High shelf filters, corner by corner

Lifts or drops everything above f0 by a fixed amount and leaves the bottom 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 5 corners at and below 50 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	0.997802	diverges
25 Hz	+3.00 dB	0.997254	diverges
31.5 Hz	+3.00 dB	0.996541	diverges
40 Hz	+3.00 dB	0.995609	diverges
50 Hz	+3.00 dB	0.994515	diverges
60 Hz	+3.00 dB	0.993421	holds · ≤ 1.303 dB
80 Hz	+3.00 dB	0.991238	holds · ≤ 0.732 dB
100 Hz	+3.00 dB	0.989060	holds · ≤ 0.053 dB
125 Hz	+3.00 dB	0.986343	holds · ≤ 2.484 dB
160 Hz	+3.00 dB	0.982553	holds · ≤ 1.923 dB
200 Hz	+3.00 dB	0.978239	holds · ≤ 1.157 dB
250 Hz	+3.00 dB	0.972874	holds · ≤ 0.754 dB
315 Hz	+3.00 dB	0.965943	holds · ≤ 0.011 dB
400 Hz	+3.00 dB	0.956955	holds · ≤ 0.285 dB
500 Hz	+3.00 dB	0.946489	holds · ≤ 0.004 dB
630 Hz	+3.00 dB	0.933057	holds · ≤ 0.025 dB
800 Hz	+3.00 dB	0.915786	holds · ≤ 0.011 dB
1 kHz	+3.00 dB	0.895887	holds · ≤ 0.049 dB
1.25 kHz	+3.00 dB	0.871640	holds · ≤ 0.003 dB
1.6 kHz	+3.00 dB	0.838843	holds · ≤ 0.020 dB
2 kHz	+3.00 dB	0.802961	holds · ≤ 0.001 dB
2.5 kHz	+3.00 dB	0.760439	holds · ≤ 0.002 dB
3.15 kHz	+3.00 dB	0.708912	holds · ≤ 0.006 dB
4 kHz	+3.00 dB	0.647725	holds · ≤ 0.004 dB
5 kHz	+3.00 dB	0.584520	holds · ≤ 0.000 dB
6.3 kHz	+3.00 dB	0.516530	holds · ≤ 0.000 dB
8 kHz	+3.00 dB	0.452761	holds · ≤ 0.001 dB
10 kHz	+3.00 dB	0.416689	holds · ≤ 0.000 dB
12.5 kHz	+3.00 dB	0.430322	holds · ≤ 0.000 dB
16 kHz	+3.00 dB	0.534310	holds · ≤ 0.001 dB
20 kHz	+3.00 dB	0.731048	holds · ≤ 0.001 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.

Low shelf

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