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	b0 · b1 · b2 · a1 · a2 @ 48 kHz
20 Hz	+3.00 dB	0.997802	diverges	1.993980642 · -3.981749673 · 1.987778691 · -1.995599695 · 0.995609355
25 Hz	+3.00 dB	0.997254	diverges	1.993660354 · -3.979557444 · 1.985912175 · -1.994499624 · 0.994514710
31.5 Hz	+3.00 dB	0.996541	diverges	1.993244057 · -3.976708450 · 1.983488325 · -1.993069538 · 0.993093471
40 Hz	+3.00 dB	0.995609	diverges	1.992699803 · -3.972984386 · 1.980323139 · -1.991199438 · 0.991237993
50 Hz	+3.00 dB	0.994515	diverges	1.992059697 · -3.968605378 · 1.976605858 · -1.988999342 · 0.989059519
60 Hz	+3.00 dB	0.993421	holds · ≤ 1.303 dB	1.991419804 · -3.964228800 · 1.972895556 · -1.986799277 · 0.986885837
80 Hz	+3.00 dB	0.991238	holds · ≤ 0.732 dB	1.990140655 · -3.955482948 · 1.965495841 · -1.982399261 · 0.982552808
100 Hz	+3.00 dB	0.989060	holds · ≤ 0.053 dB	1.988862363 · -3.946746861 · 1.958123891 · -1.977999435 · 0.978238828
125 Hz	+3.00 dB	0.986343	holds · ≤ 2.484 dB	1.987265718 · -3.935840531 · 1.948947844 · -1.972499991 · 0.972873022
160 Hz	+3.00 dB	0.982553	holds · ≤ 1.923 dB	1.985032719 · -3.920597494 · 1.936173618 · -1.964801563 · 0.965410406
200 Hz	+3.00 dB	0.978239	holds · ≤ 1.157 dB	1.982484060 · -3.903213954 · 1.921677073 · -1.956004771 · 0.956951950
250 Hz	+3.00 dB	0.972874	holds · ≤ 0.754 dB	1.979303336 · -3.881540461 · 1.903709069 · -1.945011414 · 0.946483358
315 Hz	+3.00 dB	0.965943	holds · ≤ 0.011 dB	1.975177045 · -3.853458463 · 1.880601871 · -1.930725451 · 0.933045904
400 Hz	+3.00 dB	0.956955	holds · ≤ 0.285 dB	1.969796255 · -3.816896692 · 1.850807987 · -1.912055208 · 0.915762758
500 Hz	+3.00 dB	0.946489	holds · ≤ 0.004 dB	1.963488505 · -3.774118353 · 1.816361008 · -1.890110549 · 0.895841709
630 Hz	+3.00 dB	0.933057	holds · ≤ 0.025 dB	1.955326148 · -3.718890956 · 1.772538225 · -1.861622642 · 0.870596061
800 Hz	+3.00 dB	0.915786	holds · ≤ 0.011 dB	1.944718992 · -3.647333157 · 1.716825740 · -1.824452358 · 0.838663933
1 kHz	+3.00 dB	0.895887	holds · ≤ 0.049 dB	1.932340509 · -3.564118722 · 1.653523430 · -1.780867407 · 0.802612624
1.25 kHz	+3.00 dB	0.871640	holds · ≤ 0.003 dB	1.917026163 · -3.461589559 · 1.577673140 · -1.726646713 · 0.759756458
1.6 kHz	+3.00 dB	0.838843	holds · ≤ 0.020 dB	1.895893565 · -3.320841558 · 1.477299403 · -1.651306618 · 0.703658028
2 kHz	+3.00 dB	0.802961	holds · ≤ 0.001 dB	1.872195181 · -3.163970496 · 1.370386176 · -1.566135570 · 0.644746431
2.5 kHz	+3.00 dB	0.760439	holds · ≤ 0.002 dB	1.843266331 · -2.973789321 · 1.247553880 · -1.461236992 · 0.578267882
3.15 kHz	+3.00 dB	0.708912	holds · ≤ 0.006 dB	1.806817697 · -2.736116060 · 1.104159493 · -1.327694957 · 0.502556086
4 kHz	+3.00 dB	0.647725	holds · ≤ 0.004 dB	1.761095829 · -2.440864482 · 0.941185608 · -1.158131060 · 0.419548015
5 kHz	+3.00 dB	0.584520	holds · ≤ 0.000 dB	1.709988787 · -2.114436921 · 0.780064095 · -0.966047773 · 0.341663734
6.3 kHz	+3.00 dB	0.516530	holds · ≤ 0.000 dB	1.647514094 · -1.720317350 · 0.611846064 · -0.727760377 · 0.266803184
8 kHz	+3.00 dB	0.452761	holds · ≤ 0.001 dB	1.571670189 · -1.248879927 · 0.448691733 · -0.433510142 · 0.204992136
10 kHz	+3.00 dB	0.416689	holds · ≤ 0.000 dB	1.489304746 · -0.745543633 · 0.322025467 · -0.107842975 · 0.173629555
12.5 kHz	+3.00 dB	0.430322	holds · ≤ 0.000 dB	1.394021804 · -0.174820897 · 0.243053864 · 0.277077672 · 0.185177099
16 kHz	+3.00 dB	0.534310	holds · ≤ 0.001 dB	1.269517186 · 0.550348575 · 0.260241040 · 0.794619594 · 0.285487207
20 kHz	+3.00 dB	0.731048	holds · ≤ 0.001 dB	1.132966351 · 1.312123521 · 0.475333783 · 1.385991859 · 0.534431796

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.