the low-pass filter

Imagine standing outside a nightclub with the door shut. You can still hear the deep thud of the bass, but the singer's voice and the cymbals are gone. The wall lets the low notes through and blocks the high ones. A low-pass filter does exactly that to an electrical signal: it passes low frequencies almost untouched and increasingly throws away the high ones.

The simplest version is a resistor feeding a capacitor to ground, with the output taken across the capacitor. At low frequencies the capacitor has very high reactance, so it barely loads the signal and the output follows the input. As frequency rises, the capacitor's reactance falls (reactance is 1/(2 times pi times f times C)), so it shunts more and more of the signal to ground, and the output shrinks. The crossover point, where the output has dropped to about 70.7 percent of the input (that is the -3 dB point), is the cutoff frequency f_c = 1/(2 times pi times R times C). For example, R = 1.6 kilohm and C = 100 nF give f_c of roughly 1 kHz.

Low-pass filters are everywhere: smoothing the lumpy output of a DAC or a PWM signal into a clean voltage, removing hiss and high-frequency noise from a sensor reading, and as the anti-alias filter that must sit in front of every ADC. The honest caveat is that the transition is gradual, not a wall; a simple first-order low-pass only rolls off at 6 dB per octave, so frequencies just above cutoff still get through, just quieter.