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Chebyshev type-I filters have equiripple (equal-magnitude oscillations) in the passband and monotonic stopband, achieving sharper transitions than Butterworth at the cost of passband ripple. Chebyshev type-II (inverse) ripple in stopband instead. The ripple magnitude is a design parameter. For fixed order and ripple specifications, Chebyshev provides the sharpest transition band, making it optimal when passband ripple is acceptable.
Design Chebyshev type-I filter with varying ripple specifications (0.5 dB, 1 dB, 3 dB). Observe the trade-off between passband ripple magnitude and stopband transition sharpness.
From filter order and transition band, you know that a Butterworth filter achieves a maximally flat passband but pays for it with a gradual roll-off: to get a steep transition you need a high-order filter, meaning more poles and more computational cost. The Chebyshev filter takes a different bargain: instead of insisting that every point in the passband be as flat as possible, it distributes small, equal-magnitude oscillations evenly across the passband. By accepting this equiripple behavior, it achieves a dramatically steeper roll-off for the same filter order.
The mathematics come from Chebyshev polynomials T_n(x) = cos(n arccos x), which have the remarkable property of oscillating between exactly ±1 for |x| ≤ 1 while growing faster than any other polynomial outside that interval for a given degree. The Type I frequency response is |H(jω)|² = 1 / (1 + ε² T_n²(ω/ω_c)). In the passband (ω ≤ ω_c), T_n(ω/ω_c) oscillates between ±1, so |H|² oscillates between 1 and 1/(1 + ε²). The parameter ε² = 10R_p/10 − 1 sets the ripple: choosing passband ripple R_p in decibels determines ε, which determines how tightly the response hugs 0 dB in the passband. In the stopband, T_n grows rapidly — polynomially in frequency — causing the magnitude to fall steeply.
For a concrete comparison: a 5th-order Butterworth and a 5th-order Chebyshev Type I (with 1 dB ripple) both reach −3 dB at ω_c. At twice the cutoff frequency, the Butterworth is down roughly 30 dB; the Chebyshev with the same order is down perhaps 45–50 dB. The extra attenuation comes directly from the equiripple trade-off. Increasing the allowed ripple (say, from 0.5 dB to 3 dB) widens the range of ε and pushes the poles further from the imaginary axis, resulting in an even faster roll-off but with larger passband variation. This is the central design lever: ripple tolerance sets the transition sharpness for a fixed order.
Type II Chebyshev (inverse Chebyshev) reverses the location of the ripple: the passband is monotonically decreasing (like Butterworth) while the stopband has equiripple. This is preferable when the passband must be smooth — audio applications where the listener can detect amplitude variation — but some residual signal in the stopband is acceptable. The two types represent complementary points in the design space, and knowing which domain (passband or stopband) has the binding constraint tells you which to choose. When neither ripple location is tolerable, the elliptic (Cauer) filter places equiripple in both bands simultaneously and achieves the steepest possible roll-off for any given order — at the cost of greater phase nonlinearity and more complex design equations.