41-EFT.WP.Comms.Navigation v1.0 | Chapter 6: Time Reference & Synchronization Chain

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Chapter 6: Time Reference & Synchronization Chain

I. Objectives & Applicability

• Provide minimal usable models, protocols, and error propagation for time references and synchronization chains in comms/navigation systems: clock physics, GNSS/terrestrial-network/two-way ranging (TWR) time transfer, and variance mapping into TOA/TDOA/FOA/CarrierPhase with covariance integration.
• All formulas/symbols/definitions are in English and wrapped in backticks. Whenever arrival-time terms are involved, the selected T_arr convention must be used, with explicit gamma(ell) and d ell.

II. Clock Physical Model & Frequency Stability

• S60-1 (Clock state)
  b_t (bias, s), \dot b_t (drift, s/s), optional \ddot b_t. Reception time: t_rx = t_tx + T_prop + b_t + ε_t.
• S60-2 (Continuous/discrete models)
  Continuous: \dot b_t = w_b, \ddot b_t = w_d.
  Discrete: b_{k+1} = b_k + \dot b_k Δt + w_b, \dot b_{k+1} = \dot b_k + w_d, with w_* ~ 𝒩(0,Q).
• S60-3 (Allan relation)
  Given Allan deviation σ_y(τ), calibrate process noise Q by mapping PSD types (white FM, flicker FM, random-walk FM) to discrete variances.
• S60-4 (CFO & FOA)
  f_CFO = f_c · \dot b_t (approx.); FOA variance includes var(f_CFO) and can be suppressed via reference chains/two-way schemes.

III. Synchronization & Time-Transfer Protocols

• S60-5 (GNSS common-view / differential)
  Common-view: Δb_{A,B} = (t_A − t_{GNSS}) − (t_B − t_{GNSS}) + ε. Differential/PPP-RTK reduce ε via iono/tropo/satellite-clock corrections.
• S60-6 (PTP / gPTP / NTP)
  PTP (IEEE 1588) timestamp exchange: t_2 − t_1 − (t_4 − t_3) → 2·delay; link asymmetry b_asym enters as a systematic. gPTP (802.1AS) provides sub-µs sync on Ethernet.
• S60-7 (UWB-TWR / two-way)
  Single round: T_prop ≈ ( (t_{r2}−t_{t1}) − (t_{t2}−t_{r1}) ) / 2 − b_asym. Multiple rounds suppress jitter; can jointly estimate b_t and T_prop.
• S60-8 (5G NR time/frequency sync)
  Use SSB/PRS + PTP over fronthaul; DL/UL asymmetry enters via calibration tables into Σ_y.
• S60-9 (Multi-source fusion)
  h_sync(z) = [h_GNSS(z); h_PTP(z); h_TWR(z)], jointly estimating b_t, \dot b_t and link biases with LS/factor-graph methods.

IV. Mapping Sync Errors to Observation Variances

• S60-10 (TOA/TDOA mapping)
  var(y_TOA) = var(T_arr) + var(b_t); var(y_TDOA) = var(ΔT) + var(b_t − b_t^0). var(T_arr) must match the selected T_arr convention.
• S60-11 (FOA mapping)
  var(y_FOA) = var(f_D) + var(f_CFO), with var(f_CFO) ≈ f_c^2 · var(\dot b_t).
• S60-12 (Carrier-phase mapping)
  var(φ) = (2π/λ)^2 ( var(ρ) + c_ref^2 var(b_t) ) + var(ε_φ); double differencing removes common terms.
• S60-13 (Covariance incorporation)
  The sync chain provides cov([b_t, \dot b_t]) as a prior block to be injected into observation covariance Σ_y or the state prior P_0.

V. Reference Architectures & Implementation Hints

• S60-14 (Master–slave + GNSS disciplining)
  A central master clock distributes via PTP/gPTP; local slaves are periodically disciplined by GNSS, with holdover during GNSS outages.
• S60-15 (Edge common-view)
  Edge nodes common-view the same GNSS satellite/terrestrial reference to form precise Δb for TDOA network alignment.
• S60-16 (UWB-TWR scans)
  Periodic TWR calibration over a grid refreshes b_asym and \dot b_t estimates to reduce short-term drift.

VI. Synchronization Calibration Workflows (Data-Contract Aligned)

• M6-1 (Sync calibration)
  Input {GNSS, PTP/gPTP, TWR} raw timestamps with quality flags → estimate b_t, \dot b_t, b_asym → produce cov([b_t, \dot b_t]) and correction tables.
• M6-2 (Link-asymmetry estimation)
  Model UL/DL timing differences vs. temperature/load; output b_asym(T, load) for runtime correction.
• M6-3 (Covariance distribution)
  Deliver sync priors as independent blocks or joint with observations into Σ_y and the dataset card.

VII. Data Contract (Required/Recommended Fields for This Chapter)

unit_system: "SI"

clock:

state: {b_t: "<s>", dot_b_t: "<s/s>"}

cov: {bb: "<s^2>", bd: "<s^2/s>", dd: "<s^2/s^2>"}

holdover: {stability: "<σ_y(τ)>", τ: "<s>"}

sync_sources:

gnss: {mode: "CV|PPP-RTK|diff", quality: "<C/N0>", mask: "<elev>"}

ptp: {profile: "IEEE1588|802.1AS", asym_corr: "<s>", delay_msmt: "E2E|P2P"}

twr: {rounds: n, scheme: "single|double|asym-comp"}

links:

asymmetry: {b_asym: "<s>", model: "table|poly", valid: "<ttl>"}

arrival_time:

convention: "pulled_const|integrand"

delta_form: "c_ref^-1 * ∫ n_eff dℓ" # or "∫ (n_eff/c_ref) dℓ"

gamma: "piecewise: free|fixture|substrate|device|environment"

d_ell: "m"

covariance:

Σ_y: "<block-diagonal or sparse>" # includes sync blocks when merged

references:

- "EFT.WP.Core.Equations v1.1:Ch.2 S20-*"

- "EFT.WP.Core.Metrology v1.0:Ch.1–3,5"

VIII. Implementation Bindings (Interface Prototypes)

• I6-1 estimate_clock(sync_obs) -> {b_t, dot_b_t, cov}
• I6-2 calibrate_asymmetry(path_logs, env) -> {b_asym_model, ci}
• I6-3 fuse_sync_sources(gnss, ptp, twr, priors) -> {post, cov}
• I6-4 propagate_sync_cov(cov_clock, channels) -> {Σ_y_updated}
• I6-5 holdover_predict(clock_state, horizon) -> {pred_state, pred_cov}

IX. Quality Gates (This Chapter)

• Q1 Convention consistency: wherever T_arr/TOA/TDOA/FOA/CarrierPhase appears, the dataset must include convention/delta_form/gamma(ell)/d_ell.
• Q2 Units & dimensions: SI columns for s/Hz complete; all expressions pass check_dim.
• Q3 Covariance integration: sync priors and link-asymmetry corrections must update the corresponding blocks of Σ_y.
• Q4 Robustness: for PTP asymmetry/congestion, provide intervals for b_asym and a “degraded mode”; on GNSS loss, enter holdover and tag validity.
• Q5 Traceability: retain raw and corrected timestamps, source identifiers, and time-base lineage for audit.

X. Cross-Volume References & Anchors (This Chapter)

1. Cross-volume (fixed style): this volume Ch. 2 (metrology baseline), Ch. 3 (path/arrival/frequency), Ch. 4 (observation models); EFT.WP.Core.Metrology v1.0 Ch.1–3,5 (units/uncertainty); EFT.WP.Core.Equations v1.1 Ch.2 S20-* (two T_arr conventions and path/measure declarations).
2. Anchors in this chapter:
   • Minimal statements: S60-1—S60-16
   • Workflows: M6-1—M6-3
   • Interfaces: I6-1—I6-5

XI. Summary
Starting from clock states and frequency stability, this chapter systematizes GNSS/network/PTP and UWB-TWR synchronization mechanisms and rigorously maps their errors into TOA/TDOA/FOA/CarrierPhase variances and covariance structure. With data contracts and interfaces, sync priors can be distributed, link asymmetry corrected, and Chapter 8 fusion and Chapter 10 measurement-matrix/design consume these inputs directly.