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
20 Hz	+3.00 dB	0.998444	diverges
25 Hz	+3.00 dB	0.998055	diverges
31.5 Hz	+3.00 dB	0.997550	diverges
40 Hz	+3.00 dB	0.996890	diverges
50 Hz	+3.00 dB	0.996114	holds · ≤ 5.870 dB
60 Hz	+3.00 dB	0.995338	holds · ≤ 5.821 dB
80 Hz	+3.00 dB	0.993789	holds · ≤ 3.464 dB
100 Hz	+3.00 dB	0.992242	holds · ≤ 0.072 dB
125 Hz	+3.00 dB	0.990312	holds · ≤ 1.353 dB
160 Hz	+3.00 dB	0.987616	holds · ≤ 0.020 dB
200 Hz	+3.00 dB	0.984545	holds · ≤ 1.176 dB
250 Hz	+3.00 dB	0.980718	holds · ≤ 0.047 dB
315 Hz	+3.00 dB	0.975765	holds · ≤ 0.424 dB
400 Hz	+3.00 dB	0.969326	holds · ≤ 0.027 dB
500 Hz	+3.00 dB	0.961805	holds · ≤ 0.070 dB
630 Hz	+3.00 dB	0.952112	holds · ≤ 0.021 dB
800 Hz	+3.00 dB	0.939582	holds · ≤ 0.016 dB
1 kHz	+3.00 dB	0.925046	holds · ≤ 0.021 dB
1.25 kHz	+3.00 dB	0.907182	holds · ≤ 0.005 dB
1.6 kHz	+3.00 dB	0.882729	holds · ≤ 0.008 dB
2 kHz	+3.00 dB	0.855551	holds · ≤ 0.002 dB
2.5 kHz	+3.00 dB	0.822689	holds · ≤ 0.004 dB
3.15 kHz	+3.00 dB	0.781734	holds · ≤ 0.004 dB
4 kHz	+3.00 dB	0.731048	holds · ≤ 0.001 dB
5 kHz	+3.00 dB	0.675412	holds · ≤ 0.000 dB
6.3 kHz	+3.00 dB	0.609406	holds · ≤ 0.000 dB
8 kHz	+3.00 dB	0.534310	holds · ≤ 0.001 dB
10 kHz	+3.00 dB	0.465000	holds · ≤ 0.000 dB
12.5 kHz	+3.00 dB	0.417558	holds · ≤ 0.001 dB
16 kHz	+3.00 dB	0.452761	holds · ≤ 0.001 dB
20 kHz	+3.00 dB	0.647725	holds · ≤ 0.004 dB

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.