R1 — ToA CRLB: the precision floor for WiFi multistatic localisation

Status: closed-form CRLB analysis + numpy demo · 2026-05-22

Why this thread exists

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.

WiFi gives us two ways to estimate range:

1. Time-of-Arrival (ToA) — measure the absolute travel time of a known pulse. Limited by bandwidth.
2. Phase-based ranging — measure the carrier phase change between samples. Limited by phase noise; needs integer-ambiguity resolution.

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.

ToA precision floor (Cramér-Rao)

For a matched-filter ToA estimator at bandwidth B and SNR ρ:

σ_ToA  ≥  1 / (2π · β_rms · √ρ)        (Kay 1993, eq. 3.14)
σ_d    =  c · σ_ToA

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.

Single-shot range CRLB (m, 1σ)

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.

That's the absolute best WiFi-bandwidth ToA can ever do for room-scale localisation. Below that floor is physically forbidden.

Phase-based ranging precision

The same demo computes single-subcarrier phase-derived ranging. At carrier f_c with phase noise σ_φ (radians):

σ_d_phi = (c / 2π · f_c) · σ_φ = λ · σ_φ / 2π

Single-subcarrier phase range precision (mm, 1σ)

The reference 5° phase-noise figure is what ESP32-S3 typically achieves after phase_align.rs's LO-offset correction.

Headline comparison

Same scenario: 20 MHz HT20, 20 dB SNR, 100 averaged frames.

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.

The catch: integer ambiguity

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.

Resolution methods:

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.
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.
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.

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.

Multistatic 4-anchor geometry

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:

σ_pos = σ_range · √(GDOP / N_anchors)

Practical result (20 MHz, 20 dB SNR, single-shot):

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.

What this means for ADR-029 (multistatic sensing)

The current multistatic.rs uses learned attention weights over raw CSI. The CRLB analysis suggests an explicit decomposition would do better:

1. ToA stage: get coarse range per Tx-Rx pair (~25 cm precision).
2. Phase stage: unwrap phase against the ToA gate, get mm-precision range.
3. Multistatic stage: solve for 3D position via weighted least squares over the high-precision ranges.

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.

Honest scope

• 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.
• 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×.
• 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.
• 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.

What this DOES enable

• 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.
• 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.
• 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.

What this DOES NOT enable

• Sub-mm ranging — that's microwave-photonics territory, not WiFi.
• Multipath-free assumption — every real deployment is multipath-rich.
• 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.

Connection back

• 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.
• 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.
• 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.
• ADR-029 (multistatic) — strongest concrete architectural lever this loop has surfaced.

Next ticks (R1 follow-ups)

• Implement multi-subcarrier wide-lane phase unwrap as a Rust module; measure how often cycle-slip resolution succeeds vs the ToA gate width.
• Empirical CRLB test: log 1000 ranging measurements from a known-position scatterer, check whether observed σ_d hits ~2× CRLB.
• 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.