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arxiv: 2201.07109 · v8 · pith:N373RJ2Lnew · submitted 2022-01-18 · ⚛️ physics.ins-det · physics.atom-ph· physics.optics· quant-ph

The Companion of Enrico's Chart for Phase Noise and Two-Sample Variances

Enrico Rubiola , Francois Vernotte This is my paper

Pith reviewed 2026-05-24 13:02 UTC · model grok-4.3

classification ⚛️ physics.ins-det physics.atom-phphysics.opticsquant-ph
keywords phase noisefrequency stabilityAllan varianceHadamard variancetwo-sample variancemetrologyfrequency fluctuationsFourier frequency
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The pith

A single reference chart collects the essential formulas and plots linking phase noise to two-sample variances.

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

The paper functions simultaneously as a tutorial on phase noise and frequency stability, a review of core relations among fluctuation measures, and a user guide to a compact chart. It addresses the practical difficulty of working with signals whose instabilities span fourteen orders of magnitude in relative frequency and twelve to fifteen in Fourier frequency or integration time. The chart itself condenses the most frequently needed conversions, spectral shapes, and variance definitions onto one A4 sheet, with the article supplying the surrounding explanations and recent corrections. By making these tools immediately accessible, the work aims to reduce errors that arise when practitioners move between different stability metrics.

Core claim

The central claim is that Enrico's Chart of Phase Noise and Two-Sample Variances, accompanied by this explanatory article, assembles the principal concepts, formulas, and graphical relations that connect phase-noise spectra to Allan, Hadamard, and other two-sample variances, including updated normalization factors, so that users can move rapidly among the measures without consulting scattered sources.

What carries the argument

Enrico's Chart, a single-page reference card that tabulates the functional relations, power-law slopes, and numerical factors connecting phase noise to the family of two-sample variances.

If this is right

  • Practitioners obtain immediate conversion rules between phase-noise spectra at any Fourier frequency and the corresponding Allan or Hadamard variance at any averaging time.
  • The chart supplies the correct scaling factors needed when switching among different two-sample estimators for the same underlying noise process.
  • Users can identify the dominant noise type from the slope observed on either a phase-noise plot or a variance plot without additional derivation.
  • The consolidated presentation supports consistent application of stability metrics across mechanical, electronic, and atomic frequency standards.

Where Pith is reading between the lines

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

  • Similar single-sheet compilations could be constructed for related quantities such as timing jitter or phase-locked-loop transfer functions.
  • Automated software that encodes the chart's conversion rules would allow real-time translation between spectral and variance domains during data acquisition.
  • The wide dynamic-range emphasis points to utility in emerging applications that combine optical and microwave frequency references.

Load-bearing premise

The formulas, plots, and normalization corrections assembled in the chart correctly reproduce the accepted definitions and relations in the field.

What would settle it

A numerical mismatch between the Hadamard-variance normalization factor printed on the chart and the factor obtained by direct integration of the original Hadamard definition against a standard reference implementation.

Figures

Figures reproduced from arXiv: 2201.07109 by Enrico Rubiola, Francois Vernotte.

Figure 1
Figure 1. Figure 1: QR codes to download Enrico’s Chart and this article. [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Enrico’s Chart of Phase Noise and Two-Sample Variances, front side. [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Enrico’s Chart of Phase Noise and Two-Sample Variances, back side. [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Clock signal (1), observed with an oscilloscope trig￾gered by an external noise-free reference. The shadowed (pink) arrows indicate where the signal is stationary, unaffected by noise. The elliptical shadowed (cyan) regions emphasize where noise shows up most. of phase noise and frequency stability after the early ideas. See also [35]. Turning our attention to books and booklets, the Riley handbook [36] is… view at source ↗
Figure 5
Figure 5. Figure 5: Basic phase-noise behavior of components and systems, [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Phase noise mechanisms inside an oscillator. (a) [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: Example of phase noise of an oscillator. The original [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 7
Figure 7. Figure 7: Typical noise spectra occurring in oscillators. (a) Type [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: Evaluation of the overlapped Allan variance. [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Example of MDEV obtained with the SigmaTheta [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Conversion from phase noise to Allan variance. The [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Beat method, applied (a) to electrical signals, and (b) [PITH_FULL_IMAGE:figures/full_fig_p018_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Traditional phase noise analyzer. The mixers, satu [PITH_FULL_IMAGE:figures/full_fig_p020_13.png] view at source ↗
Figure 15
Figure 15. Figure 15: Scheme of the bridge (interferometric) phase noise [PITH_FULL_IMAGE:figures/full_fig_p022_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Digital phase noise and Allan variance analyzer. [PITH_FULL_IMAGE:figures/full_fig_p023_16.png] view at source ↗
Figure 18
Figure 18. Figure 18: Multichannel analog Allan variance analyzer. [PITH_FULL_IMAGE:figures/full_fig_p024_18.png] view at source ↗
read the original abstract

Phase noise and frequency (in)stability both describe the fluctuation of stable periodic signals, from somewhat different standpoints. Frequency is unique compared to other domains of metrology, in that its fluctuations of interest span at least 14 orders of magnitude, from $10^{-4}$ in a mechanical watch to $10^{-18}$ in atomic clocks. The frequency span of interest is some 12-15 orders of magnitude, from $\mu$Hz to GHz Fourier frequency for phase noise, while the time span over which the fluctuations occur ranges from sub-$\mu$s to years integration time for variances. Because this domain is ubiquitous in science and technology, a common language and tools suitable to the variety mentioned are a challenge. This article is at once (1) a tutorial, (2) a review covering the most important facts about phase noise, frequency noise and two-sample (Allan and Allan-like) variances, and (3) a user guide to "Enrico's Chart of Phase Noise and Two-Sample Variances." In turn, the Chart is a reference card collecting the most useful concepts, formulas and plots in a single A4/A-size sheet, intended to be a staple on the desk of whoever works with these topics. The Chart is available under Creative Commons 4.0 CC-BY-NC-ND license from Zenodo, DOI 10.5281/zenodo.4399218. A wealth of auxiliary material is available for free on the Enrico's home page http://rubiola.org. This version includes the corrections for an unfortunate error in the normalization of the Hadamard version, and some corrections for trivial (albeit sometimes subtle) mistakes.

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

0 major / 2 minor

Summary. The manuscript is a tutorial, review, and user guide to 'Enrico's Chart of Phase Noise and Two-Sample Variances.' It compiles established concepts, formulas, and plots on phase noise, frequency noise, and two-sample variances (Allan and Allan-like, including Hadamard), spanning 12-15 orders of magnitude in Fourier frequency and integration times from sub-μs to years. The work explicitly notes and applies corrections to prior normalization errors in the Hadamard variance and makes the chart available under CC-BY-NC-ND from Zenodo (DOI 10.5281/zenodo.4399218), with auxiliary material at rubiola.org.

Significance. If the compilation is accurate, the manuscript offers practical value as a consolidated desk reference and teaching aid in frequency metrology and instrumentation. Credit is due for the explicit identification and correction of Hadamard variance normalization errors and for releasing the chart and supporting material under an open license. No new derivations, data, or predictions are presented; the contribution lies in faithful aggregation of known results across a wide dynamic range.

minor comments (2)
  1. [Abstract] Abstract: the statement that corrections have been applied to the Hadamard variance normalization would be strengthened by a one-sentence description of the nature of the prior error (e.g., the incorrect factor or missing term) so readers can immediately assess the change.
  2. The manuscript would benefit from an explicit, short section or table listing the specific corrections made in this version versus the previous chart, including the affected equations or plots, to increase transparency of the compilation process.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of the manuscript as a consolidated tutorial, review, and desk reference for phase noise and two-sample variances. We appreciate the acknowledgment of the Hadamard variance normalization corrections and the open licensing of the chart and supporting materials. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; explicit compilation of established results

full rationale

The manuscript is presented as a tutorial, review, and user guide compiling formulas, plots, and corrections from prior literature on phase noise and two-sample variances. No novel derivations, predictions, or first-principles claims are advanced that could reduce to self-definition or fitted inputs. Corrections to Hadamard normalization are noted as fixes to existing errors rather than new results. Self-citation of the authors' chart is load-bearing only for the compilation itself, not for any derived claim. The work is self-contained as a reference card and does not generate circularity under the enumerated patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is a review and tutorial paper; no new models, derivations, or empirical claims are made, so no free parameters, axioms, or invented entities are introduced by the work itself.

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