# ⚛  L1 Principle — Acoustic Beamforming — DOA & imaging from array

**ID:** `L1-265` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Acoustics — *Sensor-array DOA estimation*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** angular power spectrum
> **📡 Carrier:** acoustic · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41554049

---

## 🧠 1. Introduction

**Acoustic Beamforming — DOA & imaging from array** is a **linear inverse problem** whose unknown lives in **angular power spectrum** space, within the **Sensor-array DOA estimation** sub-domain of **Acoustics**.

Measurements consist of acoustic pressure waves recorded by transducers via a **microphone array** sensing mechanism.

The forward operator applies, in order: S · array capture operator; L · steering vector operator; L · power map operator; pixel-level spatial averaging on the detector.

Observations are corrupted by additive Gaussian noise. Conditional stability; mismatch parameters dominate at Omega bounds.

## ⚙ 2. Forward Model

Physical chain: **x** → S · array capture → L · steering vector → L · power map → Spatial integration → **y** (detector).

```
y = ∫_A dA `L.power_map` `L.steering_vector` `S.array_capture` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `S.array_capture` | S · array capture operator |
| `L.steering_vector` | L · steering vector operator |
| `L.power_map` | L · power map operator |
| `int.spatial` | Pixel-level spatial averaging on the detector |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Acoustics |
| Sub domain | Sensor-array DOA estimation |
| Carrier | acoustic |
| Problem class | linear_inverse |
| Solution space | angular_power_spectrum |
| Noise model | gaussian |
| Integration axis | spatial |
| Difficulty delta | 3 |
| L dag | 3 |

## 📡 4. Measurement Model

Conditional stability; mismatch parameters dominate at Omega bounds.

| Metric | Value |
|---|---|
| Metric | DOA_MAE_deg |
| Secondary | SSIM |

## 📏 5. Operating Range (Ω)

**Center problem class:** `doa_beamforming` · **Forward operator:** `doa_beamforming_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N mic | — | 16 |
| Snr db | dB | 20 |
| N sources | — | 2 |
| N snapshots | — | 256 |
| Sensor pos error | — | 0 |
| Element gain error | — | 0 |
| Element phase error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N mic | — | 4 – 256 |
| Snr db | dB | -10 – 40 |
| N sources | — | 1 – 8 |
| N snapshots | — | 16 – 8192 |
| Sensor pos error | — | 0.0 – 0.1 |
| Element gain error | — | 0.0 – 0.2 |
| Element phase error | — | 0.0 – 0.3 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 1.5

| Metric | Range |
|---|---|
| Doa deg | 0.1 – 10.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **DOA_MAE_deg**, with κ = `240` and δ = `3`.

## 💾 8. Reference Dataset

- **primary** · weight 1.0 · IPFS _(not pinned yet)_

## 9. On-chain Registration

- **Chain hash:** `0x7f79f6012053e88be1f95fb54349778fd6dfeeb0ee937d05f0ee560cc6add08f`
- **Chain tx hash:** `0x4b234590397c21b94b417cdc38890342e0c837d8584dc707ff7f3a0b02259262`
- **Chain block:** `41554049`

---

## File Mapping

This bundle consists of: `L1-265.md`, `L1-265.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-265.md` | Source of truth — edit this | Human or LLM |
| `L1-265.json` | Structured metadata for the registry | LLM regenerates from the sections above |

**Prompt for your LLM after editing this Markdown:**

> Read the attached Markdown. Regenerate the sibling `.json` so every field matches.
> Preserve the schema documented in the rows above.
> Output each file in its own fenced code block tagged with the filename.
> Output only the JSON object.

_This Markdown was auto-synthesized from the catalog row for `L1-265`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._