41-EFT.WP.Comms.Navigation v1.0 | Chapter 4: Observation Models — TOA/TDOA/AOA/Doppler/RSS/Carrier Phase

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Chapter 4: Observation Models — TOA/TDOA/AOA/Doppler/RSS/Carrier Phase

I. Objectives & Applicability

• Present minimal usable statements (S4x-*), Jacobians & noise models, observability notes, and data-contract fields for six observation types: TOA/TDOA/AOA/FOA(Range-Rate)/RSS/CarrierPhase.
• All formulas/symbols/definitions are in English and wrapped in backticks. Any term involving arrival time must explicitly declare the path gamma(ell) and the measure d ell. Exactly one T_arr convention is used—no mixing.

II. Unified Observation Vector & Modeling Conventions

1. S40-1 (Unified observation stack)
   y = h(x, θ, ν; x_design) + ε, where x = [p_WB, v_WB, q_WB, b_t, \dot b_t, ...], θ includes geometry/channel/array/path-loss parameters, ν are instrument/environment nuisances.
2. S40-2 (Arrival-time conventions — restated)
   • T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell )
   • T_arr = ( ∫ ( n_eff / c_ref ) d ell )
     Equivalent but must not be mixed; must record gamma(ell) and d ell.
3. S40-3 (Range proxy)
   ρ = c_ref · T_arr.
4. S40-4 (Noise model)
   ε ~ 𝒟(0, Σ_y) (Gaussian / Gaussian mixture / Student-t); robust loss ρ(·) (Huber/Tukey) supported.

III. TOA (Time of Arrival)

• S40-5 (Measurement) y_TOA = T_arr + b_t + ε_t.
• S40-6 (Jacobian) ∂y_TOA/∂p_WB = (1/c_ref) n̂^T, ∂y_TOA/∂b_t = 1; derivatives w.r.t. v_WB, q_WB follow from n̂(R_WB) via chain rule.
• S40-7 (Observability) Single-station TOA lacks absolute bearing; multi-anchor geometry and sync quality jointly set GDOP and cond(F).

IV. TDOA (Time Difference of Arrival)

• S40-8 (Definition) y_TDOA,i0 = ΔT_i0 = (T_arr,i − T_arr,0) + (b_t − b_t^0) + ε (0 is the reference).
• S40-9 (Jacobian) ∂y_TDOA/∂p_WB = (1/c_ref) ( n̂_i^T − n̂_0^T ); for shared-clock refs, ∂/∂b_t = 1, ∂/∂b_t^0 = −1.
• S40-10 (Note) TDOA is more geometry-sensitive and less dependent on a global clock; requires good anchor placement (see Chapter 5).

V. AOA (Angle of Arrival)

• S40-11 (Measurement)
  y_AOA = [θ, φ]^T + ε, with θ = atan2(n̂_y, n̂_x), φ = atan2(n̂_z, √(n̂_x^2 + n̂_y^2)).
• S40-12 (Jacobian)
  Derivatives w.r.t. p_WB follow from n̂ = (p_anchor − p_WB)/‖·‖; derivatives w.r.t. q_WB come from the directional derivative of n̂(R_WB).
• S40-13 (Array errors)
  Array phase-center, mutual coupling, and calibration errors enter as bias and covariance inflation for θ, φ.

VI. FOA / Doppler (Frequency Shift / Range-Rate)

• S40-14 (Ideal model)
  y_FOA = f_D + f_CFO + ε, with f_D = - ( f_c / c_ref ) ( v_rel · n̂ ).
• S40-15 (Jacobian)
  ∂y_FOA/∂v_WB = - ( f_c / c_ref ) n̂^T. Estimate f_D via phase-diff/short-window frequency; suppress f_CFO via two-way or reference chains.
• S40-16 (Observability)
  Fusing FOA with TOA strengthens vertical information, mitigating 2D→3D degeneracy.

VII. RSS (Received Signal Strength)

• S40-17 (Log-distance path loss)
  P_rx(dBm) = P_0 − 10·η·log10( ρ/ρ_0 ) + X_σ, where path-loss exponent η, shadowing X_σ ~ 𝒩(0, σ^2).
• S40-18 (Jacobian)
  ∂P_rx/∂p_WB = − (10 η / ln 10) · (1/ρ) · ∂ρ/∂p_WB, with ∂ρ/∂p_WB = n̂^T.
• S40-19 (Remark)
  RSS is environment-sensitive and best used for aiding/initialization.

VIII. Carrier Phase

• S40-20 (Single-difference measurement)
  y_CP = φ = (2π/λ) ( ρ + c_ref b_t ) + 2π N + ε_φ, where λ = c_ref/f_c, N ∈ 𝕫 is the integer ambiguity.
• S40-21 (Double difference / clock removal)
  Double differencing across satellites/anchors and receivers removes common clock and reduces iono/hardware residuals.
• S40-22 (Jacobian)
  ∂φ/∂p_WB = (2π/λ) n̂^T, ∂φ/∂b_t = (2π/λ) c_ref, ∂φ/∂N = 2π.
• S40-23 (Ambiguity resolution)
  Support integer fixing (LAMBDA/ILS) or Bayesian discrete priors; on failure, fall back to float and inflate uncertainty.

IX. Noise, Covariance & Robust Modeling

• S40-24 (Covariance structure)
  Σ_y = blkdiag(Σ_TOA, Σ_TDOA, Σ_AOA, Σ_FOA, Σ_RSS, Σ_CP). Multipath/NLOS enter as equivalent bias b_NLOS and variance inflation in corresponding blocks.
• S40-25 (Robust likelihood)
  p(y|x) = ∑_c π_c 𝒩( h(x); μ_c, Σ_c ) or M-estimator reweighting ρ(·); use NLOS_flag for soft/hard gating.
• S40-26 (Units & dimensions)
  TOA/TDOA in s, FOA in Hz, AOA in rad, RSS in dBm or W, CarrierPhase in rad; all expressions must pass check_dim.

X. Jacobians & Fisher Information (Linked to Experiment Design)

• S40-27 (Unified Jacobian)
  H = ∂h/∂x = [H_TOA; H_TDOA; H_AOA; H_FOA; H_RSS; H_CP].
• S40-28 (Information & conditioning)
  F = H^T Σ_y^{-1} H; use cond(F), rank(F), and block diagonals for observability; D/A/E-optimal design in Chapter 10.

XI. Data Contract (Required/Recommended 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"

y_channels: ["TOA","TDOA","AOA","FOA","RSS","CP"]

aoa:

array: {pc: "<phase_center>", cal: "<file|hash>", model: "<ULA|URA|...>"}

foa:

carrier: "f_c [Hz]"

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

rss:

P0_dBm: "<ref>", eta: "<prior>", rho0: "<m>"

cp:

lambda: "c_ref/f_c", dd_mode: "single|double", ambiguity: "int|float"

covariance:

Σ_y: "<block-diagonal or sparse>"

robust:

loss: "Huber|Tukey", nlos_flag: "<bool|score>", mixture: {pi: [...], Σc: [...]}

references:

- "EFT.WP.Comms.Navigation v1.0:Ch.2 S20-*"

- "EFT.WP.Comms.Navigation v1.0:Ch.3 S30-*"

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

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

XII. Implementation Bindings (Interface Prototypes)

• I4-1 build_observation(y_channels, config, cal_bundle) -> {y, meta}
• I4-2 build_jacobian(x, y_channels, geometry) -> {H, blocks}
• I4-3 update_covariance(y, flags, models) -> {Σ_y, robust_weights}
• I4-4 fix_ambiguity_cp(y_CP, H_CP, mode) -> {N_hat, status, cov}
• I4-5 fuse_channels(y, H, Σ_y, priors) -> {x̂, P, innovations}

XIII. Quality Gates (This Chapter)

• Q1 Convention consistency: do not mix the two T_arr conventions within a data stream; dataset must contain convention/delta_form/gamma(ell)/d_ell.
• Q2 Units/dimensions: SI columns for s/Hz/rad/W(dBm)/m are complete and pass check_dim.
• Q3 Ambiguity handling: CarrierPhase integer/float status must be explicit in dataset and interface returns.
• Q4 Covariance consistency: Σ_y block structure matches y_channels; NLOS/multipath edits are reflected.
• Q5 Collision bans: never mix T_fil/T_trans or n/n_eff.

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

• Cross-volume (fixed style): this volume Ch. 2 (terminology/metrology baseline), Ch. 3 (path/arrival time/frequency shift); EFT.WP.Core.Equations v1.1 Ch.2 S20-* (path/measure); EFT.WP.Core.Metrology v1.0 Ch.1–3,5 (units/uncertainty).
• Anchors: S40-1—S40-28; I4-1—I4-5.

XV. Summary
This chapter formalizes six observation types under a unified convention, providing equations and interfaces to build the observation vector y, Jacobian H, and covariance Σ_y, with clear observability and robust-modeling guidance. Its outputs feed directly into Chapter 10 (measurement matrix & experiment design), Chapter 8 (fusion estimation), and Chapter 13 (case reproductions).