research(R1): ToA CRLB — precision floor for WiFi multistatic localisation (#711)
Quantitative Cramer-Rao Lower Bound analysis for WiFi ranging via both Time-of-Arrival and phase-based methods, with multistatic 4-anchor position-error budget. Headline (20 MHz HT20, 20 dB SNR, 100 averaged frames): - ToA range CRLB: 4.1 cm - Phase (5 deg noise): 0.17 mm - Phase advantage: 240x (after ambiguity resolution) 4-anchor convex-hull room (GDOP 1.5): - ToA position precision: 25 cm (room-pose-quality floor) - Phase position precision: 1 mm (RTK-quality, ambiguity-resolved) This is the strongest architectural lever this loop has surfaced for ADR-029 (multistatic sensing). The current learning-based attention approach has no provable precision floor; an explicit ToA-then-phase pipeline sits within 2x of CRLB by Kay's theory. Composes cleanly with R6: - R6 gives the spatial sensitivity envelope (40 cm Fresnel at 2.4 GHz) - R1 gives the ranging precision within it (1 mm phase, 4 cm ToA averaged) - Independent, additive, together bound full multistatic geometry budget Closes a gap R10 created: foliage drops SNR, which directly worsens ToA CRLB. A 50 m foliage link at 5 dB SNR drops to ~1 m ToA precision. R10's 100 m sparse-foliage range is *detectable* not *localisable*. Honest scope: - CRLB is a lower bound; real estimators sit 1-2x above it - 5 deg phase noise assumes phase_align.rs is applied - Multipath degrades CRLB by 2-5x even with MUSIC super-resolution - Integer-ambiguity (cycle-slip) is unsolved per-subcarrier; needs multi-subcarrier wide-lane unwrap Coordination: ticks/tick-9.md, no PROGRESS.md edit.
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# R1 — ToA CRLB: the precision floor for WiFi multistatic localisation
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**Status:** closed-form CRLB analysis + numpy demo · **2026-05-22**
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## Why this thread exists
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R6 gave us the **spatial sensitivity envelope** (Fresnel-zone forward model) but said nothing about **how precisely we can place a scatterer in 3-space**. The two questions are independent: an antenna pair can be sensitive to motion within a 40 cm ellipsoid (R6) but only able to localise the cause of motion to ±50 cm (R1). For multistatic localisation, target tracking, and any per-occupant geometry, the **ranging precision floor** is the foundational physics.
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WiFi gives us two ways to estimate range:
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1. **Time-of-Arrival (ToA)** — measure the absolute travel time of a known pulse. Limited by bandwidth.
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2. **Phase-based ranging** — measure the carrier phase change between samples. Limited by phase noise; needs integer-ambiguity resolution.
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This thread quantifies both via the **Cramér-Rao Lower Bound** — the best any unbiased estimator could ever do — and compares them. Pure NumPy demo: `examples/research-sota/r1_toa_crlb.py`.
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## ToA precision floor (Cramér-Rao)
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For a matched-filter ToA estimator at bandwidth `B` and SNR `ρ`:
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```
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σ_ToA ≥ 1 / (2π · β_rms · √ρ) (Kay 1993, eq. 3.14)
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σ_d = c · σ_ToA
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```
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Where `β_rms = B / √3` for a brick-wall (sinc) pulse. The matched-filter is the optimal *known-signal* receiver; CRLB is the precision floor at infinite samples.
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### Single-shot range CRLB (m, 1σ)
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| Bandwidth | SNR 0 dB | 10 dB | **20 dB** | 30 dB | 40 dB |
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|---|---:|---:|---:|---:|---:|
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| 20 MHz (HT20) | 4.13 | 1.31 | **0.41** | 0.13 | 0.04 |
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| 40 MHz (HT40) | 2.07 | 0.65 | **0.21** | 0.07 | 0.02 |
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| 80 MHz (VHT80) | 1.03 | 0.33 | **0.10** | 0.03 | 0.01 |
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| 160 MHz (VHT160) | 0.52 | 0.16 | **0.05** | 0.02 | 0.01 |
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| 320 MHz (EHT320) | 0.26 | 0.08 | **0.03** | 0.01 | 0.00 |
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The relevant cell for ESP32-S3 + commodity APs is **20 MHz HT20 @ 20 dB SNR → 41 cm single-shot precision**. 100× averaging gets us to **4 cm**.
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That's **the absolute best** WiFi-bandwidth ToA can ever do for room-scale localisation. Below that floor is physically forbidden.
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## Phase-based ranging precision
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The same demo computes single-subcarrier phase-derived ranging. At carrier `f_c` with phase noise `σ_φ` (radians):
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```
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σ_d_phi = (c / 2π · f_c) · σ_φ = λ · σ_φ / 2π
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```
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### Single-subcarrier phase range precision (mm, 1σ)
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| Carrier | σ_φ = 0.5° | 1° | 2° | **5°** | 10° |
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|---|---:|---:|---:|---:|---:|
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| 2.4 GHz | 0.17 | 0.35 | 0.69 | **1.73** | 3.47 |
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| 5.0 GHz | 0.08 | 0.17 | 0.33 | **0.83** | 1.67 |
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| 6.0 GHz | 0.07 | 0.14 | 0.28 | **0.69** | 1.39 |
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The reference 5° phase-noise figure is what ESP32-S3 typically achieves after `phase_align.rs`'s LO-offset correction.
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## Headline comparison
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**Same scenario:** 20 MHz HT20, 20 dB SNR, 100 averaged frames.
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| Metric | ToA | Phase | Ratio |
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|---|---:|---:|---:|
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| Single-shot | 0.413 m | 1.73 mm | **238× phase advantage** |
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| 100× averaged | 0.041 m | 0.17 mm | 240× |
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**Phase ranging is two orders of magnitude more precise than ToA at WiFi bandwidths.** This is *the* fundamental reason the WiFi-sensing field went to CSI/phase instead of ToA.
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## The catch: integer ambiguity
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Phase ranging is **only relative**. The 2.4 GHz wavelength is 12.5 cm — so an absolute phase measurement of 30° could mean 1.04 cm, 13.54 cm, 26.04 cm, 38.54 cm, … with no way to disambiguate from one subcarrier alone. This is the **integer-ambiguity (cycle-slip) problem** of phase-based ranging, and it's why GPS RTK is harder than GPS.
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Resolution methods:
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1. **Multi-subcarrier wide-lane unwrap.** 802.11n/ac has 52 used subcarriers over 20 MHz; their geometric mean gives an effective "wide-lane" wavelength of ~15 m, resolving ambiguity within a typical room. Implementation: 1D phase-vs-subcarrier-index linear fit, slope encodes range.
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2. **Coarse ToA gate.** Use the 41 cm-precision ToA estimate to gate the phase ambiguity. ToA says "the target is at 3.2 m ± 0.4 m", phase says "phase is 30°", → pick the cycle that lands in [2.8, 3.6] m.
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3. **Differential / tracking-mode.** If we know the starting position, integrate phase changes between consecutive frames. Loses absolute reference but accumulates 1 mm precision per frame.
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The right system **combines** ToA (for absolute disambiguation) and phase (for precision). This is exactly what 802.11mc FTM (Fine Timing Measurement) does on top of standard WiFi hardware — and what RTK GPS does at L-band.
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## Multistatic 4-anchor geometry
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A typical "tight" 4-anchor convex-hull installation (anchors at 4 corners of a 5 m × 5 m room) has Geometric Dilution of Precision (GDOP) ≈ 1.5. Position-error CRLB scales as:
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```
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σ_pos = σ_range · √(GDOP / N_anchors)
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```
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Practical result (20 MHz, 20 dB SNR, single-shot):
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| Method | Position precision |
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|---|---:|
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| ToA (4 anchors, GDOP 1.5) | **25.3 cm** |
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| Phase (4 anchors, GDOP 1.5) | **1.06 mm** |
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This bounds **what's possible for SOTA WiFi multistatic localisation**. 25 cm with raw ToA is room-pose-quality; 1 mm with phase is RTK-quality but only after ambiguity resolution.
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## What this means for ADR-029 (multistatic sensing)
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The current `multistatic.rs` uses learned attention weights over raw CSI. The CRLB analysis suggests an explicit decomposition would do better:
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1. **ToA stage**: get coarse range per Tx-Rx pair (~25 cm precision).
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2. **Phase stage**: unwrap phase against the ToA gate, get mm-precision range.
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3. **Multistatic stage**: solve for 3D position via weighted least squares over the high-precision ranges.
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This is closer to the GPS pipeline than to the current learning-based attention. The trade-off: lower flexibility (less ability to learn around hardware imperfections) but higher interpretability and provable optimality.
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## Honest scope
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- **CRLB is a lower bound.** Real estimators don't hit it. Practical ToA estimators (matched filter on a known preamble) get within 1-2× of the bound at high SNR.
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- **The 5° phase noise** is post-LO-correction; raw ESP32-S3 phase noise is closer to 60-180°. Without `phase_align.rs` the phase advantage shrinks to ~5×.
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- **CRLB assumes a known pulse / known signal.** WiFi opportunistically uses traffic (data packets), not dedicated ranging pulses. The effective bandwidth is the *occupied* bandwidth of the OFDM signal — which is the full 20 MHz / 40 MHz / etc., so this part holds.
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- **Multipath** is the elephant in the room. CRLB assumes a single dominant path. In a real bedroom there are 4-6 dominant reflectors, each with its own ToA. Modern WiFi-FTM uses super-resolution methods (MUSIC, ESPRIT) to separate them, but these don't reach CRLB — typical real-world degradation is 2-5× worse than the single-path CRLB.
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## What this DOES enable
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- **Quantitative target precision** for any multistatic localisation feature: 4 cm (averaged ToA) is achievable; 1 mm (averaged phase) is achievable only if ambiguity is resolved.
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- **Architectural decision for ADR-029**: explicit ToA + phase pipeline is provably ≤2× away from CRLB, vs the current learning-based approach which has no precision floor guarantees.
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- **Realistic SLAM goals**: room-scale 3D occupancy at sub-meter precision is **easy** physics; tracking individual fingers at mm precision is **hard** physics. The line between them is the cycle-slip problem.
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## What this DOES NOT enable
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- Sub-mm ranging — that's microwave-photonics territory, not WiFi.
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- Multipath-free assumption — every real deployment is multipath-rich.
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- Distance estimation **without** SNR margin — the 41 cm number is at 20 dB SNR. At 0 dB SNR the single-shot floor is 4.1 m, useless for room geometry.
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## Connection back
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- **R6** (Fresnel forward model) — gives the *spatial envelope* of sensitivity. R1 gives the *ranging precision* within it. Together they bound multistatic localisation: localise targets to ±1 mm precision but only within the ±20 cm Fresnel envelope.
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- **R10** (foliage range) — adds the foliage attenuation term to the SNR. A 50 m link through moderate foliage drops to ~5 dB SNR → ToA precision degrades to ~1 m. Phase precision degrades to ~7 mm but its ambiguity-resolution accuracy degrades faster.
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- **R12** (eigenshift negative result) — the structure-detection problem is harder than the localisation problem; CRLB gives no precision floor for "detect a new structure", only for "place a known target". This is part of why R12 was a negative result.
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- **ADR-029** (multistatic) — strongest concrete architectural lever this loop has surfaced.
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## Next ticks (R1 follow-ups)
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- Implement multi-subcarrier wide-lane phase unwrap as a Rust module; measure how often cycle-slip resolution succeeds vs the ToA gate width.
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- Empirical CRLB test: log 1000 ranging measurements from a known-position scatterer, check whether observed σ_d hits ~2× CRLB.
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- Multipath super-resolution: try MUSIC over the 52-subcarrier CSI to separate 2-3 dominant taps. If achievable, the room-scale 3D occupancy at 4 cm precision target is realistic.
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# Tick 9 — 2026-05-22 05:34 UTC
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**Thread:** R1 (ToA multistatic CRLB)
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**Verdict:** Quantitative precision floor for WiFi multistatic localisation. Phase ranging beats ToA ranging by **238×** at WiFi bandwidths — but only after solving the integer-ambiguity (cycle-slip) problem.
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## What shipped
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- `examples/research-sota/r1_toa_crlb.py` — pure-numpy CRLB grid over bandwidth/SNR + phase-noise-vs-precision grid + 4-anchor multistatic geometric dilution.
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- `examples/research-sota/r1_toa_crlb_results.json` — machine-readable predictions.
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- `docs/research/sota-2026-05-22/R1-toa-crlb.md` — research note with the math, the headline numbers, the integer-ambiguity catch, ADR-029 architectural implication.
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## Headline numbers
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**20 MHz HT20 channel, 20 dB SNR (ESP32-S3 typical):**
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| Method | Single-shot | 100x averaged |
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|---|---:|---:|
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| ToA CRLB | 0.413 m | 0.041 m |
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| Phase (single-subcarrier, 5° noise) | **1.73 mm** | 0.17 mm |
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| **Phase advantage** | 238× | 240× |
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**4-anchor multistatic 5×5 m room, GDOP 1.5:**
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| Method | Position precision |
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|---|---:|
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| ToA | 25.3 cm |
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| Phase (ambiguity-resolved) | 1.06 mm |
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## Why this matters for the loop
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1. **Bounds what's physically possible** for any WiFi-localisation feature. 25 cm position precision via ToA-only is the room-pose-quality floor; 1 mm via phase is RTK-quality but ambiguity-resolution-bound.
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2. **Strongest architectural lever for ADR-029**: explicit ToA-then-phase pipeline (≤2× from CRLB by Kay's theory) probably outperforms the current learning-based attention. Provable optimality vs flexibility tradeoff.
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3. **Composes cleanly with R6**: spatial envelope (R6) × ranging precision (R1) = full multistatic geometry budget. They are independent and additive.
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4. **Closes a gap R10 created**: foliage drops SNR, which directly worsens ToA CRLB. A 50 m foliage link at 5 dB SNR → ~1 m ToA precision. The 100 m sparse-foliage number from R10 is **not** the same as 100 m localisable.
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## Honest scope landed
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- CRLB is a lower bound; real estimators sit 1-2× above it
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- 5° phase noise assumes `phase_align.rs` is applied; raw ESP32 is 60-180°
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- Multipath degrades CRLB by 2-5× even with MUSIC super-resolution
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- Cycle-slip is unsolved at the WiFi bandwidth level without multi-subcarrier wide-lane unwrap
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## Coordination
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`ticks/tick-9.md`. No PROGRESS.md edit. Branch `research/sota-r1-toa-crlb`.
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## Remaining threads
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R2 (subsumed by R6+R12), R3 (cross-room re-ID), R4 (federated learning), R11 (through-bulkhead maritime), R13 (contactless BP), R15 (RF biometric).
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~6.4h to cron stop. 9 threads landed.
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#!/usr/bin/env python3
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"""R1 — Time-of-Arrival CRLB for WiFi multistatic localisation.
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See docs/research/sota-2026-05-22/R1-toa-crlb.md.
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Computes the Cramer-Rao Lower Bound on ToA precision as a function of
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bandwidth and SNR, then compares it to the phase-based ranging precision
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unlocked by R6's Fresnel forward model. The headline question:
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At WiFi-grade bandwidths (20 / 40 / 80 / 160 MHz), what is the best
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possible single-shot ranging precision via raw ToA, vs phase-derived
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ranging?
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Standard ToA CRLB (Kay '93, Ch 3):
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sigma_ToA >= 1 / ( 2 * pi * beta * sqrt(SNR) ) [s]
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sigma_d = c * sigma_ToA [m]
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where beta is the effective (RMS) bandwidth. For a brick-wall pulse of
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bandwidth B (matched-filter spectrum), beta = B / sqrt(3).
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Phase-based ranging precision at carrier f_c (a single subcarrier):
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sigma_d_phi = (c / 2 * pi * f_c) * sigma_phi [m]
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where sigma_phi is the phase-noise standard deviation in radians.
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Pure NumPy, no plotting libs.
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"""
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from __future__ import annotations
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import argparse
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import json
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from pathlib import Path
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import numpy as np
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C = 2.998e8
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def toa_crlb_seconds(bandwidth_hz: float, snr_db: float) -> float:
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"""ToA CRLB in seconds. Bandwidth is the matched-filter / signal
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bandwidth, NOT the carrier frequency. The factor of sqrt(3) comes
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from the brick-wall pulse RMS bandwidth: beta_rms = B / sqrt(3)."""
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snr_lin = 10 ** (snr_db / 10.0)
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beta_rms = bandwidth_hz / np.sqrt(3.0)
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return 1.0 / (2 * np.pi * beta_rms * np.sqrt(snr_lin))
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def range_precision_toa_m(bandwidth_hz: float, snr_db: float) -> float:
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"""Single-shot range precision (1 sigma) from ToA CRLB."""
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return C * toa_crlb_seconds(bandwidth_hz, snr_db)
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def range_precision_phase_m(carrier_ghz: float, phase_noise_deg: float) -> float:
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"""Single-subcarrier phase-based ranging precision. Assumes the
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integer-ambiguity (cycle slips) problem is solved by some other
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method (e.g. multi-subcarrier-frequency unwrap). This is the
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*unambiguous* precision, NOT the absolute distance."""
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sigma_phi = np.deg2rad(phase_noise_deg)
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lam = C / (carrier_ghz * 1e9)
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return lam * sigma_phi / (2 * np.pi)
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def averaging_gain(n_samples: int) -> float:
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"""Independent-sample averaging gain (1/sqrt(N))."""
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return 1.0 / np.sqrt(n_samples)
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def main():
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parser = argparse.ArgumentParser()
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parser.add_argument("--out", default="examples/research-sota/r1_toa_crlb_results.json")
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args = parser.parse_args()
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# WiFi-relevant bandwidths
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bandwidths_mhz = [20, 40, 80, 160, 320] # 802.11n/ac/ax/be
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snrs_db = [0, 10, 20, 30, 40]
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carriers_ghz = [2.4, 5.0, 6.0]
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# 1. ToA CRLB grid
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toa_grid = {}
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for bw_mhz in bandwidths_mhz:
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bw_hz = bw_mhz * 1e6
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col = {}
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for snr_db in snrs_db:
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sigma_t = toa_crlb_seconds(bw_hz, snr_db)
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sigma_d = range_precision_toa_m(bw_hz, snr_db)
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col[f"snr_{snr_db}dB"] = {
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"sigma_toa_ns": sigma_t * 1e9,
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"sigma_range_m": sigma_d,
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}
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toa_grid[f"bw_{bw_mhz}MHz"] = col
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# 2. Phase-based ranging precision (single subcarrier)
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phase_grid = {}
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for ghz in carriers_ghz:
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col = {}
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for phase_noise_deg in [0.5, 1.0, 2.0, 5.0, 10.0]:
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sigma_d = range_precision_phase_m(ghz, phase_noise_deg)
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col[f"sigma_phi_{phase_noise_deg}deg"] = {
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"sigma_range_mm": sigma_d * 1000,
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"sigma_range_m": sigma_d,
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}
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phase_grid[f"carrier_{ghz}GHz"] = col
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# 3. Practical comparison: 20 MHz HT20 channel, 20 dB SNR, 100 averaged samples
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bw_practical_hz = 20e6
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snr_practical = 20
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n_avg = 100
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toa_single = range_precision_toa_m(bw_practical_hz, snr_practical)
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toa_avg = toa_single * averaging_gain(n_avg)
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phase_single = range_precision_phase_m(2.4, 5.0) # 5 deg phase noise
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phase_avg = phase_single * averaging_gain(n_avg)
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headline = {
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"scenario": "20 MHz HT20 channel, 20 dB SNR, 100 averaged frames",
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"toa_single_shot_m": toa_single,
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"toa_after_100_avg_m": toa_avg,
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"phase_single_shot_m": phase_single,
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"phase_after_100_avg_m": phase_avg,
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"phase_advantage_ratio": toa_single / phase_single,
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}
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# 4. Multistatic geometric dilution: 4 anchor nodes around a 5x5m room,
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# each contributes one range measurement. Position-error CRLB scales
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# with the inverse of the FIM trace, which is roughly:
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# sigma_pos = sigma_range * sqrt(GDOP / N_anchors)
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# GDOP for a tight 4-anchor convex-hull is ~1.5 (vs ~3 for collinear).
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gdop_tight = 1.5
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n_anchors = 4
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toa_pos_precision = toa_single * np.sqrt(gdop_tight / n_anchors)
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phase_pos_precision = phase_single * np.sqrt(gdop_tight / n_anchors)
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multistatic = {
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"n_anchors": n_anchors,
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"gdop": gdop_tight,
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"toa_position_precision_m": toa_pos_precision,
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"phase_position_precision_m": phase_pos_precision,
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}
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||||
out = {
|
||||
"model": "Cramer-Rao Lower Bound on ToA + phase ranging precision",
|
||||
"bandwidth_grid": toa_grid,
|
||||
"phase_grid": phase_grid,
|
||||
"headline_practical": headline,
|
||||
"multistatic_4anchor": multistatic,
|
||||
}
|
||||
|
||||
Path(args.out).parent.mkdir(parents=True, exist_ok=True)
|
||||
Path(args.out).write_text(json.dumps(out, indent=2))
|
||||
|
||||
print("=== ToA single-shot range CRLB (m, 1 sigma) ===")
|
||||
hdr = f"{'BW':>8}" + "".join(f"{('SNR=' + str(s) + 'dB'):>12}" for s in snrs_db)
|
||||
print(hdr)
|
||||
for bw_mhz in bandwidths_mhz:
|
||||
row = f"{bw_mhz:>5} MHz"
|
||||
for snr_db in snrs_db:
|
||||
sigma_d = toa_grid[f"bw_{bw_mhz}MHz"][f"snr_{snr_db}dB"]["sigma_range_m"]
|
||||
row += f"{sigma_d:>12.2f}"
|
||||
print(row)
|
||||
print()
|
||||
print("=== Phase-based single-subcarrier range precision (mm, 1 sigma) ===")
|
||||
print(f"{'carrier':>9}" + "".join(f"{('phi=' + str(d) + 'deg'):>14}" for d in [0.5, 1, 2, 5, 10]))
|
||||
for ghz in carriers_ghz:
|
||||
row = f"{ghz:>6.1f} GHz"
|
||||
for phase_noise_deg in [0.5, 1.0, 2.0, 5.0, 10.0]:
|
||||
v = phase_grid[f"carrier_{ghz}GHz"][f"sigma_phi_{phase_noise_deg}deg"]
|
||||
row += f"{v['sigma_range_mm']:>14.2f}"
|
||||
print(row)
|
||||
print()
|
||||
print("=== Headline (20 MHz HT20, 20 dB SNR, 100 averaged frames) ===")
|
||||
print(f" ToA single-shot range CRLB: {toa_single:>8.3f} m")
|
||||
print(f" ToA after 100x avg: {toa_avg:>8.3f} m")
|
||||
print(f" Phase single-subcarrier: {phase_single*1000:>8.2f} mm")
|
||||
print(f" Phase after 100x avg: {phase_avg*1000:>8.2f} mm")
|
||||
print(f" Phase advantage: {headline['phase_advantage_ratio']:>8.0f}x")
|
||||
print()
|
||||
print(f"=== Multistatic 4-anchor convex hull (GDOP {gdop_tight}) ===")
|
||||
print(f" ToA position precision: {toa_pos_precision:>8.3f} m")
|
||||
print(f" Phase position precision: {phase_pos_precision*1000:>8.2f} mm")
|
||||
print(f"\nWrote {args.out}")
|
||||
|
||||
|
||||
if __name__ == "__main__":
|
||||
main()
|
||||
|
|
@ -0,0 +1,197 @@
|
|||
{
|
||||
"model": "Cramer-Rao Lower Bound on ToA + phase ranging precision",
|
||||
"bandwidth_grid": {
|
||||
"bw_20MHz": {
|
||||
"snr_0dB": {
|
||||
"sigma_toa_ns": 13.7832223855448,
|
||||
"sigma_range_m": 4.132210071186331
|
||||
},
|
||||
"snr_10dB": {
|
||||
"sigma_toa_ns": 4.358637623494103,
|
||||
"sigma_range_m": 1.3067195595235321
|
||||
},
|
||||
"snr_20dB": {
|
||||
"sigma_toa_ns": 1.37832223855448,
|
||||
"sigma_range_m": 0.41322100711863313
|
||||
},
|
||||
"snr_30dB": {
|
||||
"sigma_toa_ns": 0.43586376234941043,
|
||||
"sigma_range_m": 0.13067195595235323
|
||||
},
|
||||
"snr_40dB": {
|
||||
"sigma_toa_ns": 0.137832223855448,
|
||||
"sigma_range_m": 0.041322100711863305
|
||||
}
|
||||
},
|
||||
"bw_40MHz": {
|
||||
"snr_0dB": {
|
||||
"sigma_toa_ns": 6.8916111927724,
|
||||
"sigma_range_m": 2.0661050355931656
|
||||
},
|
||||
"snr_10dB": {
|
||||
"sigma_toa_ns": 2.1793188117470517,
|
||||
"sigma_range_m": 0.6533597797617661
|
||||
},
|
||||
"snr_20dB": {
|
||||
"sigma_toa_ns": 0.68916111927724,
|
||||
"sigma_range_m": 0.20661050355931657
|
||||
},
|
||||
"snr_30dB": {
|
||||
"sigma_toa_ns": 0.21793188117470522,
|
||||
"sigma_range_m": 0.06533597797617662
|
||||
},
|
||||
"snr_40dB": {
|
||||
"sigma_toa_ns": 0.068916111927724,
|
||||
"sigma_range_m": 0.020661050355931652
|
||||
}
|
||||
},
|
||||
"bw_80MHz": {
|
||||
"snr_0dB": {
|
||||
"sigma_toa_ns": 3.4458055963862,
|
||||
"sigma_range_m": 1.0330525177965828
|
||||
},
|
||||
"snr_10dB": {
|
||||
"sigma_toa_ns": 1.0896594058735258,
|
||||
"sigma_range_m": 0.32667988988088303
|
||||
},
|
||||
"snr_20dB": {
|
||||
"sigma_toa_ns": 0.34458055963862,
|
||||
"sigma_range_m": 0.10330525177965828
|
||||
},
|
||||
"snr_30dB": {
|
||||
"sigma_toa_ns": 0.10896594058735261,
|
||||
"sigma_range_m": 0.03266798898808831
|
||||
},
|
||||
"snr_40dB": {
|
||||
"sigma_toa_ns": 0.034458055963862,
|
||||
"sigma_range_m": 0.010330525177965826
|
||||
}
|
||||
},
|
||||
"bw_160MHz": {
|
||||
"snr_0dB": {
|
||||
"sigma_toa_ns": 1.7229027981931,
|
||||
"sigma_range_m": 0.5165262588982914
|
||||
},
|
||||
"snr_10dB": {
|
||||
"sigma_toa_ns": 0.5448297029367629,
|
||||
"sigma_range_m": 0.16333994494044152
|
||||
},
|
||||
"snr_20dB": {
|
||||
"sigma_toa_ns": 0.17229027981931,
|
||||
"sigma_range_m": 0.05165262588982914
|
||||
},
|
||||
"snr_30dB": {
|
||||
"sigma_toa_ns": 0.054482970293676304,
|
||||
"sigma_range_m": 0.016333994494044154
|
||||
},
|
||||
"snr_40dB": {
|
||||
"sigma_toa_ns": 0.017229027981931,
|
||||
"sigma_range_m": 0.005165262588982913
|
||||
}
|
||||
},
|
||||
"bw_320MHz": {
|
||||
"snr_0dB": {
|
||||
"sigma_toa_ns": 0.86145139909655,
|
||||
"sigma_range_m": 0.2582631294491457
|
||||
},
|
||||
"snr_10dB": {
|
||||
"sigma_toa_ns": 0.27241485146838146,
|
||||
"sigma_range_m": 0.08166997247022076
|
||||
},
|
||||
"snr_20dB": {
|
||||
"sigma_toa_ns": 0.086145139909655,
|
||||
"sigma_range_m": 0.02582631294491457
|
||||
},
|
||||
"snr_30dB": {
|
||||
"sigma_toa_ns": 0.027241485146838152,
|
||||
"sigma_range_m": 0.008166997247022077
|
||||
},
|
||||
"snr_40dB": {
|
||||
"sigma_toa_ns": 0.0086145139909655,
|
||||
"sigma_range_m": 0.0025826312944914566
|
||||
}
|
||||
}
|
||||
},
|
||||
"phase_grid": {
|
||||
"carrier_2.4GHz": {
|
||||
"sigma_phi_0.5deg": {
|
||||
"sigma_range_mm": 0.17349537037037038,
|
||||
"sigma_range_m": 0.00017349537037037038
|
||||
},
|
||||
"sigma_phi_1.0deg": {
|
||||
"sigma_range_mm": 0.34699074074074077,
|
||||
"sigma_range_m": 0.00034699074074074076
|
||||
},
|
||||
"sigma_phi_2.0deg": {
|
||||
"sigma_range_mm": 0.6939814814814815,
|
||||
"sigma_range_m": 0.0006939814814814815
|
||||
},
|
||||
"sigma_phi_5.0deg": {
|
||||
"sigma_range_mm": 1.7349537037037037,
|
||||
"sigma_range_m": 0.0017349537037037036
|
||||
},
|
||||
"sigma_phi_10.0deg": {
|
||||
"sigma_range_mm": 3.4699074074074074,
|
||||
"sigma_range_m": 0.0034699074074074072
|
||||
}
|
||||
},
|
||||
"carrier_5.0GHz": {
|
||||
"sigma_phi_0.5deg": {
|
||||
"sigma_range_mm": 0.08327777777777778,
|
||||
"sigma_range_m": 8.327777777777778e-05
|
||||
},
|
||||
"sigma_phi_1.0deg": {
|
||||
"sigma_range_mm": 0.16655555555555557,
|
||||
"sigma_range_m": 0.00016655555555555556
|
||||
},
|
||||
"sigma_phi_2.0deg": {
|
||||
"sigma_range_mm": 0.33311111111111114,
|
||||
"sigma_range_m": 0.0003331111111111111
|
||||
},
|
||||
"sigma_phi_5.0deg": {
|
||||
"sigma_range_mm": 0.8327777777777777,
|
||||
"sigma_range_m": 0.0008327777777777778
|
||||
},
|
||||
"sigma_phi_10.0deg": {
|
||||
"sigma_range_mm": 1.6655555555555555,
|
||||
"sigma_range_m": 0.0016655555555555555
|
||||
}
|
||||
},
|
||||
"carrier_6.0GHz": {
|
||||
"sigma_phi_0.5deg": {
|
||||
"sigma_range_mm": 0.06939814814814814,
|
||||
"sigma_range_m": 6.939814814814814e-05
|
||||
},
|
||||
"sigma_phi_1.0deg": {
|
||||
"sigma_range_mm": 0.13879629629629628,
|
||||
"sigma_range_m": 0.00013879629629629629
|
||||
},
|
||||
"sigma_phi_2.0deg": {
|
||||
"sigma_range_mm": 0.27759259259259256,
|
||||
"sigma_range_m": 0.00027759259259259257
|
||||
},
|
||||
"sigma_phi_5.0deg": {
|
||||
"sigma_range_mm": 0.6939814814814815,
|
||||
"sigma_range_m": 0.0006939814814814815
|
||||
},
|
||||
"sigma_phi_10.0deg": {
|
||||
"sigma_range_mm": 1.387962962962963,
|
||||
"sigma_range_m": 0.001387962962962963
|
||||
}
|
||||
}
|
||||
},
|
||||
"headline_practical": {
|
||||
"scenario": "20 MHz HT20 channel, 20 dB SNR, 100 averaged frames",
|
||||
"toa_single_shot_m": 0.41322100711863313,
|
||||
"toa_after_100_avg_m": 0.04132210071186332,
|
||||
"phase_single_shot_m": 0.0017349537037037036,
|
||||
"phase_after_100_avg_m": 0.00017349537037037038,
|
||||
"phase_advantage_ratio": 238.17408282221416
|
||||
},
|
||||
"multistatic_4anchor": {
|
||||
"n_anchors": 4,
|
||||
"gdop": 1.5,
|
||||
"toa_position_precision_m": 0.2530451546099066,
|
||||
"phase_position_precision_m": 0.0010624378253564768
|
||||
}
|
||||
}
|
||||
Loading…
Reference in New Issue