Responses
| Response | Passband ripple | Stopband | Phase / group delay | Use |
|---|---|---|---|---|
| Butterworth | None (maximally flat) | Slow roll-off | Moderate | General — smooth magnitude |
| Chebyshev I | Ripples | Faster roll-off | Worse than Butter | When sharp cut beats flatness |
| Chebyshev II | None | Ripples in stop | Similar | Alternative Chebyshev |
| Elliptic (Cauer) | Ripples | Ripples | Worst | Sharpest transition per order |
| Bessel | None | Slowest roll-off | Best (flat delay) | Audio, pulse — preserves shape |
| Gaussian | None | Soft | No overshoot | Matched filters |
Topologies
| Topology | Notes |
|---|---|
| Passive RC / RLC | Simple, lossy, poor selectivity |
| Active (op-amp) Sallen-Key | Common 2nd-order; low output Z |
| Multiple Feedback (MFB) | Inverting topology; easy to cascade |
| State-variable | Three outputs (LP, BP, HP) from one circuit |
| Biquad | Second-order building block; used for equalizers |
| Switched-capacitor | Tunable via clock; good for integrated |
| FIR digital | Linear phase, stable, straightforward |
| IIR digital | Efficient, can reach analog-like response |
Order vs roll-off
| First-order | −20 dB/decade (−6 dB/octave) |
|---|---|
| Second-order | −40 dB/decade |
| n-th order | −20·n dB/decade |
Notes
- For audio crossovers, match order between high-pass and low-pass. For anti-aliasing, Butterworth or Bessel preferred (no passband ripple).
- Digital filters are often specified by normalized cutoff (0.5 = Nyquist).
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