22-EFT.WP.Metrology.Instrument v1.0 | Chapter 8 — Triggering, Sampling, and Timebase (Aligned with TimeBase)

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Chapter 8 — Triggering, Sampling, and Timebase (Aligned with TimeBase)

One-Sentence Objective
Unify trigger conventions, sampling, and timebase expressions at the instrument layer; define measurable Fs/T_s/T_ap/J/offset/skew and their workflows; and align strictly with the TimeBase volume on tau_mono/ts and the two T_arr formulations, with persisted records.

I. Scope and Objects

1. Scope
   • Triggering: analog/digital/photonic triggers; threshold/window/mode triggers; pre-/post-trigger and re-arm.
   • Sampling: sample/hold aperture T_ap, sampling rate Fs, oversampling/decimation, anti-aliasing AA(f).
   • Timebase: sample timestamp semantics, estimation of offset/skew/J, and mapping tau_mono → ts.
2. Objects
   Single-/multi-channel synchronous sampling; cross-device time alignment; hardware/software/hybrid timestamp chains.
3. Outputs
   Fs, T_s, T_ap, J, offset, skew, timestamp semantics for t_k, AA(f), decim(R, H_aa), and manifest.instrument.time.*.

II. Terms and Variables

• Sampling & timebase: Fs, T_s = 1 / Fs, tau_mono, ts, offset, skew, J (rms jitter), phi_clk(t) (phase), S_phi(f) (phase-noise PSD).
• Triggering: t_trig (trigger instant, aperture-center semantics), th (threshold), hys (hysteresis), dead_time, rearm.
• Aperture & timestamping: T_ap (aperture width), t_k_ap_center (aperture-center time of the k-th sample).
• Frequency bounds: f_N = Fs / 2, AA(f) (anti-alias amplitude response), R (decimation factor).
• Arrival time: two T_arr formulations, delta_form, path gamma(ell) and measure d ell.
• Units & dimensions: unit(Fs)="Hz", unit(T_s|T_ap|J|offset|skew•T)= "s", check_dim(all).

III. Postulates P708-*

• P708-1 (Consistent sample-time semantics): use aperture-center semantics for sample times: t_k = t_k_ap_center. If an edge-based convention is published, declare it explicitly and provide the mapping.
• P708-2 (Anti-alias first): apply H_aa before any downsample ↓R, and record the equivalent group delay tau_filt.
• P708-3 (Timebase alignment first): compute internally on tau_mono; publish on ts, carrying offset/skew/J.
• P708-4 (Jitter budget): at the specification frequency f_in_max, jitter-induced SNR_j must not fall below the threshold; J shall be published as a metrological quantity with uncertainty.
• P708-5 (Reproducible triggering): characterize and publish trigger time-walk and re-arm time.
• P708-6 (Arrival-time consistency): for event arrival–gated sampling/windowing, record both T_arr formulations and delta_form in parallel.
• P708-7 (Cross-device sync): synchronize across devices using TimeBase’s sync_ref; estimate/publish offset/skew/J via a unified algorithm.

IV. Minimal Equations S708-*

• S708-1 (Sampling and time mapping)
  T_s = 1 / Fs；t_k_ap_center = t_0 + k * T_s；
  t_k( tau_mono ) = tau_0 + offset + ( 1 + skew ) * k * T_s + epsilon_k，where epsilon_k is zero-mean jitter with std(epsilon_k) = J.
• S708-2 (Nyquist and anti-aliasing)
  f_N = Fs / 2；prior to decimation, the low-pass satisfies |AA(f)| ≤ attn_stop for |f| ≥ f_N / R.
• S708-3 (Jitter-limited SNR and ENOB)
  SNR_j(dB) = - 20 * log10( 2 * pi * f_in * J )；
  ENOB_j = ( SNR_j - 1.76 ) / 6.02 (sine-input convention).
• S708-4 (Aperture equivalence)
  x_meas[k] = ( 1 / T_ap ) * ( ∫_{t_k - T_ap/2}^{t_k + T_ap/2} x(t) dt )；
  equivalently, ideal sampling followed by convolution with a sinc filter; group delay ≈ T_ap / 2 (declare if compensated in the manifest).
• S708-5 (Resampling and decimation)
  Band-limited reconstruction: x_hat(t) = ( ∑_k x[k] * sinc( Fs * ( t - t_k ) ) )；
  Decimation: x_R[n] = ( x * h_aa )[ n R ], where h_aa implements AA(f) in time; for linear-phase FIR, tau_filt = (L-1)/2 * T_s.
• S708-6 (Restating the two arrival-time forms)
  T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell ) and T_arr = ( ∫ ( n_eff / c_ref ) d ell )；
  delta_form = | ( 1 / c_ref ) * ( ∫ n_eff d ell ) - ( ∫ ( n_eff / c_ref ) d ell ) |.

V. Metrology Procedure M70-8 (Trigger → Sampling → Timebase → Resample → Harmonize → Persist)

• Readiness & configuration
  Set mode ∈ {edge, window, pattern}, th/hys, N_pre/N_post, Fs, and RefCond; choose AA(f) and R.
• Timebase alignment
  Following TimeBase Chapters 5–8, measure offset/skew/J and Fs_meas; determine the sample-time mapping k → tau_mono; record the timestamp chain (hardware/software/hybrid) and the delay budget.
• Trigger characterization
  Sweep amplitude/slew to obtain time-walk curves and the t_trig distribution; measure dead_time/rearm; produce u(t_trig) and coverage U = k * u_c.
• Aperture & jitter
  Estimate T_ap (pulse/edge methods); measure J (zero-cross/phase-noise mapping); compute SNR_j(f_in_max) and ENOB_j.
• Anti-aliasing & decimation
  Design/load H_aa; verify pass/stop-band specs and group delay; execute ↓R, compensate tau_filt, and re-label sample times on tau_mono.
• Arrival-time harmonization
  For event-gated/timing measurements, compute both T_arr forms and delta_form, and record the path gamma(ell).
• Contracts & rollback
  Enforce C70-8*; on failure, lower f_in_max, raise R, or revert to the prior Fs/AA pairing; record impact scope.
• Persistence & signing
  Emit manifest.instrument.time.*: Fs/Fs_meas/T_s/T_ap/J/offset/skew, trigger characteristics, AA/decim, T_arr/delta_form, uncertainties, and signatures.

VI. Contracts & Assertions C70-8*

• C70-81 ts.monotone: strictly monotone sample times, t_{k+1} - t_k > 0.
• C70-82 fs.tolerance: | Fs_meas - Fs_nom | / Fs_nom ≤ tol_Fs.
• C70-83 jitter.bound: J_p95 ≤ J_max and SNR_j(f_in_max) ≥ snr_min.
• C70-84 aperture.spec: | T_ap - T_ap_nom | ≤ tol_ap, and declare whether T_ap/2 group delay is compensated.
• C70-85 aa.spec: ripple_pass ≤ rp_max_dB, attn_stop ≥ as_min_dB, tau_filt stable and compensated.
• C70-86 decim.alias: post-decimation alias energy ratio alias_ratio ≤ tol_alias.
• C70-87 trigger.walk: max| dt/dA | ≤ walk_slope_max, rearm ≤ rearm_max.
• C70-88 sync.crossdev: cross-device |offset| ≤ off_max, |skew| ≤ skew_max.
• C70-89 arrival.forms: delta_form ≤ tol_Tarr.
• C70-8A dim.check: check_dim(all)=true.

VII. Implementation Bindings I70-8* (Interface Prototypes)

• configure_trigger(instr, mode, th, hys, pre, post) -> cfg
• measure_trigger_latency(instr, sweep) -> {t_trig_stats, walk_curve, dead_time, rearm}
• estimate_timebase(ds, sync_ref) -> {Fs_meas, offset, skew, J, tau_map}
• measure_aperture(ds, method) -> {T_ap, U}
• design_aa_filter(Fs, spec) -> H_aa, tau_filt
• decimate(ds, R, H_aa, tau_filt) -> ds'
• resample_to_tau(ds, tau_map, semantics="aperture-center") -> ds_tau
• compute_snr_jitter(f_in, J) -> SNR_j_dB, ENOB_j
• enforce_arrival_time_convention(ds) -> ds_with_Tarr (and delta_form)
• assert_time_contracts(ds, rules) -> report
• emit_time_manifest(results, policy) -> manifest.instrument.time
  Invariants: sample order and timestamps are consistent; AA matches R; sources/algorithms for offset/skew/J are explicit; tau_mono/ts fields are complete.

VIII. Cross-References

• Timebase modeling and jitter: EFT.WP.Metrology.TimeBase v1.0, Chapters 2, 6, 7, 8.
• Timestamp chains and delay budgeting: TimeBase, Chapter 4; sync protocols and servo: TimeBase, Chapter 5.
• Arrival-time cleaning and publication contracts: EFT.WP.Methods.Cleaning v1.0, Chapters 5, 6, 10.
• Distributed synchronization and topology: EFT.WP.Metrology.Sync v1.0, Chapters 5, 11.

IX. Quality & Risk Control

• SLI/SLO
  Fs_error_ppm_p95, J_rms_p95, offset_p95, skew_p95, alias_ratio_p95, trigger_timewalk_p95, latency_p99.
• Drift monitoring
  Track ΔFs/Δt, ΔJ/Δt, Δoffset/Δt with temperature/load covariates; step changes trigger re-calibration.
• Rollback strategy
  Narrow bandwidth / increase R; deploy stronger AA(f); switch to hardware timestamps; enable sync redundancy or enter holdover; preserve impact assessment and re-publication window.

Summary
Through P708-* / S708-* / M70-8 / C70-8* / I70-8*, this chapter standardizes an integrated convention spanning trigger → sampling → timebase → resampling → arrival time, ensuring cross-device and cross-volume temporal consistency that is comparable, traceable, and auditable.