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R6 — Fresnel-zone forward model: making CSI sensitivity predictable

Status: working forward model + numpy demo · 2026-05-22

The gap this fills

The entire wifi-densepose-signal DSP pipeline — vital_signs, multistatic, pose_tracker — operates on CSI windows whose physical meaning is taken for granted. We measure complex per-subcarrier amplitudes, treat them as input features, and learn classifiers. Nobody in the repo has written down the forward model: given a known scatterer position + size + reflectivity, what does the CSI look like?

Without a forward model:

R12 (eigenshift) was forced to invent its own subspace basis from data — and discovered it was indistinguishable from natural drift.
R7 (multi-link consistency) had to bootstrap an adversarial detector from scratch instead of comparing against a physics-grounded expectation.
R10 (foliage range) had to use ITU-R + FSPL alone, ignoring the fact that an obstacle larger than the first Fresnel zone causes diffraction loss that no FSPL model captures.

This tick makes the forward model explicit. Self-contained numpy; no dependencies on the workspace.

The model

For a Tx-Rx link of length L, the first Fresnel zone is the prolate ellipsoid where most of the diffracted RF energy travels. Its radius at fractional position p ∈ [0, 1] along the LOS is:

r_1(p) = sqrt(λ · L · p · (1 − p))      [metres]

A point scatterer at perpendicular offset x from the LOS, at link position d_1 from Tx (so d_2 = L − d_1 from Rx), introduces a path-length delta:

Δℓ(x) = sqrt(d_1² + x²) + sqrt(d_2² + x²) − (d_1 + d_2)

Phase shift on subcarrier k with centre frequency f_k:

φ_k = 2π · f_k · Δℓ / c

That's it. Six lines that the entire workspace's DSP secretly assumes.

What the demo computes

examples/research-sota/r6_fresnel_zone.py runs four canonical scenarios and emits per-subcarrier phase predictions for 802.11n/ac 20 MHz channels (52 used subcarriers, 312.5 kHz spacing):

First Fresnel radii (the basic envelope)

Link length	2.4 GHz @ midpoint	5 GHz @ midpoint
2 m	25.0 cm	17.3 cm
5 m	39.5 cm	27.4 cm
10 m	55.9 cm	38.7 cm

These are measurable, physical envelopes: a 5 m WiFi link in a typical bedroom has a roughly 40 cm wide "channel of maximum sensitivity" centered on the LOS, narrowing toward each antenna. A human standing inside that ellipsoid moves the entire CSI vector; a human standing outside it perturbs only edge subcarriers.

Single-scatterer predictions

Scenario	Offset	Position	Zone @ 2.4 GHz	Phase spread
Human standing at midpoint	10 cm	2.5 m	zone-1	0.077°
Human walking into Fresnel	25 cm	2.5 m	zone-1	0.477°
Scatterer outside Fresnel	1.5 m	2.5 m	far-field	15.9°
Scatterer near Tx	5 cm	0.5 m	zone-1	0.053°

Key insight (concrete now): the phase spread across subcarriers grows monotonically with Δℓ, which grows quadratically with offset x. A scatterer in the far field (15.9° spread across 52 subcarriers) is the regime where multi-tap channel estimation works well. A scatterer inside the first Fresnel zone (<0.5° spread) is essentially uniform across subcarriers — which is why R5's saliency revealed band-spread top subcarriers (the scatterer effectively excites the whole band) rather than tight clusters.

This unifies R5 and R6: the saliency band-spread we measured experimentally is exactly what the Fresnel forward model predicts for inside-zone-1 occupancy.

Why this matters for the workspace

Existing module	What R6 gives it
`vital_signs` (breathing/HR)	Predicts that chest-wall motion at ~1 cm amplitude inside zone-1 produces 0.01–0.05° phase change per breath — sets the floor SNR for HR detection
`multistatic.rs` (attention-weighted fusion)	Provides ground-truth weights: scatterers in different Fresnel zones contribute different per-subcarrier phase signatures, so the attention weights have a closed-form prior
`tomography.rs` (RF tomography)	Forward operator A in `Ax = y` was a black box; R6 makes A explicit (per-voxel position → per-subcarrier phase contribution) so the L1-ISTA inverse problem becomes properly conditioned
`pose_tracker.rs` (17-keypoint Kalman)	The "sensitivity to limb position" prior is now derivable from the Fresnel geometry — distal limbs (hands, feet) often sit outside the first Fresnel zone for indoor links, explaining why they're harder to track than torso/head

Connection to R12

R12 (eigenshift) failed because the SVD spectrum is a 1-D summary that loses the spatial structure the Fresnel forward model preserves. The right revision is:

y_predicted = sum_voxels  A(voxel) · reflectivity(voxel)
residual = y_observed − y_predicted
PABS = norm(residual)   # the structure-detection signal

where A(voxel) is exactly the per-subcarrier phase prediction from R6. This is essentially RF tomography, but used as a structure-detection prior rather than as inverse reconstruction. PABS-over-Fresnel-grounded-basis is the right next step that R12 explicitly identified — R6 supplies the basis.

Connection to R10 (the wildlife angle)

R10's range estimates used FSPL + ITU foliage attenuation. But foliage also blocks the first Fresnel zone, and an obstacle filling >60% of the zone produces diffraction loss that FSPL alone misses. For the 2.4 GHz / 100 m sparse case, the first Fresnel zone at midpoint is sqrt(0.125 · 100 · 0.5 · 0.5) = 1.77 m wide — large enough that a tree trunk in the middle of the link cuts deeply into it.

A more honest sparse-foliage range, accounting for partial zone obstruction: probably closer to 70 m than 100 m for canopies with ~1.5 m vertical clearance. Documented here as a known under-estimate of the range we should retract toward in any field deployment.

Honest scope

Point scatterer. Real bodies are distributed scatterers (limbs, chest, head — all at different positions in the zone). The full forward model is a volume integral over body-mounted RCS, not the scalar Δℓ here. The scalar version is the correct first-order approximation.
First Fresnel only. Real diffraction includes contributions from zones 2..N (the Cornu spiral). For obstacle classification (presence/absence/size) zone-1 dominates and the model is enough. For phase-precise reconstruction (millimeter-wave-style imaging) we'd need to sum over more zones.
Frequency-flat scatterers. We assume the scatterer's reflectivity is constant across the 20 MHz channel. Real biological tissue has frequency-dependent permittivity; the error is small at WiFi bands but non-zero.
LOS-only. Multipath (floor / ceiling / wall reflections) is not modeled. In a real bedroom there are typically 4-6 dominant reflectors, each contributing its own Δℓ. The full multipath model is just a sum of single-scatterer terms with their own A matrices — additive in the forward direction, harder to invert.

What this DOES enable

Closed-form sensitivity bounds. For any specified (link length, frequency, scatterer position+size) we can predict the per-subcarrier signature analytically. Removes mystery from "why does this signal look like this?"
R12 revision path with a basis. PABS computed against a Fresnel-grounded forward operator is the right structure-detection signal.
Antenna-placement heuristics. For a given room, R6 immediately predicts where the Fresnel envelope sits and which sensor positions maximise coverage. The current installation-guide is "guess and measure"; R6 enables "compute and validate."
R10 range correction. Foliage range estimates should be discounted for partial Fresnel-zone obstruction. ~30% conservative correction in the sparse case.

What this DOES NOT enable

Without antenna calibration, the absolute phase predictions are off by a constant per-subcarrier offset (the LO phase, per-antenna delay, etc.). The relative predictions (phase spread across subcarriers; phase change between consecutive windows) survive. The existing phase_align.rs handles the calibration step.
Multipath-rich environments need the multi-scatterer extension before R6 is quantitatively useful.

Next ticks (R6 follow-ups)

PABS over Fresnel basis: implement R12's revision — observed CSI minus forward-model prediction, structure detection on the residual. Should improve R12's 0.69× signal/drift ratio.
R6.1 — multi-scatterer additive forward model: sum over a coarse voxel grid, see whether breathing-rate estimation accuracy improves vs the current vital_signs heuristic.
R6.2 — Fresnel-aware antenna placement: given a room geometry + target occupancy zones, solve for the antenna positions that maximise Fresnel-envelope coverage. Could ship as a CLI tool in wifi-densepose-cli.

Connection back

R5 (saliency) — band-spread top subcarriers are exactly what zone-1 occupancy predicts. R5 measured it; R6 explains it.
R7 (mincut adversarial) — physically inconsistent CSI is now well-defined: residual from R6's forward model exceeds noise floor across all links simultaneously. Stoer-Wagner mincut detects the violation.
R10 (foliage range) — Fresnel-zone obstruction adds ~30% range discount in sparse-foliage scenarios; the 100 m number should be retracted to ~70 m.
R12 (eigenshift) — the failed SVD-spectrum approach has a clear successor: PABS over Fresnel-grounded basis.
R14 (empathic appliances) — Fresnel-envelope sensitivity bound sets the per-room calibration floor for the V1 stress-responsive lighting use case.
ADR-029 (multistatic) — provides the closed-form attention-weight prior the current learned-weights system lacks.

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