Definition
Weak signal extraction recovers signals buried in noise:
\text{SNR} \ll 1 \xrightarrow{\text{extraction}} \text{Signal recovered}
In SCU terms: Weak signal extraction reveals χ-modes hidden far below the noise floor—using integration, correlation, and optimal filtering to accumulate signal.
The Challenge
Signal is much weaker than noise:
y(t) = \underbrace{s(t)}_{\text{tiny}} + \underbrace{n(t)}_{\text{huge}}
Direct observation shows only noise.
Key Techniques
| Technique | How It Works |
|---|---|
| Matched filter | Optimal for known χ-mode shape |
| Correlation | Exploit χ-mode coherence |
| Stacking | Average aligned χ-mode signals |
| Lock-in | Phase-sensitive χ-mode detection |
| Bayesian | Prior χ-mode knowledge |
Matched Filtering
Optimal extraction for known waveform:
T = \int h(t) y(t) dt = \int |S(f)|^2 / N(f) df
Maximizes SNR for given signal shape.
Integration Time
SNR improves with observation time:
\text{SNR} \propto \sqrt{T}
Longer integration → weaker χ-modes detectable.
Requirements
| Requirement | Why Needed |
|---|---|
| Signal knowledge | Templates for matching |
| Noise characterization | Statistical properties |
| Stability | Coherent χ-modes over integration |
| Calibration | Accurate χ-mode amplitude |
Fundamental Limits
| Limit | Source |
|---|---|
| Quantum noise | Irreducible χ-mode fluctuations |
| Systematic error | Dominates at high sensitivity |
| Integration time | Practical observation limits |
Applications
| Application | χ-Mode Challenge |
|---|---|
| Gravitational waves | Strain 10⁻²² in seismic noise |
| Exoplanets | Star overwhelms planet χ-modes |
| Neutrinos | Rare interaction events |
| Dark matter | Unknown χ-mode signature |
| Pulsars | Periodic in radio noise |
The Key Insight
Weak signal extraction pushes χ-mode detection limits.
Recovering buried signals:
- Signals far below noise floor
- Use signal properties (templates)
- Integrate to accumulate χ-mode power
- Quantum limits set ultimate floor
When we extract weak signals, we're using everything we know about χ-mode structure to accumulate evidence—pulling coherent information from apparently random noise.