Chebyshev Filter

Summary

  • The Chebyshev filter is an analog filter with equiripple characteristics in either the passband or the stopband
  • Type I: equiripple in the passband, monotonically decreasing in the stopband, all-pole topology for simpler circuits
  • Type II: flat passband, equiripple in the stopband, transmission zeros require more components
  • For the same specifications, it can be realized with a lower order than Butterworth
  • Main applications: communication systems, RF circuits, and measurement instrument filters

The Chebyshev filter is an analog filter with equiripple characteristics in either the passband or the stopband. It is designed based on Chebyshev polynomials, studied by the Russian mathematician Pafnuty Chebyshev (1821--1894). It can achieve a steeper cutoff than the Butterworth filter, but introduces ripple in either the passband or the stopband.

Differences Between Type I and Type II

There are two types of Chebyshev filters.

Type I

Equiripple in the passband, monotonically decreasing in the stopband.

Magnitude response:

$$|H(\omega)|^2 = \frac{1}{1 + \varepsilon^2 T_n^2(\omega/\omega_c)}$$

Design parameter: passband ripple $r$ [dB]

Poles: located on an ellipse

Zeros: none (all-pole type)

Type II (Inverse Chebyshev)

Monotonically decreasing in the passband, equiripple in the stopband.

Magnitude response:

$$|H(\omega)|^2 = \frac{1}{1 + \dfrac{1}{\varepsilon^2 T_n^2(\omega_c/\omega)}}$$

Design parameter: stopband attenuation $A_s$ [dB]

Poles: located on an ellipse

Zeros: located on the imaginary axis

Comparison of Chebyshev Type I and Type II magnitude responses. Type I has equiripple in the passband and monotonically decreasing stopband; Type II has monotonically decreasing passband and equiripple in the stopband.
Figure 1: Comparison of Chebyshev Type I and Type II magnitude responses

Which One to Choose

Condition Recommendation
Passband flatness is important Type II
Monotonically decreasing stopband is desired Type I
Simple circuit required (no zeros) Type I
VSWR guarantee needed for RF applications Type I

Why Type II is less common: Type II requires more components (to realize transmission zeros) to achieve comparable steepness to Type I. Furthermore, passive LC ladder synthesis is more naturally suited to Type I, and when a flat passband with equiripple stopband is needed, elliptic filters are often used instead. For these reasons, Type I is overwhelmingly more common in practice, and "Chebyshev filter" typically refers to Type I.

Comparison with Other Filters

Filter Passband Stopband Transition Band Phase Response
Butterworth Monotonically decreasing (maximally flat) Monotonically decreasing Gradual Better than Chebyshev
Chebyshev Type I Equiripple Monotonically decreasing Steep Somewhat inferior
Chebyshev Type II Monotonically decreasing Equiripple Steep Somewhat inferior
Elliptic Equiripple Equiripple Steepest Inferior
Bessel Monotonically decreasing Monotonically decreasing Most gradual Best (near-linear phase)

Order Comparison for the Same Specifications

Required filter order to meet the following specifications: passband ripple 1 dB, stopband attenuation 40 dB, transition band ratio $\omega_s/\omega_p = 1.5$:

Filter Required Order
Bessel 25
Butterworth 14
Chebyshev Type I 7
Chebyshev Type II 7
Elliptic 5

In this example, Chebyshev Type I achieves equivalent transition characteristics with half the order of a Butterworth filter. The degree of order reduction depends on the specifications, but generally a 30--60% reduction is possible, which translates to fewer components and reduced computational cost in digital implementations.

Applications

  • Communication systems: channel filters, IF filters
  • RF/microwave: impedance matching circuits (VSWR guarantees)
  • Measurement instruments: anti-aliasing filters

Detailed Topics

Theory (Type I)

Properties of Chebyshev polynomials, squared magnitude response, and mathematical derivation of elliptical pole placement.

Theory (Type II)

Magnitude response of the inverse Chebyshev filter, zero locations, pole placement, and transfer function construction.

Frequency Response

Detailed magnitude, phase, and group delay characteristics. Comparison graphs with Butterworth.

Design

Order determination from specifications, design procedure, and Python implementation examples.

Circuits

Implementation using Sallen-Key, MFB, state-variable filters, and LC ladders, with component selection guidelines.

Component Value Tables

Normalized LC ladder filter element value tables (orders 1--10, various ripple levels).

Online Design Tool

Design in your browser. Enter parameters to automatically generate coefficients and frequency response plots.

Summary

The Chebyshev filter is an analog filter that achieves steeper transition characteristics than the Butterworth filter through its equiripple property.

  • Choose Type I when: passband ripple is acceptable and a steep cutoff with a simple circuit (all-pole type) is needed
  • Choose Type II when: a flat passband is required and stopband ripple is acceptable
  • Consider other filters when: phase response is critical (Bessel), or minimum order is needed (elliptic)

In practice, Type I is overwhelmingly more common, and "Chebyshev filter" typically refers to Type I. During design, compute the required order from the passband ripple, stopband attenuation, and transition band ratio, then obtain the coefficients from element value tables or a design tool.

References

  • A. B. Williams and F. J. Taylor, Electronic Filter Design Handbook, 4th ed., McGraw-Hill, 2006.
  • A. I. Zverev, Handbook of Filter Synthesis, Wiley, 1967.
  • R. Schaumann, M. S. Ghausi, and K. R. Laker, Design of Analog Filters, Prentice Hall, 1990.