41-EFT.WP.Comms.Navigation v1.0 | Chapter 10: Metrology Chain & Measurement Matrix

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Chapter 10: Metrology Chain & Measurement Matrix

I. Objectives & Applicability

• Establish a unified framework for the experimental metrology chain and the measurement matrix in comms/navigation systems, standardizing the end-to-end flow from data contracts → de-embedding/synchronization → observation construction → linearization → information & identifiability → experiment design.
• All formulas/symbols/definitions are in English and wrapped in backticks. Whenever arrival time is involved, use the selected T_arr convention and explicitly declare the path gamma(ell) and measure d ell.

II. Metrology Chain & Traceability

• S100-1 (Metrology elements)
  Time/frequency references, link asymmetry, path de-embedding, phase unwrapping, sync priors, device geometry and array calibration; constants c_ref, k_B, f_c.
• S100-2 (De-embedding & baselines)
  Per Chapter 3 M3-1/2/3, freeze non-DUT segments T_arr^{(k≠segment)} and n_bg(ω, geom); record cal_bundle and segmentwise unwrapping uncertainty.
• S100-3 (Sync integration)
  Inject Chapter 6 cov([b_t,\dot b_t]) as an independent prior block or a joint observation block into Σ_y.
• M10-1 (Metrology calibration workflow)
  Standards / common-view / two-way measurements → estimate cal_bundle, b_asym, cov_clock → write into dataset/pipeline cards.

III. Measurement Matrix Definition & Linearization

• S100-4 (Observation & parameter vectors)
  y = [ y_TOA, y_TDOA, y_AOA, y_FOA, y_RSS, y_CP, ... ]^T
  θ = [ geometry, clocks, channel, array, loss, ... ]^T
  Nuisances ν (de-embedding/asymmetry/unwrapping residuals, etc.) enter as noise or explicit perturbations.
• S100-5 (Mapping & linearization)
  y = M(θ, ν; x_design); about nominal θ0,
  Δy ≈ J(θ0) · Δθ + B · Δν, with J = ∂y/∂θ, B = ∂y/∂ν.
• S100-6 (Separating geometry/state Jacobians)
  Estimator-internal still uses H = ∂h/∂x; this chapter’s J targets experiment design & system parameters.
• S100-7 (Multi-band / multi-geometry stacking)
  J_stack = blkstack{ J(ω_k, geom_m) }; within the coherence window, weight by w(ω) ∝ L_coh(ω) or Fisher-information density.

IV. Observation Geometry & Sensitivities

• S100-8 (Geometry parametrization)
  geom = {anchors, mode∈{TOA,TDOA,AOA,FOA,RSS,CP}, array, θ, φ, L, mask}; provide equivalent n̂(·) and directional derivatives for arrays/antennas.
• S100-9 (Channel standardization)
  Avoid scale pathologies via dimensional/amplitude normalization: ỹ = W_y · y, J̃ = W_y · J.
• M10-2 (Sensitivity spectra)
  Output per-channel/band |J̃|, principal singular directions, and recommended geom* combos (cross-checked with Chapter 5 GDOP).

V. Error Propagation & Uncertainty Budget

• S100-10 (Linear propagation)
  cov(Δθ) ≈ (J^T Σ_y^{-1} J)^{-1}; with nuisances, use Σ_y ← Σ_y + B Σ_ν B^T.
• S100-11 (Robust & mixture cases)
  If Σ_y comes from mixture/robust reweighting (Chapter 9), propagate weights and inflated block covariances.
• M10-3 (Error-budget tables)
  Produce parameter CIs, channel contribution breakdown, and sensitivity Sankey; align with the dataset’s Error-Budget section.

VI. Identifiability & Conditioning

• S100-12 (Fisher information)
  F = J^T Σ_y^{-1} J; use cond(F), rank(F), correlation matrix C = diag(F)^{-1/2} F diag(F)^{-1/2} for pathology & correlation.
• S100-13 (Priors & constraints)
  Add physical/engineering priors F ← F + F_prior (array/geometry, sync, bounds); report posterior approximation cov(Δθ) ≈ F^{-1}.
• M10-4 (Identifiability assessment)
  Output scores, weakest directions, and reinforcement advice (add anchors/channels/bands; adjust look angles/height; improve sync).

VII. Experiment Design & Resource Optimization

• S100-14 (Criteria)
  D-/A-/E-optimal: Φ_D = -log det(F), Φ_A = tr(F^{-1}), Φ_E = -λ_min(F); combine with cost(x_design) (time/power/BW/anchor count/backhaul).
• S100-15 (Design space)
  x_design = {ω, geom, θ, φ, anchors, power, dwell, schedule}; support scene_tag (Chapter 9) to select configuration sets.
• M10-5 (Design-optimization workflow)
  ① Build J, Σ_y, F → ② choose objective/constraints → ③ heuristic/convex approximation/MIP → ④ output x_design*, expected cov(Δθ), and channel quotas.

VIII. 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"

cal_bundle:

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

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

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

channels:

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

Σ_y: "<block-diagonal or sparse>" # includes robustly reweighted version

design:

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

geometry: {anchors:[{id,p_S,cov}], array:"<model>", view:{θ,φ}}

resources: {power:"<W>", dwell:"<s>", bandwidth:"<Hz>", slots:n}

measurement_matrix:

J_shape: [Ny, Nθ]

stacking: {modes:["freq","geom"], weights:"coherence|info_density"}

outputs: ["J","B","Σ_y","F","cond(F)","scores"]

priors:

F_prior: "<matrix or blocks>", bounds: {θ_min, θ_max}

logging:

repro_bundle: {scripts:"...", env:"...", anchors:["S100-*","M10-*","I10-*"]}

references:

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

- "EFT.WP.Comms.Navigation v1.0:Ch.4 S40-*"

- "EFT.WP.Comms.Navigation v1.0:Ch.6 S60-*"

- "EFT.WP.Comms.Navigation v1.0:Ch.9 S90-*"

IX. Implementation Bindings (Interface Prototypes)

• I10-1 build_measurement_matrix(config, cal_bundle, contracts) -> {J, B, Σ_y, meta}
• I10-2 compute_fisher(J, Σ_y, priors=None) -> {F, cond, rank, C}
• I10-3 assess_identifiability(J, Σ_y, priors=None) -> {scores, weak_dirs, advice}
• I10-4 optimize_experiment(design_space, budget, criterion) -> {x_design*, F*, covθ*}
• I10-5 propagate_uncertainty(J, Σ_y, Σ_ν=None, priors=None) -> {covθ, ci}
• I10-6 stack_multi_band(mats, weights) -> {J_stack, Σy_stack}
• I10-7 emit_repro_bundle(J, Σ_y, F, config) -> {bundle, hash}

X. Quality Gates (This Chapter)

• Q1 Convention consistency: if a channel includes T_arr, the dataset must include convention/delta_form/gamma(ell)/d_ell.
• Q2 Units/dimensions: all expressions pass check_dim; SI/engineering columns complete (s/Hz/rad/m/W).
• Q3 SPD & conditioning: keep Σ_y and prior blocks SPD; if cond(F) > κ_th, flag non-identifiable/ill-conditioned and return reinforcement advice.
• Q4 Coherence-window weighting: W_coh weights must align with Σ_y block structure; out-of-window data act as low-weight or systematic terms only.
• Q5 Traceability/reproducibility: output a repro_bundle containing scripts/environment/anchors & hashes; retain decision trails for de-embedding/sync/reweighting.

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

1. Cross-volume (fixed style): this volume Ch. 3 (de-embedding/unwrapping/conventions), Ch. 4 (observation models & H,R), Ch. 5 (geometry & GDOP), Ch. 6 (sync chain & priors), Ch. 9 (robust reweight/deflation); EFT.WP.Core.Metrology v1.0 Ch.1–3,5 (units/uncertainty).
2. Anchors in this chapter:
   • Minimal statements: S100-1—S100-15
   • Workflows: M10-1—M10-5
   • Interfaces: I10-1—I10-7

XII. Summary
Grounded in metrological traceability, this chapter defines the measurement matrix y = M(θ), its linearization and stacking across bands/geometries, and Fisher-driven identifiability and D/A/E-optimal experiment design. With data contracts, implementation bindings, and quality gates, it unifies characterization and optimization across bands/geometry/protocols, providing standardized inputs to Chapter 8 (fusion estimation) and Chapter 11 (simulation stack/benchmarks).