a Bessel filter
/ BESS-uhl /
Sometimes you care less about how the filter treats each frequency's loudness and more about whether a pulse comes out the same shape it went in. That is Bessel's specialty. It is the gentle approximation that preserves the wave's shape in time, keeping square waves square and pulses crisp, at the cost of a lazy, gradual cutoff. If your signal is in the time domain (digital edges, video, control loops), Bessel is your friend.
Bessel's defining property is maximally flat group delay, which is another way of saying it has the most nearly linear phase of the classic filters. Linear phase means every frequency in a pulse is delayed by the same amount, so they all arrive together and the pulse keeps its shape, no overshoot, no ringing. The cost is the price of that gentleness: for a given order the Bessel has the slowest roll-off of the three, so its stopband rejection lags well behind Butterworth and far behind Chebyshev. Each second-order Bessel section uses a low Q (around 0.577), which is why it never peaks and never rings.
Bessel is the choice wherever waveform fidelity beats frequency selectivity: anti-aliasing for sampled data where edges matter, reconstructing a DAC output cleanly, video, and any control loop that must not ring. Its honest tradeoff is exactly its gentleness: if you also need a sharp cutoff you must use a much higher order than Butterworth or Chebyshev would need, or accept that frequencies just past the cutoff are barely attenuated. It is the best phase and step response of the three, and the worst amplitude selectivity.
Send a 1 kHz square wave through three 4th-order low-pass filters with the same cutoff. The Chebyshev output rings and overshoots the most, the Butterworth overshoots a little, and the Bessel comes through with rounded but clean edges and almost no overshoot, the waveform's shape preserved.
Bessel preserves pulse shape (linear phase, no ringing) but has the gentlest roll-off of the three.
Bessel optimizes phase and step response, not amplitude selectivity. For a sharp cutoff it needs a much higher order than Butterworth or Chebyshev, so do not choose it when stopband sharpness is the priority.