# ⚛  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._