# ⚛  L1 Principle — Spectral Estimation (line spectrum / parametric PSD)

**ID:** `L1-394` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Signal Processing — *Frequency estimation from finite-length time series*
> **🎯 Problem class:** parametric spectral estimation · **🧮 Solution space:** frequency parameters
> **📡 Carrier:** electromagnetic · **🌫 Noise:** complex gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41555199

---

## 🧠 1. Introduction

**Spectral Estimation (line spectrum / parametric PSD)** is a **parametric spectral estimation** whose unknown lives in **frequency parameters** space, within the **Frequency estimation from finite-length time series** sub-domain of **Signal Processing**.

Measurements consist of electromagnetic field measurements via a **harmonic observation** sensing mechanism.

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

Observations are corrupted by complex-valued Gaussian noise on real and imaginary parts. Fourier-limited resolution 2pi/N; subspace methods (MUSIC, ESPRIT) achieve super-resolution given SNR; atomic-norm minimization (Tang-Bhaskar-Shah-Recht) guarantees exact recovery when delta_omega > 4pi/N.

## ⚙ 2. Forward Model

Physical chain: **x** → L · synthesis · harmonic → Temporal integration → **y** (detector).

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

**Measurement DAG:**

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

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Signal Processing |
| Sub domain | Frequency estimation from finite-length time series |
| Carrier | electromagnetic |
| Problem class | parametric_spectral_estimation |
| Solution space | frequency_parameters |
| Noise model | complex_gaussian |
| Integration axis | temporal |
| Difficulty delta | 3 |
| L dag | 2.3 |

## 📡 4. Measurement Model

Fourier-limited resolution 2pi/N; subspace methods (MUSIC, ESPRIT) achieve super-resolution given SNR; atomic-norm minimization (Tang-Bhaskar-Shah-Recht) guarantees exact recovery when delta_omega > 4pi/N.

| Metric | Value |
|---|---|
| Metric | frequency_RMSE_cyc |
| Secondary | amplitude_RMSE |

## 📏 5. Operating Range (Ω)

**Center problem class:** `line_spectrum_estimation` · **Forward operator:** `uniform_sampling_complex_exp`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| K | — | 3 |
| N | — | 256 |
| Snr db | dB | 20 |
| Sampling | — | uniform |
| Delta omega | — | 0.05 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| K | — | 1 – 64 |
| N | — | 32 – 16384 |
| Snr db | dB | -10 – 40 |
| Delta omega | — | 0.001 – 0.5 |
| Windowing error | — | 0.0 – 0.5 |
| K overestimation | — | 0 – 30 |
| Sampling irregularity | — | 0.0 – 0.5 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 1e-3 cyc

| Metric | Range |
|---|---|
| Rmse cyc | 1e-06 – 0.5 |

## ⚖ 7. Hardness Function

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

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x8b7b9b8bb8e50340050b1b6979796ad5c35f3aa0b15fafd5955c2a74536ec656`
- **Chain tx hash:** `0x40c36b2c90f20beb2ccd799e355c82e9382e6a85e40abd8d246905c9e4574667`
- **Chain block:** `41555199`

---

## File Mapping

This bundle consists of: `L1-394.md`, `L1-394.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-394.md` | Source of truth — edit this | Human or LLM |
| `L1-394.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 `L1-394`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._