L1 ⚛ L1-394 ⊙ Testnet

Spectral Estimation (line spectrum / parametric PSD)

Signal Processing · Frequency estimation from finite-length time series

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⬇ principle.md ⬇ principle.json

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⬇ L1-394.md ⬇ L1-394.json

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You are helping me submit a MODIFICATION of L1-394 (Spectral Estimation (line spectrum / parametric PSD)) to the PWM Protocol — a Principle (L1) artifact.

I will paste a Markdown template (or the current L1-394.md).
1. Rewrite the Markdown so the science is correct and clearly explained for my change.
2. Regenerate the sibling L1-394.json so EVERY field matches the Markdown.
3. Keep the schema in the "File Mapping" footer at the bottom of the MD.
4. Keep the parent reference unchanged unless I ask otherwise.
Rules: the Markdown is the source of truth; use SI units; do NOT invent benchmark scores.
Output each file in its own fenced code block tagged with the filename.

Here is my template:
[PASTE THE .md HERE]

⚙ Forward Model

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

world state x (frequency parameters)

→

L · synthesis · harmonic `L.synthesis.harmonic`

→

Temporal integration ∫_t dt

→

observation y (electromagnetic detector)

Noise: complex-valued Gaussian noise on real and imaginary parts

Markdown — human-readable source of truth

⚙ auto-generated

⬇ L1-394.md

# ⚛ 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._

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⌥ Hierarchy

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L2-394-001 Spectral Estimation Uniform-Sampling Spec (subspace methods)

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📋 JSON metadata

{
  "artifact_id": "L1-394",
  "chain_block": 41555199,
  "chain_hash": "0x8b7b9b8bb8e50340050b1b6979796ad5c35f3aa0b15fafd5955c2a74536ec656",
  "chain_tx_hash": "0x40c36b2c90f20beb2ccd799e355c82e9382e6a85e40abd8d246905c9e4574667",
  "domain": "Signal Processing",
  "hardness_fn": {
    "delta": 3,
    "kappa": 500,
    "metric": "frequency_RMSE_cyc",
    "type": "epsilon_fn"
  },
  "initiator_dataset": [
    {
      "ipfs_cid": null,
      "license_hash": null,
      "name": "primary",
      "weight": 1.0
    }
  ],
  "layer": "L1",
  "observable_profile": {
    "metric": "frequency_RMSE_cyc",
    "regime": "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 \u003e 4pi/N.",
    "secondary": "amplitude_RMSE"
  },
  "physics_fingerprint": {
    "L_DAG": 2.3,
    "carrier": "electromagnetic",
    "difficulty_delta": 3,
    "domain": "Signal Processing",
    "integration_axis": "temporal",
    "noise_model": "complex_gaussian",
    "primitives": [
      "L.synthesis.harmonic",
      "int.temporal"
    ],
    "problem_class": "parametric_spectral_estimation",
    "sensing_mechanism": "harmonic_observation",
    "solution_space": "frequency_parameters",
    "sub_domain": "Frequency estimation from finite-length time series",
    "title": "Spectral Estimation (line spectrum / parametric PSD)"
  },
  "size_tiers": {
    "allowed_forward_operators": [
      "uniform_sampling_complex_exp",
      "nonuniform_sampling",
      "windowed_observation"
    ],
    "allowed_omega_dimensions": [
      "N",
      "K",
      "SNR_dB",
      "delta_omega",
      "sampling",
      "windowing_error",
      "K_overestimation"
    ],
    "allowed_problem_classes": [
      "line_spectrum_estimation",
      "ar_ma_estimation",
      "atomic_norm_dereverberation"
    ],
    "center_spec": {
      "epsilon_fn_center": "1e-3 cyc",
      "forward_operator": "uniform_sampling_complex_exp",
      "input_format": "time_series",
      "omega": {
        "K": 3,
        "N": 256,
        "SNR_dB": 20,
        "delta_omega": 0.05,
        "sampling": "uniform"
      },
      "problem_class": "line_spectrum_estimation"
    },
    "epsilon_bounds": {
      "rmse_cyc": [
        1e-06,
        0.5
      ]
    },
    "omega_bounds": {
      "K": [
        1,
        64
      ],
      "K_overestimation": [
        0,
        30
      ],
      "N": [
        32,
        16384
      ],
      "SNR_dB": [
        -10,
        40
      ],
      "delta_omega": [
        0.001,
        0.5
      ],
      "sampling_irregularity": [
        0.0,
        0.5
      ],
      "windowing_error": [
        0.0,
        0.5
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Frequency estimation from finite-length time series",
  "title": "Spectral Estimation (line spectrum / parametric PSD)"
}

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