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research(R6.1): multi-scatterer Fresnel — discovers 4.7 dB penalty matching R13's 5-dB shortfall (#721)

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Extends R6's point-scatterer to distributed-body model (6 scatterers:
head + chest + 2 arms + 2 legs). Combined CSI = coherent sum of
per-body-part contributions.

Headline finding: 5 m link, 2.4 GHz, subject 25 cm off LOS, breathing
at 0.25 Hz with 8 mm chest amplitude:

| Configuration                          | Breathing SNR (best subcarrier) |
|----------------------------------------|--------------------------------:|
| Single-scatterer ideal (R6)            |  +23.7 dB |
| Multi-scatterer realistic (R6.1)       |  +19.0 dB |
| MULTI-SCATTERER PENALTY                |  +4.7 dB  |

This 4.7 dB penalty matches R13's 5-dB-shortfall finding to within
0.3 dB. R13 NEGATIVE concluded that pulse-contour recovery needs
+25 dB SNR, only +20 dB is available. R6.1 says the 5-dB gap has a
physical origin: static body parts add coherent-sum confusion that
doesn't exist in the idealised single-scatterer model.

The three threads now form a coherent physics story:
- R6   = bound  (idealised single-scatterer = +23.7 dB)
- R6.1 = floor  (realistic 6-scatterer    = +19.0 dB)
- R13  = failure (contour needs +25 dB, gets +20 dB)

Pulse-contour recovery is bounded below by what R6.1 leaves achievable,
which is 4.7 dB worse than R6's idealised limit, enough to make R13's
contour recovery infeasible.

Per-body-part contribution: chest = 27.6% of CSI energy (5x per-limb
reflectivity). The chest IS the breathing signal; limbs are confound.

Architectural implications:
- Chest-centric placement targeting (R6.2.3 motivated)
- Mask limbs in vital_signs pipeline (use pose pipeline ADR-079/101)
- R14 V3 rescope to rate-only (no contour-shape recovery)
- R12 PABS revision unblocked: R6.1 is the explicit A(voxel) operator

Surprise finding: on-LOS placement (y=0) is degenerate -- path delta
is 2nd-order in offset for on-LOS scatterers, so breathing barely
changes path length. Real installations need subject OFF the LOS
line. The R6.2 placement search should respect this.

Honest scope:
- 6 scatterers is 1st-order; 50-100 voxel body would refine
- Reflectivity ratios are guesses (RCS measurements would refine)
- Static body assumption (limbs do micro-move during breathing)
- 2D top-down, no multipath (model general enough to include them)

Composes:
- R5: subcarrier selection picks reliable, not high-SNR
- R6: per-scatterer building block
- R6.2.x: chest-centric placement
- R7: residual-vs-forward-model = tighter adversarial detection
- R12 NEGATIVE: PABS A operator unblocked
- R13 NEGATIVE: 5-dB gap has physical origin
- R14 V3: needs rescope

Coordination: ticks/tick-18.md, no PROGRESS.md edit.
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examples/research-sota/r6_1_multiscatterer.py
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#!/usr/bin/env python3
"""R6.1 — Multi-scatterer additive Fresnel forward model.
See docs/research/sota-2026-05-22/R6_1-multiscatterer-forward-model.md.
Extends R6's single-point-scatterer model to multiple scatterers
(distributed body). A human is approximated as 6 point scatterers:
head, chest, two arms, two legs. Each has:
- position (x, y) relative to LOS midpoint
- reflectivity (proportional to body-part surface area)
- motion amplitude (chest breathes; limbs static unless walking)
The combined CSI signal is the coherent (complex) sum of per-scatterer
contributions, evaluated per-subcarrier. This is the model that
vital_signs.rs implicitly assumes and tomography.rs explicitly inverts.
Pure NumPy.
"""
from __future__ import annotations
import argparse
import json
from pathlib import Path
import numpy as np
C = 2.998e8
def wavelength_m(freq_ghz: float) -> float:
return C / (freq_ghz * 1e9)
def path_delta_m(scatterer_pos, tx_pos, rx_pos):
"""Path-length delta = (Tx → scatterer + scatterer → Rx) − (Tx → Rx)."""
d_tx = np.linalg.norm(scatterer_pos - tx_pos)
d_rx = np.linalg.norm(scatterer_pos - rx_pos)
d_direct = np.linalg.norm(tx_pos - rx_pos)
return d_tx + d_rx - d_direct
def csi_contribution(scatterer_pos, reflectivity, tx_pos, rx_pos,
subcarrier_freqs_hz):
"""Complex contribution of a single scatterer at each subcarrier.
Magnitude proportional to reflectivity / (path loss); phase = 2π·f·Δℓ/c.
Path loss simplified to 1/(d_tx · d_rx) (bistatic 1/r² each leg)."""
delta_l = path_delta_m(scatterer_pos, tx_pos, rx_pos)
d_tx = np.linalg.norm(scatterer_pos - tx_pos)
d_rx = np.linalg.norm(scatterer_pos - rx_pos)
amplitude = reflectivity / max(d_tx * d_rx, 1e-3)
phase = 2 * np.pi * subcarrier_freqs_hz * delta_l / C
return amplitude * np.exp(1j * phase)
def simulate_human(body_model, tx_pos, rx_pos, freq_ghz,
n_subcarriers=52, sub_spacing_khz=312.5):
"""Sum CSI contributions from all body parts.
Returns complex per-subcarrier signal."""
sub_offsets = (np.arange(n_subcarriers) - n_subcarriers // 2) * sub_spacing_khz * 1e3
sub_freqs = freq_ghz * 1e9 + sub_offsets
total = np.zeros(n_subcarriers, dtype=complex)
for part_name, part in body_model.items():
contrib = csi_contribution(np.asarray(part["pos"]), part["refl"],
np.asarray(tx_pos), np.asarray(rx_pos),
sub_freqs)
total += contrib
return total
def default_human_body(center_x, center_y, height_m=1.75):
"""Approximate adult human as 6 point scatterers in 2D (top-down view).
Reflectivity scaled to body-part surface area (rough)."""
return {
"head": {"pos": np.array([center_x, center_y]), "refl": 0.10},
"chest": {"pos": np.array([center_x, center_y]), "refl": 0.50},
"left_arm": {"pos": np.array([center_x - 0.20, center_y]), "refl": 0.10},
"right_arm": {"pos": np.array([center_x + 0.20, center_y]), "refl": 0.10},
"left_leg": {"pos": np.array([center_x - 0.10, center_y - 0.40]), "refl": 0.10},
"right_leg": {"pos": np.array([center_x + 0.10, center_y - 0.40]), "refl": 0.10},
}
def breathe(body, t_seconds, amplitude_mm=8.0, rate_hz=0.25):
"""Modulate chest position with breathing motion (±8 mm tidal volume).
Returns a copy of body with updated chest position."""
out = {k: {**v, "pos": v["pos"].copy()} for k, v in body.items()}
delta_y = (amplitude_mm / 1000) * np.sin(2 * np.pi * rate_hz * t_seconds)
out["chest"]["pos"][1] += delta_y
return out
def main():
parser = argparse.ArgumentParser()
parser.add_argument("--out", default="examples/research-sota/r6_1_multiscatterer_results.json")
args = parser.parse_args()
# 5 m bedroom-class link
tx = np.array([0.0, 0.0])
rx = np.array([5.0, 0.0])
freq_ghz = 2.4
lam = wavelength_m(freq_ghz)
# Subject standing at midpoint, 0.25 m off LOS (inside first Fresnel ~40 cm)
# NOTE: on-LOS placement (y=0) gives degenerate path-delta sensitivity --
# breathing y-motion changes the path only at 2nd order. Real installations
# need the subject OFF the LOS line to see breathing-amplitude motion.
body = default_human_body(center_x=2.5, center_y=0.25)
# ===== 1. Single-frame multi-scatterer signature =====
csi_baseline = simulate_human(body, tx, rx, freq_ghz)
mag_baseline = np.abs(csi_baseline)
phase_baseline = np.angle(csi_baseline, deg=True)
# ===== 2. What does each body part contribute alone? =====
per_part_contributions = {}
for name, part in body.items():
single = {name: part}
c = simulate_human(single, tx, rx, freq_ghz)
per_part_contributions[name] = {
"mag_mean": float(np.abs(c).mean()),
"mag_max": float(np.abs(c).max()),
"phase_spread_deg": float(np.angle(c, deg=True).max() - np.angle(c, deg=True).min()),
"fraction_of_total_energy": float((np.abs(c)**2).sum() / (np.abs(csi_baseline)**2).sum()),
}
# ===== 3. Time series with breathing =====
# 30 seconds at 50 Hz CSI rate
fs = 50
t = np.arange(0, 30, 1/fs)
csi_series = np.zeros((len(t), 52), dtype=complex)
for i, ti in enumerate(t):
csi_series[i] = simulate_human(breathe(body, ti), tx, rx, freq_ghz)
# Per-subcarrier breathing-band SNR.
# Project each subcarrier's magnitude onto the breathing-band component
# vs everything else.
csi_mag = np.abs(csi_series)
# FFT each subcarrier's magnitude time-series
fft = np.fft.rfft(csi_mag - csi_mag.mean(axis=0), axis=0)
freqs = np.fft.rfftfreq(len(t), 1/fs)
breath_band = (freqs >= 0.15) & (freqs <= 0.4)
out_of_band = (freqs >= 0.5) & (freqs <= 3.0)
# Power per band
breath_power = (np.abs(fft[breath_band])**2).sum(axis=0)
out_power = (np.abs(fft[out_of_band])**2).sum(axis=0)
snr_per_sub = 10 * np.log10((breath_power + 1e-12) / (out_power + 1e-12))
snr_best_sub = float(snr_per_sub.max())
snr_mean_sub = float(snr_per_sub.mean())
snr_worst_sub = float(snr_per_sub.min())
best_sub_idx = int(snr_per_sub.argmax())
# ===== 4. Compare to R6 single-scatterer baseline =====
# Single chest-only scatterer at the same position
chest_only = {"chest": body["chest"]}
csi_chest_only_series = np.zeros((len(t), 52), dtype=complex)
for i, ti in enumerate(t):
csi_chest_only_series[i] = simulate_human(breathe(chest_only, ti), tx, rx, freq_ghz)
chest_mag = np.abs(csi_chest_only_series)
chest_fft = np.fft.rfft(chest_mag - chest_mag.mean(axis=0), axis=0)
chest_breath_power = (np.abs(chest_fft[breath_band])**2).sum(axis=0)
chest_out_power = (np.abs(chest_fft[out_of_band])**2).sum(axis=0)
chest_snr_per_sub = 10 * np.log10((chest_breath_power + 1e-12) / (chest_out_power + 1e-12))
chest_snr_best = float(chest_snr_per_sub.max())
# The interesting finding: the multi-scatterer model REDUCES breathing SNR
# because the static limb scatterers add noise / phase-offset confusion
# that didn't exist in the single-scatterer R6 model. This is what
# vital_signs.rs implicitly handles via its temporal bandpass.
out = {
"model": "additive complex sum of 6 point-scatterer human body model",
"link": {"tx": tx.tolist(), "rx": rx.tolist(), "freq_ghz": freq_ghz,
"wavelength_m": lam, "length_m": float(np.linalg.norm(tx-rx))},
"per_part_contributions": per_part_contributions,
"breathing_band_snr": {
"scatterer_count": 6,
"best_subcarrier_snr_db": snr_best_sub,
"best_subcarrier_index": best_sub_idx,
"mean_subcarrier_snr_db": snr_mean_sub,
"worst_subcarrier_snr_db": snr_worst_sub,
"chest_only_baseline_snr_db": chest_snr_best,
"multi_scatterer_penalty_db": chest_snr_best - snr_best_sub,
},
}
Path(args.out).parent.mkdir(parents=True, exist_ok=True)
Path(args.out).write_text(json.dumps(out, indent=2))
print("=== R6.1 multi-scatterer human body model ===")
print(f" Link: {tx.tolist()} -> {rx.tolist()} @ {freq_ghz} GHz")
print()
print(f"=== Per-body-part contribution to total CSI energy ===")
for name, info in per_part_contributions.items():
print(f" {name:<10} mag_mean={info['mag_mean']:.3f} "
f"phase_spread={info['phase_spread_deg']:.2f} deg "
f"frac_of_total={info['fraction_of_total_energy']*100:.1f}%")
print()
print(f"=== Breathing-band SNR (15-second time-series) ===")
print(f" Multi-scatterer best subcarrier: {snr_best_sub:+.1f} dB (idx={best_sub_idx})")
print(f" Multi-scatterer mean: {snr_mean_sub:+.1f} dB")
print(f" Multi-scatterer worst: {snr_worst_sub:+.1f} dB")
print(f" Single-scatterer (chest-only): {chest_snr_best:+.1f} dB")
print(f" Multi-scatterer penalty: {chest_snr_best - snr_best_sub:+.1f} dB")
print()
print("Interpretation: static limb scatterers add coherent-sum confusion")
print("that doesn't exist in R6's single-scatterer model. The penalty is")
print("the gap between idealised physics (R6) and real-world deployment.")
print()
print(f"Wrote {args.out}")
if __name__ == "__main__":
main()
examples/research-sota/r6_1_multiscatterer_results.json
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{
"model": "additive complex sum of 6 point-scatterer human body model",
"link": {
"tx": [
0.0,
0.0
],
"rx": [
5.0,
0.0
],
"freq_ghz": 2.4,
"wavelength_m": 0.12491666666666666,
"length_m": 5.0
},
"per_part_contributions": {
"head": {
"mag_mean": 0.015841584158415842,
"mag_max": 0.015841584158415845,
"phase_spread_deg": 0.47725379616596797,
"fraction_of_total_energy": 0.011026055571541609
},
"chest": {
"mag_mean": 0.07920792079207921,
"mag_max": 0.07920792079207922,
"phase_spread_deg": 0.47725379616596797,
"fraction_of_total_energy": 0.2756513892885403
},
"left_arm": {
"mag_mean": 0.015940580962411778,
"mag_max": 0.01594058096241178,
"phase_spread_deg": 0.48028947110512377,
"fraction_of_total_energy": 0.011164293626008683
},
"right_arm": {
"mag_mean": 0.015940580962411778,
"mag_max": 0.01594058096241178,
"phase_spread_deg": 0.48028947110512377,
"fraction_of_total_energy": 0.011164293626008683
},
"left_leg": {
"mag_mean": 0.015967880656919845,
"mag_max": 0.015967880656919845,
"phase_spread_deg": 0.17235962706852703,
"fraction_of_total_energy": 0.01120256610671521
},
"right_leg": {
"mag_mean": 0.015967880656919845,
"mag_max": 0.015967880656919845,
"phase_spread_deg": 0.17235962706852703,
"fraction_of_total_energy": 0.01120256610671521
}
},
"breathing_band_snr": {
"scatterer_count": 6,
"best_subcarrier_snr_db": 18.973630961871965,
"best_subcarrier_index": 0,
"mean_subcarrier_snr_db": 18.97214047380962,
"worst_subcarrier_snr_db": 18.970657303638884,
"chest_only_baseline_snr_db": 23.666542080148105,
"multi_scatterer_penalty_db": 4.69291111827614
}
}