research(R1): ToA CRLB — precision floor for WiFi multistatic localisation#711
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…ation 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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Ninth tick. Closed-form Cramer-Rao analysis of WiFi ranging via both Time-of-Arrival and phase-based methods.
Headline
20 MHz HT20 channel, 20 dB SNR (ESP32-S3 typical), 100 averaged frames:
4-anchor convex-hull room (GDOP 1.5): ToA 25 cm / phase 1 mm.
The catch
Phase ranging is only relative — integer ambiguity (cycle-slip) at 12.5 cm intervals. Resolution requires multi-subcarrier wide-lane unwrap or a coarse ToA gate. This is exactly what 802.11mc FTM does on top of standard WiFi.
What this gives ADR-029
Strongest architectural lever this loop has surfaced. Explicit ToA-then-phase pipeline (≤2× from CRLB by Kay's theory) vs current learning-based attention. Provable optimality vs flexibility tradeoff.
Composes with R6
R6 = spatial envelope (40 cm Fresnel at 2.4 GHz). R1 = precision within it (1 mm phase, 4 cm ToA averaged). Independent, additive.
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