a Chebyshev filter
/ CHEB-i-shef /
When you need the sharpest possible cutoff for a given number of parts and you can tolerate a little imperfection in the passband, you reach for Chebyshev. It buys a steeper roll-off than Butterworth by making a deal: it allows the passband to ripple up and down by a small, fixed amount. If transition sharpness is what you are short of, Chebyshev gives you more of it for free, by paying with ripple.
A Chebyshev filter has equiripple passband: the gain wobbles between a peak and a trough by a designer-chosen amount, the ripple, specified in dB (common values are 0.1, 0.5, or 1 dB). In exchange, for the same order its transition band is narrower than Butterworth's, so it reaches a given stopband attenuation sooner. You set up its sections at higher Q values than Butterworth uses; more ripple allowed means higher section Q and a steeper cutoff. (The Type II, or inverse Chebyshev, instead keeps a flat passband and ripples the stopband.)
Chebyshev is the choice when stopband sharpness is the priority and a flat passband is not, for instance squeezing maximum rejection out of a low-order anti-alias filter. Its honest tradeoffs are significant: the passband ripple distorts amplitude, the higher section Q means worse overshoot and ringing on transients and a more non-linear phase than Butterworth, and the high-Q sections are more sensitive to component tolerances. You accept all of that in return for the steepest amplitude roll-off of the three classic shapes for a given order.
For a 4th-order 1 kHz low-pass that must be 40 dB down at 2 kHz, a Butterworth (24 dB per octave) only reaches 24 dB there and fails. A 0.5 dB-ripple Chebyshev of the same order, being steeper, clears 40 dB, at the price of a 0.5 dB wobble across the passband.
Chebyshev trades passband ripple for a steeper transition, more rejection from the same order.
The ripple is a real amplitude error across the passband, and the higher section Q worsens overshoot, ringing, phase, and tolerance sensitivity. Choose Chebyshev only when you need the steepness more than a clean passband.