Definition
Spectral analysis examines frequency content:
P(f) = |S(f)|^2 = \text{power at frequency } f
In SCU terms: Spectral analysis reveals how χ-mode energy is distributed across frequencies—showing the characteristic oscillations in a signal.
Power Spectrum
The power spectrum shows χ-mode energy distribution:
P(f) = \lim_{T \to \infty} \frac{1}{T} |S_T(f)|^2
| Feature | Physical Meaning |
|---|---|
| Peaks | Periodic χ-mode components |
| Broadband | Random χ-mode activity (noise) |
| Slope | χ-mode correlation structure |
Spectral Estimation Methods
| Method | Characteristics |
|---|---|
| Periodogram | Simple, high variance |
| Welch | Averaged segments, reduced variance |
| Multitaper | Optimal, low bias |
| AR models | Parametric, smooth |
Time-Frequency Analysis
When χ-mode content changes over time:
S(t, f) = \text{Spectrogram (time-varying spectrum)}
- Short-time Fourier transform
- Wavelets
- Hilbert-Huang transform
Resolution Trade-offs
\Delta t \cdot \Delta f \geq \frac{1}{4\pi}
Can't have arbitrarily precise time AND frequency resolution simultaneously.
What Spectra Reveal
| Spectrum Shape | χ-Mode Information |
|---|---|
| Line spectrum | Discrete periodic χ-modes |
| Continuous | Aperiodic χ-modes |
| 1/f | Scale-free χ-mode structure |
| White | Uncorrelated χ-mode noise |
Applications
| Field | What Spectral Analysis Shows |
|---|---|
| Astronomy | Star/galaxy χ-mode signatures |
| Seismology | Earth's χ-mode resonances |
| Neuroscience | Brain χ-mode rhythms |
| Climate | Periodic χ-mode patterns |
The Key Insight
Spectra reveal χ-mode structure.
Frequency distribution of energy:
- Periodic signals → spectral peaks
- Noise → broadband spectrum
- Complexity → spectral shape
- Time variation → spectrogram
When we compute a power spectrum, we're seeing how χ-mode energy is distributed across frequencies—revealing the characteristic oscillation patterns in the data.