L1 ⚛ L1-387 ⊙ Testnet

Direction-of-Arrival Estimation (narrowband array processing)

Signal Processing · Spatial spectral estimation with sensor arrays

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⬇ L1-387.md

# ⚛ 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._

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Read the attached Markdown (L1-387.md).
Regenerate L1-387.json so every field matches the Markdown.
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L2-387-001 DoA Estimation Calibration-Mismatch Spec (ULA family)

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📋 JSON metadata

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  "chain_block": 41555197,
  "chain_hash": "0xa4118413094b567041253e9362a4b1c59493c803aebca8ca8a39dd16f0818378",
  "chain_tx_hash": "0x7eb61b3751320ff4cade70b4ec32589eb593c2f4cb67d16d1cb7131650ea418c",
  "domain": "Signal Processing",
  "hardness_fn": {
    "delta": 3,
    "kappa": 500,
    "metric": "RMSE_deg",
    "type": "epsilon_fn"
  },
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  "layer": "L1",
  "observable_profile": {
    "metric": "RMSE_deg",
    "regime": "Identifiable when K \u003c= 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.",
    "secondary": "detection_probability"
  },
  "physics_fingerprint": {
    "L_DAG": 2.3,
    "carrier": "RF",
    "difficulty_delta": 3,
    "domain": "Signal Processing",
    "integration_axis": "temporal",
    "noise_model": "complex_gaussian",
    "primitives": [
      "L.project.arraymanifold",
      "int.temporal"
    ],
    "problem_class": "parametric_spectral_estimation",
    "sensing_mechanism": "sensor_array",
    "solution_space": "angle_vector",
    "sub_domain": "Spatial spectral estimation with sensor arrays",
    "title": "Direction-of-Arrival Estimation (narrowband array processing)"
  },
  "size_tiers": {
    "allowed_forward_operators": [
      "ula_narrowband",
      "uca_narrowband",
      "arbitrary_array",
      "wideband_array"
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      "K",
      "T",
      "SNR_dB",
      "d_over_lambda",
      "calib_gain_err",
      "calib_phase_err_deg",
      "position_err_lambda",
      "correlation"
    ],
    "allowed_problem_classes": [
      "narrowband_doa",
      "wideband_doa",
      "coherent_doa",
      "compressive_doa"
    ],
    "center_spec": {
      "epsilon_fn_center": "0.5 deg RMSE",
      "forward_operator": "ula_narrowband",
      "input_format": "snapshots_only",
      "omega": {
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        "L": 8,
        "SNR_dB": 10,
        "T": 100,
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        "calib_phase_err_deg": 0.0,
        "d_over_lambda": 0.5,
        "position_err_lambda": 0.0
      },
      "problem_class": "narrowband_doa"
    },
    "epsilon_bounds": {
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        0.01,
        10.0
      ]
    },
    "omega_bounds": {
      "K": [
        1,
        32
      ],
      "L": [
        4,
        128
      ],
      "SNR_dB": [
        -10,
        40
      ],
      "T": [
        1,
        10000
      ],
      "calib_gain_err": [
        0.0,
        0.2
      ],
      "calib_phase_err_deg": [
        0.0,
        15
      ],
      "correlation": [
        0.0,
        1.0
      ],
      "d_over_lambda": [
        0.25,
        2.0
      ],
      "position_err_lambda": [
        0.0,
        0.05
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Spatial spectral estimation with sensor arrays",
  "title": "Direction-of-Arrival Estimation (narrowband array processing)"
}

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