L1 ⚛ L1-395 ⊙ Testnet

Adaptive Filtering (LMS / RLS time-varying system identification)

Signal Processing · Online stochastic-gradient / recursive-least-squares

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⬇ L1-395.md

# ⚛ L1 Principle — Adaptive Filtering (LMS / RLS time-varying system identification) **ID:** `L1-395` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Signal Processing — *Online stochastic-gradient / recursive-least-squares* > **🎯 Problem class:** online linear regression · **🧮 Solution space:** time varying filter > **📡 Carrier:** electromagnetic · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 2 · **⛓ Block:** 41555199 --- ## 🧠 1. Introduction **Adaptive Filtering (LMS / RLS time-varying system identification)** is a **online linear regression** whose unknown lives in **time varying filter** space, within the **Online stochastic-gradient / recursive-least-squares** sub-domain of **Signal Processing**. Measurements consist of electromagnetic field measurements via a **reference probe online** sensing mechanism. The forward operator applies, in order: L · project · reference operator; detector accumulates flux over the exposure window. Observations are corrupted by additive Gaussian noise. LMS converges in mean for 0 < mu < 2/lambda_max; RLS converges in one step for stationary deterministic input; tracking error scales with non-stationarity rate. ## ⚙ 2. Forward Model Physical chain: **x** → L · project · reference → Temporal integration → **y** (detector). ``` y = ∫_t dt `L.project.reference` x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `L.project.reference` | L · project · reference operator | | `int.temporal` | Detector accumulates flux over the exposure window | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Signal Processing | | Sub domain | Online stochastic-gradient / recursive-least-squares | | Carrier | electromagnetic | | Problem class | online_linear_regression | | Solution space | time_varying_filter | | Noise model | gaussian | | Integration axis | temporal | | Difficulty delta | 2 | | L dag | 2.3 | ## 📡 4. Measurement Model LMS converges in mean for 0 < mu < 2/lambda_max; RLS converges in one step for stationary deterministic input; tracking error scales with non-stationarity rate. | Metric | Value | |---|---| | Metric | tracking_MSE_dB | | Secondary | convergence_time_samples | ## 📏 5. Operating Range (Ω) **Center problem class:** `lms_system_identification` · **Forward operator:** `fir_filter_online` **Center point:** | Parameter | Unit | Value | |---|---|---| | L | — | 32 | | Mu | — | 0.01 | | Sigma v | — | 0.01 | | Eigenspread | — | 10 | | Non stationarity rate | — | 0 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | L | — | 4 – 4096 | | Mu | — | 1e-05 – 1.0 | | Sigma v | — | 0.0001 – 1.0 | | Eigenspread | — | 1 – 1000 | | Step size miscal | — | 0.0 – 5.0 | | Non stationarity rate | — | 0.0 – 0.1 | | Impulsive noise fraction | — | 0.0 – 0.1 | ## 🎯 6. Tolerance (ε) **Center tolerance:** -30 dB tracking MSE | Metric | Range | |---|---| | Mse db | -60 – 10 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **tracking_MSE_dB**, with κ = `500` and δ = `2`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0x8b6ed2bd7c400d6066681e6c1b8a9eb9205e749ba055bce04eed3eefe408bed0` - **Chain tx hash:** `0x26bc3c4359680388e625126cecdd2373db19a7c5a70ce5fb781abf7f5d7917fb` - **Chain block:** `41555199` --- ## File Mapping This bundle consists of: `L1-395.md`, `L1-395.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-395.md` | Source of truth — edit this | Human or LLM | | `L1-395.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 `L1-395`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Read the attached Markdown (L1-395.md).
Regenerate L1-395.json so every field matches the Markdown.
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L2-395-001 Adaptive Filtering Non-Stationary Spec (echo cancellation family)

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📋 JSON metadata

{
  "artifact_id": "L1-395",
  "chain_block": 41555199,
  "chain_hash": "0x8b6ed2bd7c400d6066681e6c1b8a9eb9205e749ba055bce04eed3eefe408bed0",
  "chain_tx_hash": "0x26bc3c4359680388e625126cecdd2373db19a7c5a70ce5fb781abf7f5d7917fb",
  "domain": "Signal Processing",
  "hardness_fn": {
    "delta": 2,
    "kappa": 500,
    "metric": "tracking_MSE_dB",
    "type": "epsilon_fn"
  },
  "initiator_dataset": [
    {
      "ipfs_cid": null,
      "license_hash": null,
      "name": "primary",
      "weight": 1.0
    }
  ],
  "layer": "L1",
  "observable_profile": {
    "metric": "tracking_MSE_dB",
    "regime": "LMS converges in mean for 0 \u003c mu \u003c 2/lambda_max; RLS converges in one step for stationary deterministic input; tracking error scales with non-stationarity rate.",
    "secondary": "convergence_time_samples"
  },
  "physics_fingerprint": {
    "L_DAG": 2.3,
    "carrier": "electromagnetic",
    "difficulty_delta": 2,
    "domain": "Signal Processing",
    "integration_axis": "temporal",
    "noise_model": "gaussian",
    "primitives": [
      "L.project.reference",
      "int.temporal"
    ],
    "problem_class": "online_linear_regression",
    "sensing_mechanism": "reference_probe_online",
    "solution_space": "time_varying_filter",
    "sub_domain": "Online stochastic-gradient / recursive-least-squares",
    "title": "Adaptive Filtering (LMS / RLS time-varying system identification)"
  },
  "size_tiers": {
    "allowed_forward_operators": [
      "fir_filter_online",
      "iir_filter_online",
      "nonlinear_volterra"
    ],
    "allowed_omega_dimensions": [
      "L",
      "mu",
      "eigenspread",
      "non_stationarity_rate",
      "sigma_v",
      "step_size_miscal",
      "impulsive_noise_fraction"
    ],
    "allowed_problem_classes": [
      "lms_system_identification",
      "channel_equalization",
      "echo_cancellation",
      "active_noise_control"
    ],
    "center_spec": {
      "epsilon_fn_center": "-30 dB tracking MSE",
      "forward_operator": "fir_filter_online",
      "input_format": "reference_plus_desired",
      "omega": {
        "L": 32,
        "eigenspread": 10,
        "mu": 0.01,
        "non_stationarity_rate": 0.0,
        "sigma_v": 0.01
      },
      "problem_class": "lms_system_identification"
    },
    "epsilon_bounds": {
      "mse_db": [
        -60,
        10
      ]
    },
    "omega_bounds": {
      "L": [
        4,
        4096
      ],
      "eigenspread": [
        1,
        1000
      ],
      "impulsive_noise_fraction": [
        0.0,
        0.1
      ],
      "mu": [
        1e-05,
        1.0
      ],
      "non_stationarity_rate": [
        0.0,
        0.1
      ],
      "sigma_v": [
        0.0001,
        1.0
      ],
      "step_size_miscal": [
        0.0,
        5.0
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Online stochastic-gradient / recursive-least-squares",
  "title": "Adaptive Filtering (LMS / RLS time-varying system identification)"
}

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