20-EFT.WP.Metrology.TimeBase v1.0 | Chapter 7 — Phase/Time Noise and the Allan Family

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Chapter 7 — Phase/Time Noise and the Allan Family

One-line objective: Unify the representations of phase/time/frequency noise and the computation gauges for the Allan family (ADEV/MDEV/TDEV/HDEV)—including spectral type identification and uncertainty publication—and align them with the published time base ts, servo bandwidth, and the latency-budget loop.

I. Scope & Targets

1. Covered
   • Transformations and bandwidth-normalized gauges among phase-noise PSD S_phi(f), time-error series x(t)=TE(t), and fractional frequency y(t).
   • The Allan family—ADEV (with overlap), MDEV, TDEV, HDEV—definitions, estimators, and selection of tau tiles.
   • Noise-type discrimination across white PM / flicker PM / white FM / flicker FM / random-walk FM using slope criteria and segmented fits.
   • Uncertainty and equivalent degrees of freedom nu_eff( tau ): estimation and publication.
2. Inputs
   Phase or time-error samples x(t_k) (aligned to tau_mono), or fractional frequency y(t_k), or phase-noise spectrum S_phi(f) with bandwidth [f1,f2].
3. Outputs
   adev(tau_grid), mdev(tau_grid), tdev(tau_grid), hdev(tau_grid); noise-type spectrum and breakpoints; U = k * u_c and nu_eff; manifest.time.noise.*.

II. Terms & Symbols

1. Relationships among time / phase / frequency
   • x(t): time error (s); y(t) = d x(t) / d t: fractional frequency; phi(t): phase (rad); carrier f0 (Hz).
   • PSD relations: S_y(f) = ( 2 * pi * f )^2 * S_x(f ); S_phi(f) = ( 2 * pi * f0 )^2 * S_x(f ); hence S_y(f) = ( f^2 / f0^2 ) * S_phi(f).
2. Allan family notation
   • adev( tau ) = sqrt( Avar( tau ) ); mdev( tau ); tdev( tau ); hdev( tau ).
   • Sampling primitive tau0, averaging factor m = tau / tau0, sample count N, overlapped windows.
3. Units
   unit(adev)="1", unit(mdev)="1", unit(hdev)="1", unit(tdev)="s"; check_dim(expr)=pass is a pre-publication prerequisite.

III. Axioms P507- **

• P507-1 (Explicit measure & bandwidth): Any spectral integral or bandwidth-normalized statistic must declare [f1,f2], the measure df, and the grid tau_grid.
• P507-2 (Domain & dimensional integrity): Use the baseline conversions in this chapter for x/y/phi, and run check_dim before publication.
• P507-3 (Overlap preferred): Use overlapped Allan estimators whenever possible to increase nu_eff, and publish nu_eff( tau ).
• P507-4 (Consistent detrending): When deterministic drift exists, default to HDEV or first-order detrended ADEV; record the procedure in the manifest.
• P507-5 (Dual-form records): If ADEV/TDEV derives from an arrival-time path, record both T_arr forms and delta_form.
• P507-6 (Windows & alignment): Evaluate windows on tau_mono, publish on ts, and attach offset/skew/J (see Chapter 6).

IV. Minimal Equations S507- **

• S507-1 (ADEV base form—frequency domain)
  For block-averaged fractional frequency ȳ_k( tau ):

Avar( tau ) = ( 1 / 2 ) * E[ ( ȳ_{k+1}( tau ) - ȳ_k( tau ) )^2 ]

Sample estimator (overlapped):

hat{Avar}( tau ) = ( 1 / ( 2 * (N - 2m + 1) ) ) * ( ∑_{k=1}^{N-2m+1} ( ȳ_{k+m} - ȳ_k )^2 )。

• S507-2 (MDEV and TDEV)

mdev^2( tau ) = ( 1 / 2 ) * E[ ( ( 1 / m ) * ∑_{i=0}^{m-1} ( ȳ_{k+1+i} - ȳ_{k+i} ) )^2 ]

tdev( tau ) = ( tau / sqrt(3) ) * mdev( tau )。

• S507-3 (HDEV)

hdev^2( tau ) = ( 1 / 6 ) * E[ ( ȳ_{k+2}( tau ) - 2 * ȳ_{k+1}( tau ) + ȳ_k( tau ) )^2 ]。

• S507-4 (Spectral-to-time-domain jitter integral)

J_rms^2 = ( 1 / ( 2 * pi * f0 )^2 ) * ( ∫_{f1}^{f2} S_phi(f) d f ) (consistent with Chapter 6)。

• S507-5 (Equivalent degrees of freedom & intervals)

nu_eff( tau ) = nu_ovl( N, m ) (overlap approximation);

CI_adev( tau ) = [ adev / sqrt( chi2_{up} / nu_eff ), adev / sqrt( chi2_{low} / nu_eff ) ]。

Publish the approximation used for nu_eff and its parameters.

• S507-6 (Phase/time/frequency conversions)

S_x(f) = S_phi(f) / ( ( 2 * pi * f0 )^2 )，S_y(f) = ( 2 * pi * f )^2 * S_x(f)。

V. Computation Flow M50-7 (Align → Preprocess → Estimate → Classify → Publish)

• Align & ready
  Resample/align x(t_k) or phi(t_k) to tau_mono; declare tau0 and tau_grid.
• Variable transforms & detrending
  Derive y(t) from x(t), or obtain S_x/S_y from S_phi; apply linear de-drift by policy or adopt HDEV.
• Allan-family estimation
  Compute overlapped adev/mdev/tdev/hdev; report nu_eff( tau ) and U = k * u_c.
• Noise-type discrimination
  Fit piecewise lines in log tau – log adev space; infer slopes, types, and breakpoints.
• Cross-checks
  Cross-validate with spectral jitter J_rms, Chapter 6’s MTIE/TIE, and servo residual spectra.
• Persistence & publication
  Write manifest.time.noise.*: tau_grid, Allan-family curves, nu_eff, noise-type spectrum, [f1,f2], alignment/detrend policy, and signature.

VI. Contracts & Assertions

• C50-71 (Window coverage): for each tau, nu_eff( tau ) ≥ nu_min, N ≥ N_min( m ).
• C50-72 (Dimensional integrity): unit/dimension checks pass; only tdev carries unit="s".
• C50-73 (Bandwidth bridge): J_rms from S_phi vs. time-domain tdev agree within tol_jitter_bridge at tau_ref.
• C50-74 (Noise-type confidence): slope confidence conf_slope ≥ conf_min; otherwise, publish metrics only (no type labels).
• C50-75 (Dual-form difference): if a path is involved, delta_form ≤ tol_Tarr and persisted.
• C50-76 (Transparent detrend): when de-drift or HDEV is used, publish policy tags and require residual-test pval ≥ pval_min.

VII. Implementation Bindings I50-7*

• phase_to_time(phi, f0) -> x
• time_to_freq(x, tau0) -> y
• compute_adev(y, tau0, tau_grid, overlap=True) -> {adev, nu_eff}
• compute_mdev_tdev(y, tau0, tau_grid) -> {mdev, tdev, nu_eff}
• compute_hdev(y, tau0, tau_grid) -> {hdev, nu_eff}
• integrate_phase_noise(S_phi, f_band, f0) -> J_rms
• fit_noise_slopes(log_tau, log_adev, policy) -> {segments, types, breakpoints, confidence}
• compose_uncertainty_allan(adev, nu_eff, k) -> {CI, U}
• emit_noise_manifest(results, meta) -> manifest.time.noise
• Invariants: unique(TraceID); tau_grid monotone; nu_eff ≥ nu_min; check_dim(expr)=pass; f_band matches the persisted fields.

VIII. Cross-References

• With servos/synchronization: Chapter 5 (servo residuals & bandwidth).
• With offset/skew/jitter: Chapter 6 (bridging J_rms/TIE/MTIE/TDEV).
• With timestamp chains: Chapter 4 (how bandwidth and delay budgets shape the gauges).
• With statistical intervals & resampling: Methods.CrossStats v1.0 Chapters 4–5 (CI and bootstrap).
• With cleaning & publication: Methods.Cleaning v1.0 Chapters 5, 10 (manifests & contract gates).

IX. Noise Types & Slope Criteria (Table-free highlights)

1. Slopes of log10(adev) vs. log10(tau) for type discrimination:
   • white PM: slope ≈ −1 (MDEV slope steeper, ≈ −3/2, separating it from flicker PM).
   • flicker PM: ADEV slope ≈ −1, MDEV slope ≈ −1.
   • white FM: ADEV slope ≈ −1/2.
   • flicker FM: ADEV slope ≈ 0 (plateau).
   • random-walk FM: ADEV slope ≈ +1/2.
2. Recommended workflow: coarse-type with ADEV, refine PM-class with MDEV, then apply HDEV to suppress residual drifts.

Summary