How do Butterworth, Chebyshev and elliptic filters differ?
Butterworth filters have a maximally flat passband; Chebyshev filters trade passband ripple for a steeper transition; elliptic (Cauer) filters have ripple in both passband and stopband with transmission zeros, giving the steepest transition for a given order; Bessel filters maximize group-delay flatness at the cost of selectivity.[1][2]
What are the magnitude responses?
- Butterworth:
|H|² = 1/(1 + (ω/ωc)2N), −3 dB at ωc, rolloff 20N dB/decade. - Chebyshev type I:
|H|² = 1/(1 + ε²TN²(ω/ωc)), where TN is the Chebyshev polynomial and ripple = 10·log10(1 + ε²) dB. - Elliptic: uses Jacobi elliptic rational functions; finite zeros in the stopband.
How do you choose the order?
Butterworth order for passband attenuation Ap at ωp and stopband attenuation As at ωs: N ≥ log10[(10As/10 − 1)/(10Ap/10 − 1)] / (2·log10(ωs/ωp)). Chebyshev: N ≥ arccosh√[(10As/10 − 1)/(10Ap/10 − 1)] / arccosh(ωs/ωp).[3]
What are the prototype element values?
Normalized low-pass prototype values g1…gN for 1 Ω and 1 rad/s are tabulated. For Butterworth, gk = 2·sin((2k − 1)π/(2N)) with g0 = gN+1 = 1; a third-order Butterworth is 1, 2, 1. Chebyshev and elliptic values come from tables or closed forms in Matthaei, Young and Jones, and Zverev.[4][1]
Which should I use?
| Need | Approximation |
|---|---|
| Flat passband, moderate selectivity | Butterworth |
| Steep skirts, ripple acceptable | Chebyshev |
| Steepest skirts, minimum order | Elliptic |
| Pulse fidelity, flat delay | Bessel / Gaussian |
Frequently asked questions
What ripple is typical for Chebyshev?
0.01–0.5 dB is common in RF; 0.1 dB is a frequent choice.
Do elliptic filters need more components?
They need extra resonant elements to create stopband zeros, but fewer sections for a given selectivity.
Is Butterworth −3 dB at cutoff?
Yes, by definition.
See also
References
Online sources accessed October 21, 2026; printed works cited by edition. See all sources.
- “Handbook of Filter Synthesis”. A. I. Zverev, Wiley. 1967.
- “Microwave Engineering, 4th ed.”. D. M. Pozar, Wiley. 2012.
- “Analog Filter Design”. M. E. Van Valkenburg, Oxford University Press. 1982.
- “Microwave Filters, Impedance-Matching Networks, and Coupling Structures”. G. Matthaei, L. Young, E. M. T. Jones, McGraw-Hill / Artech House. 1964; Artech reprint 1980.