Low-pass filters, corner by corner

Passes everything below the corner and rolls off above it at 12 dB/octave. Every ISO third-octave centre from 20 Hz to 20 kHz, solved at 48 kHz and seven other sample rates.

All filter types Design a cascade →

Every corner, with what changes across the sweep

The pole radius column is the one worth reading down. It rises toward 1 as the corner drops relative to the sample rate, and it is what decides the last column: the 3 corners at and below 40 Hz do not survive 16-bit fixed point at 48 kHz.

Corner	At f₀	Pole radius @ 48 kHz	16-bit fixed point
20 Hz	−3.01 dB	0.998151	diverges
25 Hz	−3.01 dB	0.997689	diverges
31.5 Hz	−3.01 dB	0.997089	holds · ≤ 0.000 dB
40 Hz	−3.01 dB	0.996304	diverges
50 Hz	−3.01 dB	0.995383	holds · ≤ 0.000 dB
60 Hz	−3.01 dB	0.994462	holds · ≤ 0.054 dB
80 Hz	−3.01 dB	0.992623	holds · ≤ 6.007 dB
100 Hz	−3.01 dB	0.990787	holds · ≤ 3.474 dB
125 Hz	−3.01 dB	0.988497	holds · ≤ 0.846 dB
160 Hz	−3.01 dB	0.985300	holds · ≤ 1.159 dB
200 Hz	−3.01 dB	0.981658	holds · ≤ 0.756 dB
250 Hz	−3.01 dB	0.977126	holds · ≤ 0.091 dB
315 Hz	−3.01 dB	0.971265	holds · ≤ 0.316 dB
400 Hz	−3.01 dB	0.963653	holds · ≤ 0.169 dB
500 Hz	−3.01 dB	0.954775	holds · ≤ 0.129 dB
630 Hz	−3.01 dB	0.943355	holds · ≤ 0.024 dB
800 Hz	−3.01 dB	0.928627	holds · ≤ 0.052 dB
1 kHz	−3.01 dB	0.911595	holds · ≤ 0.027 dB
1.25 kHz	−3.01 dB	0.890744	holds · ≤ 0.013 dB
1.6 kHz	−3.01 dB	0.862354	holds · ≤ 0.005 dB
2 kHz	−3.01 dB	0.831023	holds · ≤ 0.002 dB
2.5 kHz	−3.01 dB	0.793476	holds · ≤ 0.006 dB
3.15 kHz	−3.01 dB	0.747258	holds · ≤ 0.001 dB
4 kHz	−3.01 dB	0.691080	holds · ≤ 0.002 dB
5 kHz	−3.01 dB	0.630993	holds · ≤ 0.015 dB
6.3 kHz	−3.01 dB	0.562533	holds · ≤ 0.001 dB
8 kHz	−3.01 dB	0.490314	holds · ≤ 0.011 dB
10 kHz	−3.01 dB	0.433988	holds · ≤ 0.014 dB
12.5 kHz	−3.01 dB	0.415467	holds · ≤ 0.021 dB
16 kHz	−3.01 dB	0.490314	holds · ≤ 0.000 dB
20 kHz	−3.01 dB	0.691080	holds · ≤ 0.078 dB

The other seven types

High-pass

Rejects everything below the corner at 12 dB/octave and passes what is above it.

Band-pass

Passes a band centred on f0 at unity gain and rejects everything either side.

Notch

Removes one frequency and leaves the rest of the spectrum untouched.

All-pass

Leaves every magnitude alone and shifts phase, passing through −180° at f0.

Peaking EQ

Boosts or cuts a band around f0 and leaves both ends of the spectrum at unity.

Low shelf

Lifts or drops everything below f0 by a fixed amount and leaves the top flat.

High shelf

Lifts or drops everything above f0 by a fixed amount and leaves the bottom flat.

Method. RBJ Audio EQ Cookbook coefficients at Q = 0.7071, normalised to a₀ = 1. Pole radii are the roots of the denominator quadratic, not √|a₂| — the two disagree exactly when coefficient rounding pushes the poles onto the real axis, which is the case this table's last column is about. Every figure is solved; nothing is fetched.