# ⚛  L1 Principle — Direction-of-Arrival Estimation (narrowband array processing)

**ID:** `L1-387` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Signal Processing — *Spatial spectral estimation with sensor arrays*
> **🎯 Problem class:** parametric spectral estimation · **🧮 Solution space:** angle vector
> **📡 Carrier:** RF · **🌫 Noise:** complex gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41555197

---

## 🧠 1. Introduction

**Direction-of-Arrival Estimation (narrowband array processing)** is a **parametric spectral estimation** whose unknown lives in **angle vector** space, within the **Spatial spectral estimation with sensor arrays** sub-domain of **Signal Processing**.

Measurements consist of RF coil signals (typical of MRI) via a **sensor array** sensing mechanism.

The forward operator applies, in order: L · project · arraymanifold operator; detector accumulates flux over the exposure window.

Observations are corrupted by complex-valued Gaussian noise on real and imaginary parts. Identifiable when K <= L-1 for ULA; CRLB proportional to 1/(T * SNR * array_aperture); coherent sources require spatial smoothing; sub-wavelength (super-resolution) regimes rely on MUSIC/ESPRIT subspace projection or L1-min.

## ⚙ 2. Forward Model

Physical chain: **x** → L · project · arraymanifold → Temporal integration → **y** (detector).

```
y = ∫_t dt `L.project.arraymanifold` x + n,    n ~ 𝒞𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.project.arraymanifold` | L · project · arraymanifold operator |
| `int.temporal` | Detector accumulates flux over the exposure window |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Signal Processing |
| Sub domain | Spatial spectral estimation with sensor arrays |
| Carrier | RF |
| Problem class | parametric_spectral_estimation |
| Solution space | angle_vector |
| Noise model | complex_gaussian |
| Integration axis | temporal |
| Difficulty delta | 3 |
| L dag | 2.3 |

## 📡 4. Measurement Model

Identifiable when K <= L-1 for ULA; CRLB proportional to 1/(T * SNR * array_aperture); coherent sources require spatial smoothing; sub-wavelength (super-resolution) regimes rely on MUSIC/ESPRIT subspace projection or L1-min.

| Metric | Value |
|---|---|
| Metric | RMSE_deg |
| Secondary | detection_probability |

## 📏 5. Operating Range (Ω)

**Center problem class:** `narrowband_doa` · **Forward operator:** `ula_narrowband`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| K | — | 2 |
| L | — | 8 |
| T | — | 100 |
| Snr db | dB | 10 |
| D over lambda | — | 0.5 |
| Calib gain err | — | 0 |
| Calib phase err deg | deg | 0 |
| Position err lambda | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| K | — | 1 – 32 |
| L | — | 4 – 128 |
| T | — | 1 – 10000 |
| Snr db | dB | -10 – 40 |
| Correlation | — | 0.0 – 1.0 |
| D over lambda | — | 0.25 – 2.0 |
| Calib gain err | — | 0.0 – 0.2 |
| Calib phase err deg | deg | 0.0 – 15 |
| Position err lambda | — | 0.0 – 0.05 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 0.5 deg RMSE

| Metric | Range |
|---|---|
| Rmse deg | 0.01 – 10.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **RMSE_deg**, with κ = `500` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xa4118413094b567041253e9362a4b1c59493c803aebca8ca8a39dd16f0818378`
- **Chain tx hash:** `0x7eb61b3751320ff4cade70b4ec32589eb593c2f4cb67d16d1cb7131650ea418c`
- **Chain block:** `41555197`

---

## File Mapping

This bundle consists of: `L1-387.md`, `L1-387.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-387.md` | Source of truth — edit this | Human or LLM |
| `L1-387.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 `L1-387`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._