Definition
Filtering modifies signal frequency content:
Y(f) = H(f) \cdot X(f)
In SCU terms: Filtering selectively passes or blocks χ-mode frequencies—separating desired signal components from noise and interference.
Filtering as χ-Mode Selection
Filters act in frequency domain:
\text{Input χ-modes} \xrightarrow{H(f)} \text{Output χ-modes}
| $H(f)$ | Effect |
|---|---|
| 1 | Pass χ-modes unchanged |
| 0 | Block χ-modes completely |
| 0 < H < 1 | Attenuate χ-modes |
Filter Types
| Type | Passes | Blocks |
|---|---|---|
| Low-pass | Low f χ-modes | High f χ-modes |
| High-pass | High f χ-modes | Low f χ-modes |
| Band-pass | f range χ-modes | Outside f |
| Band-stop | Outside f | f range χ-modes |
| Matched | Signal-shaped χ-modes | Others |
Time Domain: Convolution
Filtering is convolution in time:
y(t) = \int h(\tau) x(t-\tau) d\tau
The impulse response h(t) determines χ-mode modification.
Filter Design
| Parameter | Trade-off |
|---|---|
| Cutoff sharpness | Transition width |
| Passband ripple | Flatness |
| Stopband rejection | Attenuation depth |
| Phase response | Delay characteristics |
Digital Filters
| Type | Description |
|---|---|
| FIR | Finite impulse response (stable) |
| IIR | Infinite impulse response (efficient) |
| Adaptive | Self-adjusting coefficients |
Matched Filtering
Optimal for known χ-mode shapes:
h(t) = s^*(-t) \text{ (time-reversed, conjugate)}
Maximizes SNR for expected signal form.
The Key Insight
Filtering selects χ-mode frequencies.
Separating signal from noise:
- Signals and noise have different χ-mode spectra
- Filters pass desired frequencies
- Block unwanted frequencies
- Improve SNR by χ-mode selection
When we filter a signal, we're selecting which χ-mode frequencies to keep and which to discard—exploiting the fact that signal and noise often occupy different frequency bands.