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Appendix A: Symbols & Units (Incremental)
I. Notes & Dependency Anchors
- This appendix lists only symbols and units newly introduced or specialized in this volume; global items are in EFT.WP.Core.Terms v1.0 and EFT.WP.Core.Metrology v1.0.
- All formulas/symbols/definitions are in English and wrapped in backticks; cross-volume citations use the fixed style “volume + version + anchor (P/S/M/I)”.
- Dependency anchors: this volume Ch.2 S20-*, Ch.3 S30-*, Ch.4 S40-*, Ch.5 S50-*, Ch.6 S60-*, Ch.7 S70-*, Ch.8 S80-*, Ch.9 S90-*, Ch.10 S100-*, Ch.11 S110-*, Ch.12 S120-*.
II. Writing & Metrology Conventions (Restated)
- Two arrival-time conventions (choose one; do not mix):
T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell ); T_arr = ( ∫ ( n_eff / c_ref ) d ell ). Both require explicit gamma(ell) and d ell. - Units & dimensions: all expressions must pass check_dim; tables must include SI/engineering units (m, s, Hz, rad, dBm, W, etc.).
- Collision bans: T_fil ≠ T_trans; n (number density) ≠ n_eff (effective refractive index / medium proxy).
III. Symbol Table (Incremental)
Symbol | Name | Definition/Use | SI/Eng Unit | Dim. | Notes |
|---|---|---|---|---|---|
p_WB | position | body pose in world frame | m | L | — |
v_WB | velocity | body velocity | m·s⁻¹ | L T^-1 | — |
q_WB / R_WB | attitude | quaternion/rotation | — | — | right-handed |
b_t, \dot b_t | clock bias/drift | timing offsets | s / s·s⁻¹ | T / — | optional \ddot b_t |
T_arr | arrival time | one of two conventions | s | T | needs gamma(ell), d ell |
ΔT_ij | time difference | T_arr,i - T_arr,j | s | T | reference j=0 |
ρ | range | ρ = c_ref · T_arr | m | L | — |
θ, φ | azimuth/elevation | direction angles | rad | — | — |
f_c | carrier | carrier frequency | Hz | T^-1 | — |
f_D, f_CFO | Doppler/CFO | frequency shift | Hz | T^-1 | f_D = - ( f_c / c_ref ) ( v_rel · n̂ ) |
λ | wavelength | λ = c_ref / f_c | m | L | EM/acoustic |
P_rx | received power | path loss/RSS | dBm or W | M L^2 T^-3 | P_rx(dBm) common |
η | path-loss exponent | log-distance model | — | — | RSS |
K_rician | Rician factor | LOS-to-scatter ratio | dB | — | RSS feature |
φ_CP | carrier phase | single/double difference | rad | — | includes integer N |
N | integer ambiguity | CP discrete unknown | cycles | — | integer |
n_eff(ω,χ) | effective refractive index | path/band dependent | — | — | medium/env term |
α(ω) | attenuation const. | loss/dispersion | Np·m⁻¹ | L^-1 | closed-form or table |
c_ref | reference propagation speed | reference constant | m·s⁻¹ | L T^-1 | default light speed; use c_med for medium |
c_med | medium speed | acoustic/special media | m·s⁻¹ | L T^-1 | optional |
gamma(ell), d ell | path/measure | path parametrization & line element | m | L | mandatory explicit |
Σ_y | observation covariance | noise covariance | — | — | block-diagonal by channel |
H, F | Jacobian/Fisher | H=∂h/∂x, F=H^T Σ_y^{-1} H | — | — | information |
GDOP/PDOP/HDOP/VDOP | geometry precision | deployment/run metrics | — | — | dimensionless |
W_coh | coherence-window weight | cross-band fusion | — | — | aligned with Σ_y |
C/N0 | carrier-to-noise density | GNSS/5G quality | dB-Hz | — | — |
IV. Constants & Reference Values (Adopted/Extended in This Volume)
Symbol | Definition | Value/Status | Unit | Note |
|---|---|---|---|---|
c_ref | reference propagation speed | 299 792 458 (exact) | m·s⁻¹ | SI (EM) |
k_B | Boltzmann constant | exact | J·K⁻¹ | thermal noise |
N_0 | noise spectral density | by scenario | W·Hz⁻¹ | white noise |
g_W | gravity | 9.80665 | m·s⁻² | locally adjustable |
V. Typical Dimension Checks
- ρ = c_ref · T_arr: [m] = [m·s⁻¹][s] ✓
- f_D = - ( f_c / c_ref )( v_rel · n̂ ): [Hz] = [Hz][m·s⁻¹]/[m·s⁻¹] ✓
- φ_CP = (2π/λ)(ρ + c_ref b_t): [rad] = [rad·m⁻¹]([m] + [m·s⁻¹][s]) ✓
- P_rx(dBm) ↔ W: P_dBm = 10 log10( P_W / 1 mW ) (record conversion) ✓
VI. Wi-Fi/5G/UWB/Acoustic/Multi-Sensor Increments
- FTM/RTT: y_TOA in s; Wi-Fi protocol delays as b_asym systematics.
- AOA/PRS: array main-lobe/mutual coupling enter Σ_y; record array phase center pc and calibration hash cal_hash.
- UWB: CIR peak features {τ_rms, peak_ratio} enter features.toa for robustness.
- Acoustic: include c_med, α(ω); group-delay correction Δτ ≃ (1/c_med) ∫ (∂ n_eff/∂ω) dℓ.
- Multi-sensor: IMU/VO/wheel units rad/s, m/s², pixel or rad, m/s per Chapter 7.
VII. Dataset Fields (Units & Provenance — Mandatory)
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"
channels:
TOA: {unit: "s"} # or "RTT: s"
TDOA: {unit: "s"}
AOA: {unit: "rad"}
FOA: {unit: "Hz"}
RSS: {unit: "dBm"} # may alongside "W"
CP: {unit: "rad"}
carrier:
f_c: "<Hz>", lambda: "c_ref/f_c"
acoustic:
c_med: "<m/s>", alpha: "<Np/m>"
covariance:
Σ_y: "<block-diagonal or sparse>"
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"
VIII. Notation & Conventions (Fixed in This Volume)
- Fourier convention: F(k) = ∫ f(r) exp(-i k·r) d^3r, inverse uses +i.
- Frames/attitude: W/B/S triad; direction unit n̂.
- Angles in rad; frequency in Hz; power in dBm/W; phase in rad.
- Do not mix the two T_arr conventions within a dataset; any switch must be versioned.
IX. Quality Gates (This Appendix)
- Q1 Units/dimensions: table units complete, pass check_dim; explicit dBm↔W conversion.
- Q2 Convention consistency: any T_arr/TOA/TDOA/FOA/CP appearance records convention/delta_form/gamma(ell)/d_ell.
- Q3 Covariance consistency: channel Σ_y blocks match units; robust reweighting inflations reflected.
- Q4 Traceability: fields like f_c, λ, pc, cal_hash, c_med, α(ω) are traceable in the dataset.
- Q5 Collision bans: never mix T_fil/T_trans, or n/n_eff; use c_med for medium-specific speed and c_ref as the universal reference.
Copyright & License (CC BY 4.0)
Copyright: Unless otherwise noted, the copyright of “Energy Filament Theory” (text, charts, illustrations, symbols, and formulas) belongs to the author “Guanglin Tu”.
License: This work is licensed under the Creative Commons Attribution 4.0 International (CC BY 4.0). You may copy, redistribute, excerpt, adapt, and share for commercial or non‑commercial purposes with proper attribution.
Suggested attribution: Author: “Guanglin Tu”; Work: “Energy Filament Theory”; Source: energyfilament.org; License: CC BY 4.0.
First published： 2025-11-11｜Current version：v5.1
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