20-EFT.WP.Metrology.TimeBase v1.0 | Appendix D — Metrics & Drift Measures (Time-Base Specific)

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Appendix D — Metrics & Drift Measures (Time-Base Specific)

One-line objective: Standardize the core metrics, drift measures, and computational gauges for time-base & synchronization systems, yielding a unified, auditable, replayable, and benchmarkable metrology stack.

I. Purpose & Scope

• Applies to
  Clocks and sync links at endpoints/edge/core, covering NTP/PTP/SyncE/GNSS/holdover scenarios.
• Outputs
  Metric definitions S50D-*, default windows & filter gauges, threshold recommendations & alert levels, and field mappings to manifest.time.

II. Terms & Symbols

• Time error: TE(t) = ts(t) - ts_ref(t) (a.k.a. offset(t)), unit="[T]".
• Fractional frequency: y(t) = d( TE(t) ) / dt, dim="1".
• Wander–jitter decomposition: TE(t) = W(t) + J(t) where W is low-frequency wander and J is high-frequency jitter.
• Time Interval Error: TIE(t, tau) = TE(t) - TE(t - tau).
• Maximum Time Interval Error: MTIE(tau) = max_{windows} | TIE |.
• Allan family: adev(tau), mdev(tau), tdev(tau) (relations in S50D-4).
• Phase noise / spectra: S_phi(f), S_y(f); banded jitter integral per S50D-6.
• Dual-form arrival: form_const, form_general, delta_form (Appendix C).
• Drift metrics: drift_level, drift_slope, psi (population stability index), W1 (Wasserstein-1).

III. Axioms P50D- **

• P50D-1 (Time-base consistency): Compute all statistics on tau_mono, publish on ts, with unit/dim and window Delta_t attached for offset/skew/J.
• P50D-2 (Explicit filtering gauges): Wander/jitter split must declare F_HP(f_c_hp) and F_LP(f_c_lp) cutoffs and the linear-phase region.
• P50D-3 (Traceable references): ts_ref traces to upstream references (GM/NTP stratum/GNSS); record hop count and steps_removed.
• P50D-4 (Arrival mandatory): Whenever path/propagation metrics appear, record both arrival forms and enforce delta_form ≤ tol_Tarr.
• P50D-5 (Dimensional integrity): Run check_dim(expr) before publishing any metric.

IV. Minimal Equations S50D- **

1. S50D-1 (Time & frequency basics)
   • TE(t) = ts(t) - ts_ref(t)
   • y(t) = d( TE(t) ) / dt (use forward difference or Savitzky–Golay for discrete data)
2. S50D-2 (TIE/MTIE)
   • TIE(t, tau) = TE(t) - TE(t - tau)
   • MTIE(tau) = max_{t} | TIE(t, tau) |
3. S50D-3 (Allan deviation, frequency-domain form)
   • adev(tau) = sqrt( ( 1 / 2 ) * E[ ( y_{i+1} - y_i )^2 ] )
   • mdev(tau) = sqrt( ( 1 / 2 ) * E[ ( \bar{y}_{i+1}^{(m)} - \bar{y}_{i}^{(m)} )^2 ] )
4. S50D-4 (TDEV–MDEV relation)
   tdev(tau) = ( tau / sqrt(3) ) * mdev(tau) with unit(tdev) = "s"
5. S50D-5 (Time-domain wander/jitter split)
   J(t) = HP{ TE(t) ; f_c_hp }, W(t) = LP{ TE(t) ; f_c_lp }, and TE = W + J
6. S50D-6 (Spectral jitter, optional with carrier)
   J_rms = ( 1 / ( 2 * pi * f_0 ) ) * sqrt( ∫_{f1}^{f2} S_phi(f) df )
7. S50D-7 (Drift & distribution stability)
   • drift_slope = slope( W(t) ) (robust regression)
   • psi = ∑_i ( p_i - q_i ) * ln( p_i / q_i ) (reference vs. current histogram)
   • W1 = ( ∫ | F_ref(x) - F_cur(x) | dx )

V. Computational Gauges & Windows

• Sampling & windows
  Base TauSet = {1 s, 10 s, 100 s, 1000 s}; sliding step Δt_step = tau / 2.
• Filtering
  Jitter HP cutoff f_c_hp ∈ [0.1, 1] Hz; wander LP cutoff f_c_lp complementing HP; both as linear-phase FIR.
• Detrending
  Before computing adev/mdev/tdev/MTIE, remove the mean of TE within each window to avoid DC bias.

VI. Metric Set & Panel Mapping

1. Core SLIs
   • offset.mean, offset.p95, offset.p99 (unit="ns");
   • skew.mean_ppm, skew.max_ppm;
   • jitter.rms, jitter.p99 (unit="ns").
2. Stability
   adev(tau), mdev(tau), tdev(tau) for tau ∈ TauSet.
3. Path/arrival
   form_const, form_general, delta_form, tol_Tarr, L_gamma.
4. Dashboard mapping (examples)
   • TS.sli.offset.ns.p99 ← offset.p99
   • TS.sli.jitter.ns.rms ← jitter.rms
   • TS.sli.tdev.ns.tau_10 ← tdev(10 s) * 1e9
   • TS.sli.mtie.ns.tau_100 ← MTIE(100 s) * 1e9

VII. Thresholds & Alerts (Default Strategy Excerpts)

1. Latency & jitter
   • offset.p99 ≤ 1.0e-6 s (alert), offset.p99 ≤ 5.0e-7 s (target)
   • jitter.p99 ≤ 1.0e-6 s
2. Stability envelope
   MTIE(tau) ≤ MTIE_spec(tau); tdev(tau) ≤ tdev_spec(tau) for tau ∈ TauSet
3. Drift & distribution
   |drift_slope| ≤ 5.0e-9 s/s, psi(offset) ≤ 0.2, W1(offset) ≤ 150 ns
4. Dual-form arrival
   delta_form ≤ tol_Tarr (mandatory)
   Thresholds must be contextualized per tier (core/aggregation/access/edge) in the strategy cards and written to contracts.

VIII. Measurement & Audit Flow M50D-1

• Alignment & denoising
  align_timebase → compute TE(t); band-split W and J using complementary filters.
• Core statistics
  Compute offset.*, skew.*, jitter.*; evaluate TIE/MTIE and adev/mdev/tdev on TauSet.
• Drift detection
  Estimate drift_slope and compute psi/W1 against a reference window.
• Dual-form arrival
  Compute form_const/form_general in parallel; persist delta_form.
• Contract evaluation
  Apply Appendix B C50-*; produce contracts.summary / violations[].
• Manifest & signature
  Write to manifest.time.metrics.* and chain-sign.

IX. Error & Uncertainty

1. Quantification
   • U = k * u_c (default k=2); for offset.p99, use order-statistics intervals.
   • For tdev/adev, estimate uncertainty via effective independent samples
     n_eff = ( T_total / tau ) / c_corr.
2. Source decomposition
   u^2(TE) = u^2(ref) + u^2(link) + u^2(node); publish alongside manifest.time.uncertainty.

X. Arrival-Time Harmonization (Coupling with Cleaning/Arrival)

1. Dual forms restated
   • form_const = ( 1 / c_ref ) * ( ∫ n_eff d ell )
   • form_general = ( ∫ ( n_eff / c_ref ) d ell )
   • delta_form = | form_const - form_general |; persist gamma(ell), d ell, L_gamma.
2. Contract
   Any metric release involving T_arr must include delta_form with tol_Tarr.

XI. Cross-References

• Allan-family computation & noise models: Chapter 7.
• Link & latency budgets: Chapter 4.
• Sync servos & BMCA: Chapters 5 and 10.
• Manifests & fields: Appendix C.
• Contract library: Appendix B C50-*.
• Statistical uncertainty publication: Methods.CrossStats v1.0 Appendix E.

Summary