, adds a nonlinear term $G\sin(2\varphi_m)$ to the measured phase. Adding Gaussian dither before truncation smooths the sawtooth, broadens the spurs into a white floor, and restores a phase noise of about $1.3\,\mathrm{\mu rad/Hz^{1/2}}\cdot\mathrm{NSF}$ from 0.1 mHz to 1 Hz, matching the readout requirement for space gravitational wave detection.

Load-bearing premise

The experiment's excess noise at 10 MHz is blamed specifically on the internal step of dropping low-order phase bits, and not on other parts of the test chain that also differ between the 10 MHz and 10.3 MHz runs.

Editorial extensions

If this is right

  • Non-integer test frequencies such as 10.3 MHz avoid a real noise mechanism, so swept-frequency carriers that pass through integer ratios will momentarily encounter elevated low-frequency phase noise unless dither is active.
  • An LFSR-based Gaussian dither can be implemented directly in Verilog on the same FPGA, so the suppression costs little additional hardware.
  • Suppressing the truncation spurs also removes the $3\omega$ artifact, eliminating the nonlinear phase error term $G\sin(2\varphi_m)$ from the phase readout.
  • With dither, the measured phase noise floor meets the $1.3\,\mathrm{\mu rad/Hz^{1/2}}\cdot\mathrm{NSF}$ requirement from 0.1 mHz to 1 Hz, the band needed for space gravitational wave interferometry.

Reading between the lines

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

  • If the mechanism is as general as the paper's framing suggests, any direct digital synthesis system whose output frequency approaches an integer submultiple of the clock should show the same spur-concentration effect, so dithering before phase-to-amplitude conversion could serve as a general spur-mitigation recipe.
  • The experimental chain only compares one integer-ratio frequency (10 MHz) with one non-integer frequency (10.3 MHz); sweeping the carrier across the integer-ratio boundary with and without dither would test whether the predicted noise hump tracks the resonance condition.
  • The paper's spur-position calculation uses an illustrative equivalent tuning word ($\mathrm{ETW}=2674$) rather than the actual PIR of the experimental setup; recovering the real register values and re-running the aliasing calculation would quantify how tightly the 10/30 MHz spur prediction is tied to this particular experiment.
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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 / 4 minor

Summary. This manuscript studies quantization and phase-truncation errors in FPGA-based digital phasemeters for space gravitational-wave detection. It distinguishes white-noise conditions for truncation errors from non-white spurs that arise when the signal frequency is near an integer multiple of the sampling rate, explains how spur aliasing and amplitude artifacts degrade low-frequency phase measurements, and proposes suppressing the effect by adding LFSR-synthesized Gaussian dither before phase truncation. The experimental section reports that a 10.3 MHz signal meets the 1.3 µrad/Hz^1/2·NSF requirement from 0.1 mHz to 1 Hz, whereas a 10 MHz signal exceeds the requirement from 2 mHz to 0.1 Hz, and that adding dither improves the 10 MHz phase noise by 9.5 dB at 10 mHz, restoring compliance.

Significance. If substantiated, the manuscript would provide a practical, low-cost mitigation for the integer-ratio phase-noise problem in digital phasemeters, with a concrete LFSR dither implementation and a 10,000 s experimental validation. The qualitative description of spur aliasing and nonlinear phase artifacts is useful and addresses a real issue for space gravitational-wave readout. However, the central attribution of the 10 MHz excess noise to phase truncation is contradicted by the authors' own Eq. (8) for the stated fs=80 MHz and 10 MHz signal, so the claimed mechanism and the 9.5 dB improvement require major clarification and additional evidence.

major comments (4)
  1. [§3.2, Eq. (8), and §4, Fig. 10] The central claim is internally inconsistent with the stated parameters. For fs=80 MHz and a 10 MHz signal, PIR = 2^(A-3) in the NCO, and for any practical accumulator width A≥15 the lower B=12 bits of PIR are zero, so Eq. (8) gives ETW=0. The truncation word is then constant, the sawtooth in Fig. 4 has zero amplitude, and the phase-truncation error is identically zero. Under the paper's own equations, the 10 MHz excess noise shown in Fig. 10 cannot be caused by phase truncation, and the 9.5 dB dither improvement cannot be interpreted as smoothing of truncation error, unless the locked PIR contained an unreported fractional offset. Please report the NCO accumulator width, the actual PIR used in the locked loop, and a simulation of the truncation spur spectrum for the experimental parameters; if the signal was not exactly 10.000000 MHz, state the actual frequency offset.
  2. [§3.2] The ETW=2,674 example is explicitly illustrative and is never derived from the experimental phase accumulator width and PIR. The prediction that aliased truncation spurs are concentrated at 10 MHz and 30 MHz is therefore not quantitatively established for the measured configuration. This is load-bearing because it is the sole basis for attributing the dashed red curve in Fig. 10 to truncation rather than to the AWG, ADC, power splitter, or clock synchronization. The analysis should be rerun with the actual PIR and fs used in the experiment.
  3. [§4, Fig. 10] The experimental evidence for the 9.5 dB improvement is a single differential measurement without repeated runs, error bars, or a control condition that isolates the truncation mechanism, such as disabling phase truncation entirely or testing a frequency with a nonzero ETW that is not an integer multiple. Given the inconsistency identified in the first major comment, the observed improvement could equally be explained by dither acting on other quantization or clock paths. Please add repeated measurements, quantify run-to-run dispersion, and include a control measurement that verifies the absence of truncation-induced noise when ETW=0.
  4. [§4 (dither synthesis) and Eq. (6)] The manuscript does not specify the dither amplitude or standard deviation relative to the LSB, the LFSR length and polynomial used in the experiment, or the criterion for choosing them; without this information the claimed 9.5 dB result cannot be reproduced. In addition, Eq. (6) sets the phase noise density equal to the amplitude noise density of Eq. (4), omitting the |S_n|/|S_0| dependence of Eq. (5); the calculated white-noise floor of 0.03 µrad/Hz^1/2 is therefore not substantiated as stated and should be rederived with the carrier amplitude included.
minor comments (4)
  1. [Fig. 4 caption] The caption states that '1,024 discrete frequencies constitute the truncation spurs' while the complete sequence repeats after GRR=2048; please clarify the relationship between the number of unique spur lines, the GRR, and the Nyquist binning.
  2. [Fig. 10] The legend label '10MHz PTCor' is unclear; please spell out the abbreviation and clearly distinguish the measured curves from the requirement curve 1/2·NSF in the caption or legend.
  3. [Section 2, Eq. (3)] The transfer function H(z) in Eq. (3) is presented without a derivation or reference; please define all loop coefficients and state the assumptions under which the linearized DPLL transfer function is valid.
  4. [Frontmatter] The abstract appears twice at the beginning of the manuscript; please remove the duplicate.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the dither improvement is an experimental comparison against an external requirement, and the truncation-spur model comes from prior external DDS theory.

full rationale

The paper does not derive its central claim by reusing its own inputs. The 9.5 dB improvement after adding LFSR-synthesized Gaussian dither is an experimental result comparing measured phase-noise spectra at 10 MHz with and without dither against the externally stated 1.3 µrad/Hz^1/2·NSF requirement (Fig. 10 and Section 4). The non-white truncation-spur mechanism is imported from external DDS literature (Sripad & Snyder [28], Nicholas & Samueli [30]), and Eq. (8) defines ETW without fitting it to the measured data; the ETW=2674 example is explicitly illustrative (Section 3.2). The only self-citation, Ref. [17] (Feng et al. 2024), appears in a routine list of phasemeter performance improvements and is not load-bearing for the dither claim. A separate consistency concern is that with fs=80 MHz the stated 10 MHz test tone may give ETW=0 in Eq. (8), which would mean phase truncation produces no spurs under the paper's own equations; the paper never reports the NCO PIR or accumulator width used in the experiment, so the attribution of the measured excess noise to truncation is under-supported. That is a correctness/evidence gap, not a circular reduction, and does not raise the circularity score under the rubric.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central empirical result rests on standard quantization-noise theory and a cited DDS spur model; the paper introduces no new physical entities. The quantitative link between the experimental 10 MHz setup and the illustrative truncation-spur example is not established because PIR, accumulator width, and dither amplitude are not reported, so several parameters act as free knobs rather than fixed, stated inputs.

free parameters (3)
  • ETW example value 2674 = 2674 (mod 2^12, B=12)
    Introduced in Section 3.2 to make the sawtooth period 2.88 clock cycles and the aliased spurs land near 10 MHz and 30 MHz. It is not derived from the experimental PIR and phase accumulator width, which are never stated.
  • Dither amplitude or standard deviation = not stated
    The Gaussian dither is added before truncation, but its amplitude relative to the truncation LSB is never given. The reported 9.5 dB improvement and the added white noise above 0.1 Hz depend on this setting.
  • NCO phase accumulator width and PIR = not stated
    Required to compute ETW and GRR for the experimental 10 MHz case. Without these values, the claimed spur mechanism cannot be quantitatively checked against the measurement.
assumptions (4)
  • standard math Quantization noise can be treated as uniformly distributed white noise when the Sripad-Snyder conditions hold.
    Invoked in Section 3.1 after Refs. [24-28]; underpins Eq (4).
  • domain assumption Phase truncation error can be represented as a sawtooth whose period is T_t = 2^B / (ETW or 2^B - ETW), with grand repetition rate GRR.
    Section 3.2, Eqs (7)-(9), follows the Nicholas-Samueli DDS spur analysis [30]; this is the model from which non-white spurs are inferred.
  • ad hoc to paper For the experimental signal, aliased truncation spurs concentrate at 10 MHz and 30 MHz.
    Section 3.2 and Fig. 5. This depends on the illustrative ETW = 2674 example, and the paper does not show it holds for the actual PIR of the 10 MHz measurement.
  • domain assumption The mission requirement is 1.3 µrad/Hz^1/2 times NSF, with NSF = [1 + (6 mHz/f)^4]^1/2.
    Used in Section 4 and Fig. 10 as the acceptance benchmark; it is taken from space gravitational wave detection requirements, not derived in this paper.

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Pith. "Pith review of Influence Mechanism of Truncation on Low-Frequency Phase Measurement." pith.science (2026). https://pith.science/paper/MV3XP25W

@misc{pith2026250606788,
  author       = {Pith},
  title        = {Pith review of: Influence Mechanism of Truncation on Low-Frequency Phase Measurement},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MV3XP25W}},
  note         = {Machine review of arXiv:2506.06788}
}
abstract

Driven by advances in electronic technology, modern digital phasemeters have significantly improved in integration and functionality, enabling real-time measurement and analysis of dynamic signals. High-precision phase measurement is closely associated with the quantization process. This paper specifically analyzes the white and non-white noise characteristics associated with the quantization errors of phase truncation in digital phasemeters. The error can be considered white noise under specific conditions, which power correlates with the resolution of quantizer and is uniformly distributed within the Nyquist frequency. However, when the signal frequency and sampling frequency are close to an integer multiple, the non-white noise caused by truncation can result in low-frequency phase noise. Additionally, artifacts may induce nonlinear phase errors. Introducing Gaussian dither synthesized by LFSRs can smooth the truncation process, thereby mitigating its impacts on phase measurement. The results indicate that for a 10 MHz signal under test, the noise floor of the phasemeter exceeds the requirement from 2 mHz to 0.1 Hz due to the integer multiple. After adding dither, the phase noise was optimized by 9.5 dB at 10 mHz, achieving the requirement of 1.3 $\rm{\upmu rad/Hz^{1/2}} \cdot \rm{NSF}$ from 0.1 mHz to 1 Hz in space gravitational wave detection. This demonstrates that adding dither can effectively suppress the low-frequency phase noise caused by truncation.

Figures

Figures reproduced from arXiv: 2506.06788 by the authors.

Figure 1
Figure 1. Schematic diagram of a DPLL-Based Digital Phasemeter [21]. The signal digitized by the ADC enters the FPGA for phase demodulation. The NCO serves as the frequency synthesizer. The PA adds its output to the PIR to generate reference signals. Phase detection is performed through mixing and filtering. When the loop is locked, the PA and PIR can directly indicate the phase changes of the input signal. The phase data rec… view at source ↗
Figure 3
Figure 3. Complex phasors are used to describe the phase coupling mechanism of additive noise. Quantization noise phasor 𝑆𝑛 cause a phase deviation in the signal phasor 𝑆. is the signal phasor, 𝑆𝑛 is the small noise phasor, resulting in the phasor 𝑆. The angle between 𝑆𝑛 and 𝑆0 is 𝜋∕2 when the influence is maximal, the error can be expressed as: 𝜑er r = 𝜑2 − 𝜑1 ≈ tan(𝜑er r) ≈ |𝑆𝑛 | |𝑆0 | (5) So the phase noise caused by trunc… view at source ↗
Figure 2
Figure 2. The schematic diagram of phase truncation in the NCO illustrates the phase-amplitude conversion process. During this process, the PA adds its output to the PIR. The most significant bits (MSB) of the PA are then truncated and used as addresses in a Look-Up Table (LUT) to generate reference signals. In the phase-amplitude conversion process, quantization errors introduced by truncation result in periodic amplitude er… view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Overflow behavior of truncation word accumulator (Top) and spectrum of truncation spurs (Bottom). The overflow behavior exhibits a sawtooth waveform with a period of 𝑇𝑡 ≈2.88 clock cycles. Additionally, the complete sequence of truncation word values repeats after a pe…
Figure 5
Figure 5. Figure 5: Non-white noise characteristics are observed in the simulation results. When the signal frequency is 10.3 MHz, the spurs generated by truncation are evenly distributed across the Nyquist bandwidth, exhibiting the statistical characteristics of white noise. Conversely, …
Figure 7
Figure 7. Figure 7: The LFSR consists of several registers and XOR gates and can be conveniently implemented using Verilog. The schematic diagram of the Fibonacci LFSR is shown at the top. A Gaussian dither histogram with its fitting curve is presented at the bottom, with the overall prob…
Figure 6
Figure 6. Figure 6: The presence of specific artifacts introduces a nonlinear effect on the phase. The solid blue curve illustrates the linear relationship between the phase measurement results and the true phase change when 𝐺=0. In contrast, the dashed red line represents the nonlinear p…
Figure 8
Figure 8. Figure 8: The signal is generated by AWG1, an arbitrary waveform generator Keysight 33622A. This signal is evenly split into two input paths for phase measurement using a Mini-Circuits ZMSCJ-2-2 power splitter. The prototype phase meter architecture is implemented on the Terasic…
Figure 9
Figure 9. Figure 9: The spectrum analysis of the NCO output simulation results, after adding dither, reveals the following observations: the spectral linewidth of the 10 MHz reference signal has been optimized, leading to enhanced signal clarity and stability. Additionally, any artifacts …

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