a Butterworth filter
/ BUH-ter-werth /
When you ask a filter designer for a sensible default, they reach for Butterworth. It is the approximation that makes the passband as flat as mathematically possible, no ripples, no bumps, just a smooth plateau that bends gently into the roll-off. If you want the passband to leave your signal undistorted in amplitude and you have no strong reason to choose otherwise, Butterworth is the safe, classic answer.
Butterworth is one of several recipes for choosing the Q of each second-order section so the overall response has a chosen shape. Its defining property is maximally flat magnitude: the passband is flatter than any other all-pole filter of the same order, achieved by accepting a moderate (not the steepest) roll-off. For a single second-order Butterworth section the Q is exactly 0.707, which is the highest Q that still gives no peaking. Higher-order Butterworths cascade sections, each with its own tabulated Q, and the cutoff stays defined at the -3 dB point regardless of order.
Butterworth is the everyday choice for anti-alias filters, audio, and general signal conditioning, anywhere a flat passband matters and a moderate roll-off is good enough. Its honest tradeoffs: compared with Chebyshev it has a slower transition (you need a higher order for the same stopband sharpness), and compared with Bessel its step response overshoots and rings somewhat and its phase is not linear, so fast pulses get distorted. It is the balanced middle ground, not the best at any one thing.
Compare three 1 kHz low-pass filters of the same 4th order. The Butterworth has a perfectly flat passband and is about 24 dB down an octave above cutoff. The Chebyshev would be steeper there but with a rippling passband; the Bessel would be gentler but with the cleanest pulse response. Butterworth sits in the middle.
Butterworth trades ultimate steepness and best phase for the flattest possible passband, the sensible default.
Maximally flat refers to the amplitude only. Butterworth still has non-linear phase and overshoots on steps, so for clean pulse or transient response prefer Bessel, and for the steepest cutoff prefer Chebyshev.