L1 ⚛ L1-389 ⊙ Testnet

Channel Estimation (pilot-aided wireless channel identification)

Signal Processing · MIMO/OFDM channel impulse response estimation

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⬇ L1-389.md

# ⚛ L1 Principle — Channel Estimation (pilot-aided wireless channel identification) **ID:** `L1-389` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Signal Processing — *MIMO/OFDM channel impulse response estimation* > **🎯 Problem class:** linear inverse · **🧮 Solution space:** channel impulse response > **📡 Carrier:** RF · **🌫 Noise:** complex gaussian > **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41555198 --- ## 🧠 1. Introduction **Channel Estimation (pilot-aided wireless channel identification)** is a **linear inverse problem** whose unknown lives in **channel impulse response** space, within the **MIMO/OFDM channel impulse response estimation** sub-domain of **Signal Processing**. Measurements consist of RF coil signals (typical of MRI) via a **pilot probing** sensing mechanism. The forward operator applies, in order: L · project · pilot operator; detector accumulates flux over the exposure window. Observations are corrupted by complex-valued Gaussian noise on real and imaginary parts. Well-posed under Nyquist pilot spacing; sub-Nyquist recoverable via sparse CS when channel is sparse in delay-Doppler. Mismatch: CFO, phase noise, non-ideal pilot power allocation. ## ⚙ 2. Forward Model Physical chain: **x** → L · project · pilot → Temporal integration → **y** (detector). ``` y = ∫_t dt `L.project.pilot` x + n, n ~ 𝒞𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `L.project.pilot` | L · project · pilot operator | | `int.temporal` | Detector accumulates flux over the exposure window | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Signal Processing | | Sub domain | MIMO/OFDM channel impulse response estimation | | Carrier | RF | | Problem class | linear_inverse | | Solution space | channel_impulse_response | | Noise model | complex_gaussian | | Integration axis | temporal | | Difficulty delta | 3 | | L dag | 2.3 | ## 📡 4. Measurement Model Well-posed under Nyquist pilot spacing; sub-Nyquist recoverable via sparse CS when channel is sparse in delay-Doppler. Mismatch: CFO, phase noise, non-ideal pilot power allocation. | Metric | Value | |---|---| | Metric | NMSE | | Secondary | BER_postequalization | ## 📏 5. Operating Range (Ω) **Center problem class:** `pilot_aided_ofdm` · **Forward operator:** `ofdm_pilot_channel` **Center point:** | Parameter | Unit | Value | |---|---|---| | K | — | 4 | | Nr | — | 1 | | Nt | — | 1 | | L ch | — | 16 | | Snr db | dB | 10 | | Cfo hz | Hz | 0 | | F d hz | Hz | 0 | | Alpha p | — | 0.1 | | Phase noise rms | — | 0 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | K | — | 1 – 64 | | Nr | — | 1 – 128 | | Nt | — | 1 – 64 | | L ch | — | 2 – 256 | | Snr db | dB | -5 – 30 | | Cfo hz | Hz | 0 – 500 | | F d hz | Hz | 0 – 1000 | | Alpha p | — | 0.02 – 0.5 | | Doppler error | — | 0.0 – 100 | | Phase noise rms | — | 0.0 – 0.1 | ## 🎯 6. Tolerance (ε) **Center tolerance:** NMSE 1e-2 | Metric | Range | |---|---| | Nmse | 0.0001 – 0.1 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **NMSE**, with κ = `500` and δ = `3`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0xdc6a5e16a67f4523e97ea6ba091ae4a59dfd695db6757a4f1e7d3995e6d69c08` - **Chain tx hash:** `0x3aacea545fa8ef4f925c9813163f4f18e34e0ec2c0bc368b47ea1f45d9c88ebd` - **Chain block:** `41555198` --- ## File Mapping This bundle consists of: `L1-389.md`, `L1-389.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-389.md` | Source of truth — edit this | Human or LLM | | `L1-389.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 `L1-389`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Regenerate L1-389.json so every field matches the Markdown.
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L2-389-001 Channel Estimation OFDM/MIMO Spec (5G NR pilot family)

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📋 JSON metadata

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  "chain_block": 41555198,
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  "domain": "Signal Processing",
  "hardness_fn": {
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    "metric": "NMSE",
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    "regime": "Well-posed under Nyquist pilot spacing; sub-Nyquist recoverable via sparse CS when channel is sparse in delay-Doppler. Mismatch: CFO, phase noise, non-ideal pilot power allocation.",
    "secondary": "BER_postequalization"
  },
  "physics_fingerprint": {
    "L_DAG": 2.3,
    "carrier": "RF",
    "difficulty_delta": 3,
    "domain": "Signal Processing",
    "integration_axis": "temporal",
    "noise_model": "complex_gaussian",
    "primitives": [
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      "int.temporal"
    ],
    "problem_class": "linear_inverse",
    "sensing_mechanism": "pilot_probing",
    "solution_space": "channel_impulse_response",
    "sub_domain": "MIMO/OFDM channel impulse response estimation",
    "title": "Channel Estimation (pilot-aided wireless channel identification)"
  },
  "size_tiers": {
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      "ofdm_pilot_channel",
      "mimo_pilot_channel",
      "mmwave_beam_channel"
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      "alpha_p",
      "SNR_dB",
      "f_d_Hz",
      "cfo_hz",
      "phase_noise_rms",
      "doppler_error"
    ],
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      "pilot_aided_ofdm",
      "blind_channel_id",
      "semi_blind",
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    "center_spec": {
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      "omega": {
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        "Nr": 1,
        "Nt": 1,
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        "alpha_p": 0.1,
        "cfo_hz": 0,
        "f_d_Hz": 0,
        "k": 4,
        "phase_noise_rms": 0.0
      },
      "problem_class": "pilot_aided_ofdm"
    },
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      "nmse": [
        0.0001,
        0.1
      ]
    },
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      "L_ch": [
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        256
      ],
      "Nr": [
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      ],
      "Nt": [
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      ],
      "SNR_dB": [
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        30
      ],
      "alpha_p": [
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        0.5
      ],
      "cfo_hz": [
        0,
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      ],
      "doppler_error": [
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      ],
      "f_d_Hz": [
        0,
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      ],
      "k": [
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        64
      ],
      "phase_noise_rms": [
        0.0,
        0.1
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "MIMO/OFDM channel impulse response estimation",
  "title": "Channel Estimation (pilot-aided wireless channel identification)"
}

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