Low shelf filters, corner by corner

Lifts or drops everything below f0 by a fixed amount and leaves the top flat. 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 4 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	+3.00 dB	0.998444	diverges	1.000642771 · -1.996882408 · 0.996249303 · -1.996884819 · 0.996889663
25 Hz	+3.00 dB	0.998055	diverges	1.000803527 · -1.996102259 · 0.995313829 · -1.996106025 · 0.996113591
31.5 Hz	+3.00 dB	0.997550	diverges	1.001012549 · -1.995087619 · 0.994099027 · -1.995093594 · 0.995105601
40 Hz	+3.00 dB	0.996890	diverges	1.001285950 · -1.993760021 · 0.992512676 · -1.993769649 · 0.993788997
50 Hz	+3.00 dB	0.996114	holds · ≤ 5.870 dB	1.001607692 · -1.992197040 · 0.990649622 · -1.992212072 · 0.992242281
60 Hz	+3.00 dB	0.995338	holds · ≤ 5.821 dB	1.001929534 · -1.990632873 · 0.988790067 · -1.990654503 · 0.990697970
80 Hz	+3.00 dB	0.993789	holds · ≤ 3.464 dB	1.002573517 · -1.987500998 · 0.985081424 · -1.987539393 · 0.987616547
100 Hz	+3.00 dB	0.992242	holds · ≤ 0.072 dB	1.003217896 · -1.984364431 · 0.981386699 · -1.984424329 · 0.984544696
125 Hz	+3.00 dB	0.990312	holds · ≤ 1.353 dB	1.004023919 · -1.980437172 · 0.976787785 · -1.980530583 · 0.980718293
160 Hz	+3.00 dB	0.987616	holds · ≤ 0.020 dB	1.005153364 · -1.974926901 · 0.970385517 · -1.975079533 · 0.975386249
200 Hz	+3.00 dB	0.984545	holds · ≤ 1.176 dB	1.006445578 · -1.968612352 · 0.963120058 · -1.968850107 · 0.969327881
250 Hz	+3.00 dB	0.980718	holds · ≤ 0.047 dB	1.008062927 · -1.960693900 · 0.954114784 · -1.961063972 · 0.961807639
315 Hz	+3.00 dB	0.975765	holds · ≤ 0.424 dB	1.010168845 · -1.950358699 · 0.942533904 · -1.950943320 · 0.952118128
400 Hz	+3.00 dB	0.969326	holds · ≤ 0.027 dB	1.012928271 · -1.936774776 · 0.927601987 · -1.937711417 · 0.939593617
500 Hz	+3.00 dB	0.961805	holds · ≤ 0.070 dB	1.016182325 · -1.920696933 · 0.910338511 · -1.922149452 · 0.925068318
630 Hz	+3.00 dB	0.952112	holds · ≤ 0.021 dB	1.020424299 · -1.899644981 · 0.888377375 · -1.901928719 · 0.906517937
800 Hz	+3.00 dB	0.939582	holds · ≤ 0.016 dB	1.025990039 · -1.871869946 · 0.860460841 · -1.875506524 · 0.882814302
1 kHz	+3.00 dB	0.925046	holds · ≤ 0.021 dB	1.032562483 · -1.838856872 · 0.828747684 · -1.844456867 · 0.855710172
1.25 kHz	+3.00 dB	0.907182	holds · ≤ 0.005 dB	1.040811207 · -1.797113250 · 0.790763036 · -1.805708042 · 0.822979451
1.6 kHz	+3.00 dB	0.882729	holds · ≤ 0.008 dB	1.052412621 · -1.737855925 · 0.740538589 · -1.751597041 · 0.779210094
2 kHz	+3.00 dB	0.855551	holds · ≤ 0.002 dB	1.065734137 · -1.669084140 · 0.687128281 · -1.689978977 · 0.731967580
2.5 kHz	+3.00 dB	0.822689	holds · ≤ 0.004 dB	1.082460132 · -1.581730787 · 0.625951927 · -1.613325905 · 0.676816941
3.15 kHz	+3.00 dB	0.781734	holds · ≤ 0.004 dB	1.104296421 · -1.466168789 · 0.554970887 · -1.514328792 · 0.611107305
4 kHz	+3.00 dB	0.731048	holds · ≤ 0.001 dB	1.132966351 · -1.312123521 · 0.475333783 · -1.385991859 · 0.534431796
5 kHz	+3.00 dB	0.675412	holds · ≤ 0.000 dB	1.166827718 · -1.127211318 · 0.398662716 · -1.236520927 · 0.456180825
6.3 kHz	+3.00 dB	0.609406	holds · ≤ 0.000 dB	1.211074505 · -0.881372037 · 0.323118534 · -1.044189762 · 0.371375314
8 kHz	+3.00 dB	0.534310	holds · ≤ 0.001 dB	1.269517186 · -0.550348575 · 0.260241040 · -0.794619594 · 0.285487207
10 kHz	+3.00 dB	0.465000	holds · ≤ 0.000 dB	1.339727359 · -0.144480184 · 0.232616265 · -0.500598440 · 0.216225368
12.5 kHz	+3.00 dB	0.417558	holds · ≤ 0.001 dB	1.431299216 · 0.396581055 · 0.265043837 · -0.125407577 · 0.174354421
16 kHz	+3.00 dB	0.452761	holds · ≤ 0.001 dB	1.571670189 · 1.248879927 · 0.448691733 · 0.433510142 · 0.204992136
20 kHz	+3.00 dB	0.647725	holds · ≤ 0.004 dB	1.761095829 · 2.440864482 · 0.941185608 · 1.158131060 · 0.419548015

The other seven types

Low-pass

Passes everything below the corner and rolls off above it at 12 dB/octave.

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.

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 and +6 dB, 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.