The four basic filter types are defined by which frequency region they pass or attenuate

A filter modifies different frequency components of a signal differently; the standard way to characterise one is its amplitude-versus-frequency response curve. Four basic types cover most uses: low-pass (LPF) passes frequencies below its cutoff and attenuates those above; high-pass (HPF) does the reverse; band-pass (BPF) passes a band around a centre frequency and attenuates both sides; band-reject or notch (BRF) attenuates a narrow band while passing everything else. Shelving filters instead boost or cut everything above (high shelf) or below (low shelf) a threshold. Real filters have finite transition bands, not ideal sharp cutoffs, and may ripple; transition steepness is quoted in dB/octave, where a gentle 6 dB/octave rolloff is unobtrusive and a steep slope approaches a brick wall. Beyond energy shaping, all-pass filters leave amplitude unchanged but apply frequency-dependent time delay — so they still matter. A filter’s impulse response fully characterises it (convolution with any input yields the filtered output), and any device that alters a spectrum acts as a filter: instrument bodies, the vocal tract, the cochlea, a room, even an uneven loudspeaker.

Examples

A gentle LPF at 8 kHz removes hiss without dulling; a sharp notch on 60 Hz removes mains hum; a BPF makes wah/formant effects; the vocal tract filters a glottal source into vowels. In Strudel: s("white").lpf(800), .hpf(2000). In SuperCollider:

LPF.ar(WhiteNoise.ar(0.4), 4000)   // rolls off highs
HPF.ar(WhiteNoise.ar(0.4), 4000)   // rolls off lows
BPF.ar(WhiteNoise.ar(0.4), 4000, 0.01) // narrow band
BRF.ar(WhiteNoise.ar(0.4), 4000, 0.99) // notch

Assessment

Describe the frequency response of each of the four basic types. Given a curve that passes only 400 Hz–1.2 kHz, name the type; then name the type you would reach for to remove a single whistling tone. Route white noise through all four with cutoff/centre at 1 kHz and describe what you hear for each. Why does an all-pass filter matter despite passing all frequencies at equal amplitude?