First-order vs second-order low-pass filters, in code

A first-order low-pass filter, like an RC network or a one-line exponential filter, lets slow changes through and shrinks fast ones in proportion: above its cutoff, a wave ten times faster comes out about ten times smaller. Put two in a row and you have a second-order filter. Each stage cuts the fast part again, so ten times faster comes out about a hundred times smaller.

How much gets through

Above the cutoff, a first-order filter passes about cutoff ÷ frequency of a wave:

On a Bode plot that's a slope of −20 dB per decade, or −6 dB per octave. Two identical stages pass a tenth of a tenth, a hundredth: −40 dB per decade. Three stages, −60.

In code

Each stage is an exponential filter, and the second follows the first, not the raw reading:

stage1 = stage1 + (raw - stage1) * 0.025;
stage2 = stage2 + (stage1 - stage2) * 0.025;
level = stage2;

At 100 samples a second, alpha 0.025 puts each stage's cutoff near 0.4 Hz, from cutoff ≈ alpha × sample rate ÷ 2π. Alpha is per sample, so change the sample rate and the cutoff moves with it: the same alpha at 1000 samples a second cuts off ten times higher.

The trade: smooth or quick

A gauge reads a boat's tank at 60% while the fuel sloshes ±10% twice a second, and it starts from 0 when switched on. One stage slow enough to wobble less than 1% still reads only 50% after 3 s. One quick enough to arrive in time wobbles 4%. Two stages at 0.025 do both: they reach the level in about 2 s, and each passes about a fifth of the 2 Hz slosh, so together they pass a twenty-fifth, a wobble under 1%.

Not just a lower cutoff

Two stages bend more gently at the corner than one: two identical first-order stages are already 6 dB down at their own cutoff, not 3 dB. They also delay a step about twice as long as one stage. For a sharp knee, designers use a Butterworth second-order filter: a Sallen-Key op-amp circuit in hardware, or a biquad in code. Either way the slope beyond the corner is −40 dB per decade.

Where you'll meet it

A car's fuel gauge is filtered so heavily it takes minutes to settle after you fill up. Anti-aliasing filters in front of an ADC are often second order or more, and a loudspeaker's crossover uses second-order sections to split bass from treble cleanly.