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	b0 · b1 · b2 · a1 · a2 @ 48 kHz
20 Hz	−3.01 dB	0.998151	diverges	0.000001710 · 0.000003421 · 0.000001710 · -1.996297602 · 0.996304443
25 Hz	−3.01 dB	0.997689	diverges	0.000002671 · 0.000005342 · 0.000002671 · -1.995372005 · 0.995382690
31.5 Hz	−3.01 dB	0.997089	holds · ≤ 0.000 dB	0.000004238 · 0.000008476 · 0.000004238 · -1.994168733 · 0.994185685
40 Hz	−3.01 dB	0.996304	diverges	0.000006829 · 0.000013657 · 0.000006829 · -1.992595229 · 0.992622543
50 Hz	−3.01 dB	0.995383	holds · ≤ 0.000 dB	0.000010660 · 0.000021320 · 0.000010660 · -1.990744060 · 0.990786699
60 Hz	−3.01 dB	0.994462	holds · ≤ 0.054 dB	0.000015336 · 0.000030672 · 0.000015336 · -1.988892906 · 0.988954250
80 Hz	−3.01 dB	0.992623	holds · ≤ 6.007 dB	0.000027214 · 0.000054428 · 0.000027214 · -1.985190658 · 0.985299513
100 Hz	−3.01 dB	0.990787	holds · ≤ 3.474 dB	0.000042443 · 0.000084887 · 0.000042443 · -1.981488509 · 0.981658283
125 Hz	−3.01 dB	0.988497	holds · ≤ 0.846 dB	0.000066166 · 0.000132331 · 0.000066166 · -1.976861000 · 0.977125662
160 Hz	−3.01 dB	0.985300	holds · ≤ 1.159 dB	0.000108058 · 0.000216116 · 0.000108058 · -1.970382898 · 0.970815131
200 Hz	−3.01 dB	0.981658	holds · ≤ 0.756 dB	0.000168224 · 0.000336447 · 0.000168224 · -1.962980089 · 0.963652984
250 Hz	−3.01 dB	0.977126	holds · ≤ 0.091 dB	0.000261653 · 0.000523305 · 0.000261653 · -1.953727949 · 0.954774560
315 Hz	−3.01 dB	0.971265	holds · ≤ 0.316 dB	0.000412951 · 0.000825903 · 0.000412951 · -1.941702982 · 0.943354787
400 Hz	−3.01 dB	0.963653	holds · ≤ 0.169 dB	0.000660779 · 0.001321558 · 0.000660779 · -1.925983970 · 0.928627086
500 Hz	−3.01 dB	0.954775	holds · ≤ 0.129 dB	0.001023218 · 0.002046435 · 0.001023218 · -1.907501626 · 0.911594497
630 Hz	−3.01 dB	0.943355	holds · ≤ 0.024 dB	0.001605703 · 0.003211406 · 0.001605703 · -1.883495555 · 0.889918368
800 Hz	−3.01 dB	0.928627	holds · ≤ 0.052 dB	0.002550535 · 0.005101070 · 0.002550535 · -1.852146485 · 0.862348626
1 kHz	−3.01 dB	0.911595	holds · ≤ 0.027 dB	0.003916127 · 0.007832253 · 0.003916127 · -1.815341083 · 0.831005589
1.25 kHz	−3.01 dB	0.890744	holds · ≤ 0.013 dB	0.005988546 · 0.011977093 · 0.005988546 · -1.769471038 · 0.793425223
1.6 kHz	−3.01 dB	0.862354	holds · ≤ 0.005 dB	0.009525762 · 0.019051525 · 0.009525762 · -1.705552146 · 0.743655195
2 kHz	−3.01 dB	0.831023	holds · ≤ 0.002 dB	0.014401440 · 0.028802881 · 0.014401440 · -1.632993162 · 0.690598923
2.5 kHz	−3.01 dB	0.793476	holds · ≤ 0.006 dB	0.021620718 · 0.043241437 · 0.021620718 · -1.543121131 · 0.629604005
3.15 kHz	−3.01 dB	0.747258	holds · ≤ 0.001 dB	0.032653053 · 0.065306107 · 0.032653053 · -1.427782127 · 0.558394340
4 kHz	−3.01 dB	0.691080	holds · ≤ 0.002 dB	0.049489956 · 0.098979913 · 0.049489956 · -1.279632425 · 0.477592250
5 kHz	−3.01 dB	0.630993	holds · ≤ 0.015 dB	0.072230875 · 0.144461751 · 0.072230875 · -1.109228793 · 0.398152294
6.3 kHz	−3.01 dB	0.562533	holds · ≤ 0.001 dB	0.105710197 · 0.211420394 · 0.105710197 · -0.893603075 · 0.316443862
8 kHz	−3.01 dB	0.490314	holds · ≤ 0.011 dB	0.155051026 · 0.310102051 · 0.155051026 · -0.620204103 · 0.240408206
10 kHz	−3.01 dB	0.433988	holds · ≤ 0.014 dB	0.220194700 · 0.440389401 · 0.220194700 · -0.307566360 · 0.188345161
12.5 kHz	−3.01 dB	0.415467	holds · ≤ 0.021 dB	0.312326343 · 0.624652685 · 0.312326343 · 0.076692548 · 0.172612822
16 kHz	−3.01 dB	0.490314	holds · ≤ 0.000 dB	0.465153077 · 0.930306154 · 0.465153077 · 0.620204103 · 0.240408206
20 kHz	−3.01 dB	0.691080	holds · ≤ 0.078 dB	0.689306169 · 1.378612338 · 0.689306169 · 1.279632425 · 0.477592250

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.