41-EFT.WP.Comms.Navigation v1.0 | Chapter 3: Propagation & Channels — Path Terms, Arrival Time & Frequency Shift

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Chapter 3: Propagation & Channels — Path Terms, Arrival Time & Frequency Shift

I. Objectives & Applicability

• This chapter states a minimal usable equation set and executable workflows for propagation/channel modeling, unifying path terms, the two arrival-time conventions, group delay, and frequency shift (FOA), with equivalent treatment of LOS/NLOS/multipath and dispersion.
• All formulas/symbols/definitions are in English with inline backticks. The path gamma(ell) and measure d ell must be explicit; exactly one T_arr convention is chosen and must not be mixed.

II. Path Decomposition & Measure

• S30-1 (Piecewise path)
  gamma = ⊕_{k=1}^K gamma_k, where {free | fixture | substrate | device | environment}; total arrival time is the sum:
  T_arr = Σ_k T_arr^{(k)}.
• S30-2 (Differential baseline)
  To isolate sample/environment terms:
  ΔT_arr(ω) = T_arr^{case}(ω) − T_arr^{ref}(ω).
  If only the measured segment changes, ΔT_arr ≈ (1/c_ref) ∫_{segment} Δ n_eff(ω,r) d ell (subject to the chosen convention).

III. Arrival-Time Conventions & Group Delay (TOA & τg)

1. S30-3 (Two arrival-time conventions — mandatory declaration)
   • Pulled-constant: T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell )
   • Integrand form: T_arr = ( ∫ ( n_eff / c_ref ) d ell )
     Equivalent but must not be mixed; both require explicit gamma(ell) and d ell.
2. S30-4 (Phase → group delay)
   With unwrapped phase φ(ω), define group delay τ_g = ∂φ/∂ω.
3. S30-5 (When τg ≈ T_arr)
   Under single-mode, weak dispersion, and weak multipath, τ_g ≃ T_arr. First-order correction:
   Δτ ≃ (1/c_ref) ∫ ( ∂ n_eff/∂ω ) d ell.

IV. Frequency Shift & Doppler (FOA)

• S30-6 (Ideal Doppler)
  f_D = - ( f_c / c_ref ) ( v_rel · n̂ ), where n̂ is the unit direction of propagation and v_rel the relative velocity.
• S30-7 (CFO & observation)
  y_FOA = f_D + f_CFO + ε (CFO positive at receiver). Over short windows, estimate f_D by linear fit or phase differencing, and suppress f_CFO via reference chains or two-way measurements.

V. LOS/NLOS/Multipath & Dispersion (Equivalent Bias/Variance)

• S30-8 (Multipath superposition — phasor form)
  S21(ω) = Σ_m a_m(ω) exp{ -i ω T_m(ω) }, where T_m(ω) are effective path times (including reflections/diffractions) and a_m(ω) complex gains.
• S30-9 (NLOS equivalence)
  If the main path is indistinguishable, model multipath/NLOS via an equivalent bias b_NLOS and variance inflation into Σ_y, and emit an NLOS_flag (or score) for robust estimation.
• S30-10 (Medium & band terms)
  n_eff(ω,χ) = n_bg(ω,geom) ⊕ Δn_ch(ω; material, env). Include loss/dispersion when needed: attenuation α(ω) (Np·m⁻¹) affects amplitude channels directly, and impacts τ_g indirectly via ∂ n_eff/∂ω.

VI. Calibration & De-embedding Workflows (Aligned with Data Contracts)

• M3-1 (Path de-embedding)
  Using {open, short, load, thru} or {blank, substrate-only} standards, estimate non-DUT segments T_arr^{(k≠segment)} and n_bg(ω,geom); freeze in cal_bundle.
• M3-2 (Phase unwrapping & group delay)
  Unwrap S21(ω) to obtain φ(ω), τ_g; record piecewise uncertainties from unwrapping anomalies and insert into Σ_y.
• M3-3 (Arrival time & convention selection)
  Compute T_arr per the chosen convention and record in the dataset card:
• arrival_time:
• convention: "pulled_const" # or "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"
• M3-4 (FOA estimation & CFO suppression)
  Extract f_D via short-window phase differencing / time-frequency ridge; cancel (or estimate) f_CFO by two-way schemes or reference chains.

VII. Observations & Jacobians (Coupling with Geometry/Clocks)

• S30-11 (TOA observation)
  y_TOA = T_arr + b_t + ε_t; ∂y_TOA/∂p_WB = (1/c_ref) n̂^T, ∂y_TOA/∂b_t = 1.
• S30-12 (FOA observation)
  y_FOA = f_D + ε; ∂y_FOA/∂v_WB = - ( f_c / c_ref ) n̂^T. Derivatives w.r.t. attitude follow from the directional derivative of n̂(R_WB).
• S30-13 (Joint information)
  With H = ∂h/∂x, F = H^T Σ_y^{-1} H. In TOA+FOA fusion, the altitude block diagonal of Fisher increases, mitigating 2D→3D degeneracy (consistent with Chapter 5 GDOP analysis).

VIII. Data Contract (Required Fields for This Chapter)

unit_system: "SI"

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"

channel:

bands: ["microwave","UWB","5G","Wi-Fi","acoustic"]

S21: "<complex spectrum or trace>"

unwrap: {method: "phase_unwrap_v2", params: {...}}

deembed: {scheme: "OSLT|TRL", refs: ["blank","substrate-only"]}

foa:

carrier: "f_c [Hz]"

estimator: "phase_diff|ridge"

cfo: {mode: "two-way|ref-chain|free", prior: "<...>"}

quality:

nlos_flag: "<bool or score>"

coherence: {W_coh: {model: "<...>"}, weights: "L_coh|Fisher"}

covariance:

Σ_y: "<block-diagonal or sparse>"

references:

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

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

IX. Implementation Bindings (Interface Prototypes)

• I3-1 deembed_path(standards, scheme) -> {n_bg(ω), T_arr_bg, drift_model}
• I3-2 unwrap_and_delay(S21_or_trace) -> {phi(ω), τ_g, seg_uncertainty}
• I3-3 compute_arrival_time(phi_or_τg, convention, path) -> {T_arr(ω), meta}
• I3-4 estimate_doppler(trace_or_phase, f_c, mode) -> {f_D, f_CFO, ci}
• I3-5 nlos_mitigation(features, labels?) -> {NLOS_flag, b_NLOS, Σ_y_updated}

X. Quality Gates (This Chapter)

• Q1 Path/measure: any T_arr/TOA/TDOA/FOA usage must include convention/delta_form/gamma(ell)/d_ell in the dataset card.
• Q2 Dimensions & units: pass check_dim; SI columns complete (rad/Hz/s/m); power as dBm or W with explicit conversion.
• Q3 Single-convention rule: the two T_arr conventions must not be mixed within the same data stream; if switching, record version and change points.
• Q4 Covariance consistency: equivalent bias/variance from multipath/NLOS must update the corresponding channel blocks in Σ_y.
• Q5 Collision bans: never mix T_fil/T_trans or n/n_eff.

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

1. Cross-volume (fixed style): EFT.WP.Core.Equations v1.1 Ch.2 S20-* (path/measure & two conventions); EFT.WP.Core.Metrology v1.0 Ch.1–3,5 (units/dimensions/uncertainty).
2. Anchors in this chapter:
   • Minimal statements: S30-1—S30-13
   • Workflows: M3-1—M3-4
   • Interfaces: I3-1—I3-5

XII. Summary
This chapter unifies path terms—arrival time—group delay—frequency shift under a single convention, provides executable de-embedding, unwrapping, and FOA estimation workflows, and treats multipath/NLOS/dispersion via equivalent bias/variance. Together with the data contract and interfaces, it yields observation channels y = h(x,θ,ν)+ε, constructs Jacobians H and Fisher F, and supplies standardized inputs for Chapter 4 (per-channel observation models) and Chapter 10 (measurement matrix & experiment design).