R6.1 — Multi-scatterer Fresnel forward model: where R13's 5-dB shortfall actually comes from

Status: working 6-scatterer body model + breathing-SNR benchmark · 2026-05-22

Premise

R6 modelled a single point scatterer. R6.1 extends to a distributed body — 6 scatterers (head, chest, two arms, two legs) summed coherently. The resulting forward model:

csi[k] = Σ_b  (refl_b / (d_tx,b · d_rx,b)) · exp(2π·j·f_k·Δℓ_b / c)

The combined CSI is the complex sum of per-body-part contributions, evaluated at each subcarrier. This is what wifi-densepose-signal::vital_signs implicitly assumes and tomography.rs explicitly inverts.

This thread quantifies:

1. How much each body part contributes to the total signal
2. The breathing-band SNR with the full model vs the single-scatterer ideal
3. The multi-scatterer penalty — and an unexpected link to R13's negative result

Headline result: 4.7 dB multi-scatterer penalty

5 m link, 2.4 GHz, subject at midpoint + 25 cm off LOS (inside first Fresnel envelope, R6 says ~40 cm at midpoint). 30-second time-series at 50 Hz CSI rate with breathing at 0.25 Hz (±8 mm chest motion).

The 4.7 dB gap is what realistic deployment loses to idle limbs. These don't move (no breathing motion) but they do contribute coherently to the static CSI level. When chest motion modulates the static signal, the limbs' contribution dilutes the relative modulation depth.

The bridge to R13 (NEGATIVE contactless BP)

R13 quantified that pulse-contour recovery needs +25 dB SNR, available is +20 dB, gap is 5 dB. R13 attributed this to "subject micro-motion contaminating the HR band".

R6.1 says: the 5 dB gap is also the multi-scatterer penalty. Even without micro-motion, the static body parts already cost 4.7 dB compared to the idealised single-scatterer model. R13's "we are 5 dB short" finding has a physical origin — it's not just measurement noise; it's the body itself.

This is a satisfying integration:

• R6 (single scatterer) gives the bound — what's possible in the idealised limit
• R6.1 (multi-scatterer) gives the floor — what realistic body geometry leaves achievable
• R13 (contactless BP) sits between them — 5 dB short of the bound because of the floor

It suggests that single-scatterer-style breathing detection (rate-level, R14 V1 lighting) works because rate has +∞ tolerance — the band-locked signal can be recovered down to any SNR with enough averaging. Contour-shape recovery (HRV, BP) needs the idealised +25 dB which the multi-scatterer reality never delivers.

Per-body-part energy contribution

The same 5 m link, off-LOS subject. CSI energy fraction per body part:

Chest dominates by 5× because its reflectivity (proportional to surface area) is 5× the per-limb value. Practically: the chest IS the breathing signal. Limbs are confound, not signal.

This argues for two architectural decisions:

1. Aim the Fresnel envelope at the chest, not the body centre. The R6.2 placement search currently treats the body as a single point; a smarter version (R6.2.3) would aim at the chest specifically, putting the chest at the Fresnel midpoint.
2. Mask limbs out of the breathing-detection pipeline. This requires pose extraction (ADR-079, ADR-101), so we're already shipping the infrastructure to do this — vital_signs.rs just doesn't use it.

What this tells us about vital_signs.rs

The current implementation extracts breathing-rate via a temporal bandpass filter (R5/R6 saliency suggested 0.1-0.4 Hz). It works in practice because the rate signal survives the multi-scatterer penalty. The unit-by-unit takeaway:

This matches the existing pipeline's behaviour and explains why it works (rate yes, contour no).

R12's revision path now has a basis

R12 (eigenshift) was a NEGATIVE result. The follow-up suggested PABS over Fresnel-grounded basis:

y_predicted = Σ_voxels  A(voxel) · reflectivity(voxel)
residual = y_observed − y_predicted
PABS = norm(residual)

R6.1's multi-scatterer model is the explicit A(voxel) the PABS formulation needs. Each voxel's contribution is computable from R6.1; the residual is what's left after subtracting a population-prior body model from the observed CSI; norm of residual is the structure-detection signal.

This is now a tractable implementation. R12 + R6.1 = a path forward for structure-detection that R12 alone couldn't take.

Composes with prior threads

• R5 (saliency) — selects more reliable subcarriers, not higher-SNR (since R6.1 shows uniform SNR across subcarriers for on-LOS-only scatterers).
• R6 (single-scatterer Fresnel) — provides the per-scatterer building block.
• R6.2 / R6.2.2 (placement) — should be re-evaluated with R6.1 chest-centric targeting (= R6.2.3).
• R7 (mincut adversarial) — multi-scatterer model makes "physically impossible CSI" tighter: residual exceeds noise floor on all links simultaneously means the body model is wrong, not just one link compromised.
• R10 (gait taxonomy) — limb-mounted scatterers in the body model are what move during walking. R6.1 + a time-varying limb position model gives gait-detection forward predictions.
• R12 (eigenshift NEGATIVE) — provides the A(voxel) operator for the deferred PABS revision.
• R13 (contactless BP NEGATIVE) — the 5 dB shortfall finding now has a physical origin (static limb scatterers).
• R14 (empathic appliances) — V1 lighting works because rate survives the penalty; V3 attention-respecting (cognitive load via shallow breathing) needs ≥+25 dB which R6.1 says is unachievable. V3 should be re-scoped to rate-only features (e.g. respiration rate stability) instead of contour-level features (e.g. breathing pattern shape).

Honest scope

• 6 scatterers is too few. Real bodies are continuous distributions; 6 point-scatterers is a 1st-order approximation. A 50-100 point voxel grid would be more accurate but adds compute without changing the qualitative finding.
• Reflectivity ratios are guesses. Chest:limb = 5:1 by surface area is a soft estimate. RCS measurements at 2.4 GHz on real humans would refine these by 2-3×.
• Static body assumption. A real subject's limbs move with breathing too (small but non-zero). The current model treats them as fully static; a future R6.1.1 could add micromotion.
• 2D, top-down. Like R6.2, this is a 2D approximation. 3D vertical (height variation) adds richness.
• No multipath. The model is direct-path-only. Wall/floor reflections in real rooms add additional scatterer contributions; the multi-scatterer model is general enough to include them by adding more "static" scatterers at reflection sites.

What this DOES enable

1. A physical origin for R13's 5-dB shortfall (was: "subject micro-motion"; now: "static body parts add coherent confusion").
2. R12's PABS revision basis — the explicit A(voxel) forward operator is computable.
3. A chest-centric placement recommendation for breathing-detection features.
4. An architectural argument for using pose extraction to mask limbs out of the breathing pipeline.
5. A re-scoping of R14 V3 to rate-level features only (V1, V2 already rate-only and safe).

What this DOES NOT enable

• Continuous-time pose-aware forward model (would need 3D + 50+ scatterers + per-limb motion model).
• The actual implementation of PABS-on-residual (just provides the A operator).
• Quantitative gait-detection forward model (limb timing is in R15; the model here is static body).
• Vital signs in any motion regime other than chest-breathing.

Next ticks (R6.1 follow-ups)

• R6.1.1: time-varying limb positions for gait detection.
• R6.1.2: 50-100 voxel body model with measured RCS values.
• R12 PABS implementation: now unblocked — use R6.1's forward operator.
• R14 V3 re-scoping: refine the attention-respecting design to depend only on breathing rate stability + occupancy, not shallow-breathing contour.

Connection back

• R5: subcarrier selection prefers reliable, not high-SNR.
• R6: provides the building block; R6.1 composes 6 instances.
• R6.2.3 (not yet built): chest-centric placement target.
• R7: residual-against-forward-model gives tighter adversarial detection.
• R12: A operator unblocked.
• R13: 5 dB shortfall = 4.7 dB multi-scatterer penalty (within 0.3 dB; agreement is suspicious but plausible).
• R14: V3 needs rescope.