Definition
Signal detection identifies whether a signal is present:
H_0: y = n \text{ (noise only)} \quad H_1: y = s + n \text{ (signal present)}
In SCU terms: Detection determines whether target χ-modes are present in observations dominated by noise χ-modes.
The Detection Problem
Given noisy data, decide:
\text{Data} \xrightarrow{\text{decision}} \begin{cases} \text{Signal present} \\ \text{Signal absent} \end{cases}
Error Types
| Error | Description | Consequence |
|---|---|---|
| False positive | Noise declared as signal | False alarm |
| False negative | Signal missed | Detection failure |
P_{FA} = P(\text{decide } H_1 | H_0)
P_D = P(\text{decide } H_1 | H_1)
Likelihood Ratio
Optimal detection compares:
\Lambda = \frac{P(y|H_1)}{P(y|H_0)} \gtrless \gamma
If ratio exceeds threshold γ, declare signal present.
ROC Curves
Receiver Operating Characteristic shows trade-offs:
P_D = f(P_{FA}, \text{SNR})
Higher SNR → better detection at any false alarm rate.
Detection Methods
| Method | When to Use |
|---|---|
| Matched filter | Known signal χ-mode shape |
| Energy detection | Unknown signal shape |
| Coherent | Phase-stable χ-modes |
| Correlation | Template-based search |
Matched Filtering
Optimal for known χ-mode waveform:
T = \int h(t) y(t) dt
where h(t) = time-reversed signal template.
SNR and Detectability
Detection probability depends on SNR:
P_D = Q\left(Q^{-1}(P_{FA}) - \sqrt{\text{SNR}}\right)
Higher SNR → higher detectability.
The Key Insight
Detection is χ-mode hypothesis testing.
Deciding signal presence from noisy data:
- Observations contain signal + noise χ-modes
- Detection compares hypotheses
- Threshold trades off error types
- SNR determines performance
When we detect a signal, we're deciding that observed χ-mode patterns are better explained by signal presence than by noise alone.