Filter Order Rolloff⚠ unverified
Electrical / Filters · Compute the asymptotic roll-off rate of a filter
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| n | n | — | 1.0 | Filter order (number of poles) |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | rolloff | dB/decade | Roll-off rate, in decibels per decade (dB/decade) |
The science & history
Understanding the Parameters
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$n$ — the order equals the number of independent capacitors/inductors shaping the response; it sets both the steepness here and the sharpness of the transition.
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Roll‑off — how fast attenuation grows once past the cutoff; steeper is more selective but needs more components and can worsen phase/transient behaviour.
Derivation (Approaching a Proof)
Far above cutoff, an $n$‑pole low‑pass response falls as a power law, $|H(f)| \propto (f/f_c)^{-n}$. Expressed in decibels,
$$|H|_{\text{dB}} = 20\log_{10}|H| = -20\,n\,\log_{10}\!\frac{f}{f_c} + \text{const}.$$
Increasing frequency by one decade ($f \to 10f$) increases $\log_{10}(f/f_c)$ by $1$, so the magnitude drops by $20n$ dB. Hence the slope is $-20n$ dB/decade. Since a decade is $\log_2 10 \approx 3.32$ octaves, the same slope is $-20n/3.32 \approx -6n$ dB/octave.
History
The "20 dB per decade per pole" rule is a direct reading of the Bode magnitude asymptote, from Hendrik Bode's 1930s–40s feedback‑amplifier work at Bell Labs. Bode plots made filter and control‑loop response something you could sketch and reason about by straight‑line asymptotes — one slope per pole or zero.
Related Concepts: Butterworth Cutoff, RC Low-Pass Cutoff, Sallen Key Lowpass
Notes: Registry calculator filter-order-rolloff (unverified). Asymptotic (far‑from‑cutoff) slope;
near the corner the actual response is gentler.