High-pass filters, corner by corner

Rejects everything below the corner at 12 dB/octave and passes what is above it. 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	0.998150511 · -1.996301022 · 0.998150511 · -1.996297602 · 0.996304443
25 Hz	−3.01 dB	0.997689	diverges	0.997688674 · -1.995377347 · 0.997688674 · -1.995372005 · 0.995382690
31.5 Hz	−3.01 dB	0.997089	holds · ≤ 20.392 dB	0.997088604 · -1.994177209 · 0.997088604 · -1.994168733 · 0.994185685
40 Hz	−3.01 dB	0.996304	diverges	0.996304443 · -1.992608886 · 0.996304443 · -1.992595229 · 0.992622543
50 Hz	−3.01 dB	0.995383	holds · ≤ 28.275 dB	0.995382690 · -1.990765379 · 0.995382690 · -1.990744060 · 0.990786699
60 Hz	−3.01 dB	0.994462	holds · ≤ 31.364 dB	0.994461789 · -1.988923578 · 0.994461789 · -1.988892906 · 0.988954250
80 Hz	−3.01 dB	0.992623	holds · ≤ 0.982 dB	0.992622543 · -1.985245086 · 0.992622543 · -1.985190658 · 0.985299513
100 Hz	−3.01 dB	0.990787	holds · ≤ 2.814 dB	0.990786698 · -1.981573396 · 0.990786698 · -1.981488509 · 0.981658283
125 Hz	−3.01 dB	0.988497	holds · ≤ 52.505 dB	0.988496665 · -1.976993331 · 0.988496665 · -1.976861000 · 0.977125662
160 Hz	−3.01 dB	0.985300	holds · ≤ 0.100 dB	0.985299507 · -1.970599015 · 0.985299507 · -1.970382898 · 0.970815131
200 Hz	−3.01 dB	0.981658	holds · ≤ 19.717 dB	0.981658268 · -1.963316537 · 0.981658268 · -1.962980089 · 0.963652984
250 Hz	−3.01 dB	0.977126	holds · ≤ 0.074 dB	0.977125627 · -1.954251255 · 0.977125627 · -1.953727949 · 0.954774560
315 Hz	−3.01 dB	0.971265	holds · ≤ 0.020 dB	0.971264442 · -1.942528885 · 0.971264442 · -1.941702982 · 0.943354787
400 Hz	−3.01 dB	0.963653	holds · ≤ 10.091 dB	0.963652764 · -1.927305528 · 0.963652764 · -1.925983970 · 0.928627086
500 Hz	−3.01 dB	0.954775	holds · ≤ 0.007 dB	0.954774031 · -1.909548061 · 0.954774031 · -1.907501626 · 0.911594497
630 Hz	−3.01 dB	0.943355	holds · ≤ 0.019 dB	0.943353481 · -1.886706961 · 0.943353481 · -1.883495555 · 0.889918368
800 Hz	−3.01 dB	0.928627	holds · ≤ 7.814 dB	0.928623778 · -1.857247556 · 0.928623778 · -1.852146485 · 0.862348626
1 kHz	−3.01 dB	0.911595	holds · ≤ 2.840 dB	0.911586668 · -1.823173336 · 0.911586668 · -1.815341083 · 0.831005589
1.25 kHz	−3.01 dB	0.890744	holds · ≤ 2.473 dB	0.890724065 · -1.781448131 · 0.890724065 · -1.769471038 · 0.793425223
1.6 kHz	−3.01 dB	0.862354	holds · ≤ 0.004 dB	0.862301835 · -1.724603670 · 0.862301835 · -1.705552146 · 0.743655195
2 kHz	−3.01 dB	0.831023	holds · ≤ 0.829 dB	0.830898021 · -1.661796043 · 0.830898021 · -1.632993162 · 0.690598923
2.5 kHz	−3.01 dB	0.793476	holds · ≤ 0.560 dB	0.793181284 · -1.586362568 · 0.793181284 · -1.543121131 · 0.629604005
3.15 kHz	−3.01 dB	0.747258	holds · ≤ 0.381 dB	0.746544117 · -1.493088234 · 0.746544117 · -1.427782127 · 0.558394340
4 kHz	−3.01 dB	0.691080	holds · ≤ 0.273 dB	0.689306169 · -1.378612338 · 0.689306169 · -1.279632425 · 0.477592250
5 kHz	−3.01 dB	0.630993	holds · ≤ 0.001 dB	0.626845272 · -1.253690543 · 0.626845272 · -1.109228793 · 0.398152294
6.3 kHz	−3.01 dB	0.562533	holds · ≤ 0.116 dB	0.552511734 · -1.105023469 · 0.552511734 · -0.893603075 · 0.316443862
8 kHz	−3.01 dB	0.490314	holds · ≤ 0.000 dB	0.465153077 · -0.930306154 · 0.465153077 · -0.620204103 · 0.240408206
10 kHz	−3.01 dB	0.433988	holds · ≤ 0.029 dB	0.373977880 · -0.747955760 · 0.373977880 · -0.307566360 · 0.188345161
12.5 kHz	−3.01 dB	0.415467	holds · ≤ 0.000 dB	0.273980069 · -0.547960137 · 0.273980069 · 0.076692548 · 0.172612822
16 kHz	−3.01 dB	0.490314	holds · ≤ 0.014 dB	0.155051026 · -0.310102051 · 0.155051026 · 0.620204103 · 0.240408206
20 kHz	−3.01 dB	0.691080	holds · ≤ 0.002 dB	0.049489956 · -0.098979913 · 0.049489956 · 1.279632425 · 0.477592250

The other seven types

Low-pass

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

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.