22-EFT.WP.Metrology.Instrument v1.0 | Appendix D — Metrics & Drift Measures (Instrument-Specific)

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Appendix D — Metrics & Drift Measures (Instrument-Specific)

One-Sentence Objective
Define publishable performance metrics and drift measures for instrument domains; provide formulas, windows, and threshold recommendations; and map each item to manifest, contract, and dashboard fields one-to-one.

I. Scope & Targets

• Covered objects: noise, linearity, bandwidth, response, timebase-related terms, and environmental/aging drift for analog/digital/RF/photonic instruments.
• Applicable data: steady-state calibration data; open/short/standard-source alignment data; online telemetry under operating conditions.
• Output binding: write results into panel.sli.*, contracts.items["C70-*"], and manifest.instrument.* keys to support traceability and audits.
• Cross-references: arrival time & timebase in Metrology.TimeBase v1.0; sync & links in Metrology.Sync v1.0; statistical drift detection in Methods.CrossStats v1.0.

II. Terms & Variables

• Sampling & frequency domain: x(t), y(t), fs, N, PSD_x(f), BW = [f1,f2].
• Noise & resolution: Vn_rms, SNR_dB, ENOB, NF_dB, NEP_W_sqrtHz.
• Linearity: INL[k], DNL[k], LSB, code_center[k], width[k].
• Response & bandwidth: H(f), |H(f)|, phi(f), f_3dB_Hz, GD(f).
• Timebase & arrival time: tau_mono, ts, offset/skew/J, T_arr, delta_form.
• Drift & environment: theta(t) (metric of interest), theta_ref (reference), drift = theta(t) - theta_ref, drift_rel = drift / theta_ref, alpha_T = ( d theta / d T ) / theta_ref, beta_H = d theta / d RH.
• Distribution drift: W1, KL, psi (population stability index), window Delta_t.

III. Postulates P70D-*

• P70D-1 (Timebase consistency): evaluate all sequencing and windows on tau_mono, publish on ts, and persist offset/skew/J plus both T_arr formulations and delta_form.
• P70D-2 (Dimensional conservation): before publication, run check_dim(expr) and declare unit(x) and dim(x) explicitly.
• P70D-3 (Windowed stationarity): assume weak stationarity within each computation window; non-stationary excursions beyond thresholds must be labeled and window-split.
• P70D-4 (Uncertainty publication): accompany metrics with U = k * u_c (or intervals); use guardband rules for thresholding (see Chapter 11 and Appendix B).

IV. Minimal Equations S70D-*

• S70D-1 (Band-limited RMS noise):
  Vn_rms = sqrt( ∫_{f1}^{f2} PSD_v(f) df ).
• S70D-2 (SNR & effective bits):
  SNR_dB = 20 * log10( A_sig_rms / Vn_rms )；ENOB = ( SNR_dB - 1.76 ) / 6.02（sine full-scale approximation）.
• S70D-3 (Noise figure):
  NF_dB = 10 * log10( SNR_in / SNR_out ).
• S70D-4 (Dynamic range):
  DR_dB = 20 * log10( A_max_rms / Vn_rms ).
• S70D-5 (Linearity):
  INL[k] = ( code_center[k] - (k * LSB + b_fit) ) / LSB；DNL[k] = ( width[k] - LSB ) / LSB.
• S70D-6 (Magnitude response & bandwidth):
  |H(f_3dB_Hz)| = |H(0)| / sqrt(2)；ripple_pp = max| |H(f)| / |H(0)| - 1 | over the passband.
• S70D-7 (Group delay):
  GD(f) = - d phi(f) / d omega，where omega = 2 * pi * f.
• S70D-8 (Temperature coefficient):
  alpha_T = ( d theta / d T ) / theta_ref ≈ slope( theta vs T ) / theta_ref.
• S70D-9 (Temporal drift rate):
  drift_slope = d theta / d t ≈ argmin_s ∑ w_i ( theta_i - s * t_i - c )^2（robust weighted regression）.
• S70D-10 (Distribution drift):
  KL = ∫ p(x) log( p(x) / q(x) ) dx；W1 = inf_{gamma} ∫ |x-y| d gamma(x,y)；psi = ∑_i ( p_i - q_i ) * ln( p_i / q_i )（binned convention）.
• S70D-11 (Allan family — amplitude readings):
  adev_amp(tau) = sqrt( 0.5 * mean( ( y_{i+2} - 2*y_{i+1} + y_i )^2 ) )，y_i = ( x_i - mean(x) ) / mean(x).
• S70D-12 (Dual-form arrival-time consistency):
  delta_form = | ( 1 / c_ref ) * ( ∫ n_eff d ell ) - ( ∫ ( n_eff / c_ref ) d ell ) |，assert delta_form ≤ tol_Tarr（per the volume-wide convention）.

V. Metric Families & Windowing Conventions

1. Noise & resolution (open or standard source)
   • Window: Delta_t ≥ 1 s or N ≥ 2^20; band BW = [10 Hz, fs/2] or application-specific.
   • Outputs: Vn_rms, SNR_dB, ENOB, PSD key points (e.g., 1 kHz, 10 kHz, 100 kHz).
2. Linearity (code width & best-fit line)
   • Excitation: uniform code-step scan or triangle wave; estimate code widths via large-sample histograms.
   • Outputs: INL_pp_LSB, DNL_pp_LSB, res_lin_RMS_LSB.
3. Bandwidth & response (sweep or pulse)
   • Sweep: f vs |H(f)|, phi(f); pulse: step response to infer f_3dB_Hz and GD_pp_ns.
   • Outputs: f_3dB_Hz, ripple_pp, GD_pp_ns, overshoot_pct, settling_time_s.
4. Timebase-related (see TimeBase)
   Mirror: offset/skew/J, T_arr.form1/2, delta_form.
5. Environment/aging drift
   Modeling: theta ~ a_T*T + a_H*RH + a_P*P + s*t + eps; publish alpha_T, beta_H, drift_slope, and residual adev_amp.
6. Distribution drift (cross-session / cross-instrument)
   Reference ref vs current cur: publish {W1, KL, psi} and the binning/kernel bandwidth conventions.

VI. Computation & Decision Workflow M70D-*

• Data readiness & de-trending
  Remove explicit steps/saturation; slice windows on tau_mono; record RefCond.
• Noise/response estimation
  Compute Vn_rms and PSD; fit H(f) and GD(f) from sweep/pulse data.
• Linearity assessment
  Fit best-fit line b_fit; compute INL/DNL aggregates.
• Drift decomposition
  Multivariate regression to separate T/RH/P effects; take residuals to adev_amp(tau); estimate drift_slope.
• Distribution drift
  Unify bins or kernel density; compute {W1, KL, psi}; report test power and uncertainty.
• Contracts & dashboard
  Map outcomes to contracts.items["C70-*"] and panel.sli.*; apply guardband decisions and alerts.

VII. Thresholds & Strategy Cards (Recommendations)

• SNR_dB ≥ target_SNR_dB; ENOB ≥ target_ENOB - U.
• INL_pp_LSB ≤ 1.0, DNL_pp_LSB ≤ 1.0 (typical; defer to certificate scope).
• |alpha_T| ≤ alpha_T_max; |drift_slope| ≤ drift_max / day.
• W1 ≤ W1_max; psi ≤ 0.1 (small), 0.1 < psi ≤ 0.25 (watch), psi > 0.25 (alert).
• delta_form ≤ tol_Tarr is mandatory; otherwise block publication.

VIII. Dashboard & Manifest Mapping

• panel.sli.latency_ms_p99: measured from the acquisition/processing chain.
• panel.sli.error_rate: rolling contract-failure rate.
• panel.sli.rho: runtime utilization or queue occupancy.
• manifest.instrument.noise.{SNR_dB,ENOB}, response.{f_3dB_Hz,GD_pp_ns}, linearity.{INL_pp_LSB,DNL_pp_LSB}, timing.{offset/skew/J,T_arr,delta_form}, environment.RefCond, contracts.items["C70-*"].

IX. Implementation Bindings I70D-* (Prototypes)

• measure_noise(ds, BW) -> {Vn_rms, SNR_dB, ENOB, PSD_peaks}
• measure_linearity(codes) -> {INL_pp_LSB, DNL_pp_LSB, res_lin_RMS_LSB}
• measure_response(sweep|step) -> {f_3dB_Hz, ripple_pp, GD_pp_ns, overshoot_pct, settling_time_s}
• estimate_env_drift(series, env) -> {alpha_T, beta_H, drift_slope, adev_amp}
• detect_distribution_drift(ref, cur, cfg) -> {W1, KL, psi, U}
• mirror_timebase_fields(tb_manifest) -> timing_fields
• emit_instrument_metrics_manifest(results) -> manifest.instrument.metrics
  Invariants: check_dim(expr)="pass"; persist Delta_t and binning/kernel configs; delta_form ≤ tol_Tarr.

X. Cross-References

• Dual arrival-time forms & delta_form: Metrology.TimeBase v1.0, Chapter 9 & Appendix B.
• Sync-servo noise & Allan family: Metrology.Sync v1.0, Chapter 9 & Appendix D.
• Distribution-drift measures & alerting: Methods.CrossStats v1.0, Chapter 7 & Appendix D.
• Contracts & guardband: this volume’s Chapter 11 and Appendix B (C70 Suite).

Summary
This appendix defines unified conventions for instrument metrics and drift measures, the S70D-* equation family, windowing and threshold guidance, and field-level mappings to manifests, contracts, and dashboards—ensuring cross-experiment, cross-site, and cross-time comparability and auditability.