Three classical filter families — each optimising a different trade-off between passband flatness, transition sharpness, stopband attenuation, and phase linearity.
No ripple anywhere — flat passband, monotone stopband. Poles lie on a circle, equally spaced. Roll-off is −20N dB/decade. The smoothest magnitude and the most nearly linear phase.
Allows controlled ripple in the passband to gain a sharper transition than Butterworth for the same order. The ripple oscillates equally between ±ε² — hence "equiripple."
Equiripple in both passband and stopband simultaneously — the sharpest possible transition for any given order. No other filter can meet the same magnitude spec with fewer poles.
For 1 dB passband ripple, 40 dB stopband attenuation, 2× frequency ratio, the required filter orders are dramatically different:
- Butterworth: N = 8 poles required
- Chebyshev I: N = 5 poles required
- Elliptic: N = 4 poles required (half of Butterworth, 8 → 4)
- For tighter specs, elliptic savings grow to 5× or more
Sharpness and phase linearity are fundamentally at odds in IIR filters. As transition steepness increases, group delay variation worsens:
- Butterworth → best phase, gentlest roll-off
- Chebyshev I/II → sharper, moderate phase distortion
- Elliptic → sharpest, most nonlinear group delay
- For linear phase at any cost → use FIR
- Need flat passband, phase matters → Butterworth
- Need sharper transition, can tolerate passband ripple → Chebyshev I
- Need flat passband + better stopband than Butterworth → Chebyshev II
- Need minimum order, phase distortion acceptable → Elliptic
- Need linear phase → FIR, not IIR
- Butterworth = maximally flat, no ripple, best phase, highest order
- Chebyshev I = equiripple passband, sharper transition than Butterworth
- Chebyshev II = flat passband, equiripple stopband, finite zeros
- Elliptic = equiripple both bands, minimum order, worst phase
- Same spec: Butterworth N=13, Chebyshev N=7, Elliptic N=4
- Next: bilinear transform maps these analog designs to digital filters