Notch filters, corner by corner

Removes one frequency and leaves the rest of the spectrum untouched. 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	−∞ dB	0.998692	diverges	0.998692716 · -1.997378587 · 0.998692716 · -1.997378587 · 0.997385432
25 Hz	−∞ dB	0.998365	diverges	0.998366430 · -1.996722168 · 0.998366430 · -1.996722168 · 0.996732859
31.5 Hz	−∞ dB	0.997940	holds · ≤ 30.330 dB	0.997942577 · -1.995868188 · 0.997942577 · -1.995868188 · 0.995885155
40 Hz	−∞ dB	0.997385	diverges	0.997388854 · -1.994750364 · 0.997388854 · -1.994750364 · 0.994777708
50 Hz	−∞ dB	0.996733	holds · ≤ 36.018 dB	0.996738205 · -1.993433713 · 0.996738205 · -1.993433713 · 0.993476410
60 Hz	−∞ dB	0.996081	holds · ≤ 0.933 dB	0.996088410 · -1.992115377 · 0.996088410 · -1.992115377 · 0.992176820
80 Hz	−∞ dB	0.994778	holds · ≤ 14.753 dB	0.994791380 · -1.989473669 · 0.994791380 · -1.989473669 · 0.989582759
100 Hz	−∞ dB	0.993476	holds · ≤ 23.861 dB	0.993497758 · -1.986825285 · 0.993497758 · -1.986825285 · 0.986995516
125 Hz	−∞ dB	0.991852	holds · ≤ 15.647 dB	0.991885518 · -1.983505484 · 0.991885518 · -1.983505484 · 0.983771035
160 Hz	−∞ dB	0.989583	holds · ≤ 2.165 dB	0.989637300 · -1.978840513 · 0.989637300 · -1.978840513 · 0.979274600
200 Hz	−∞ dB	0.986996	holds · ≤ 14.146 dB	0.987080621 · -1.973484746 · 0.987080621 · -1.973484746 · 0.974161242
250 Hz	−∞ dB	0.983772	holds · ≤ 7.850 dB	0.983903785 · -1.966753982 · 0.983903785 · -1.966753982 · 0.967807571
315 Hz	−∞ dB	0.979597	holds · ≤ 1.709 dB	0.979805369 · -1.957945114 · 0.979805369 · -1.957945114 · 0.959610737
400 Hz	−∞ dB	0.974166	holds · ≤ 34.830 dB	0.974499323 · -1.946327611 · 0.974499323 · -1.946327611 · 0.948998646
500 Hz	−∞ dB	0.967816	holds · ≤ 0.143 dB	0.968333964 · -1.932521374 · 0.968333964 · -1.932521374 · 0.936667929
630 Hz	−∞ dB	0.959628	holds · ≤ 0.243 dB	0.960442557 · -1.914357062 · 0.960442557 · -1.914357062 · 0.920885114
800 Hz	−∞ dB	0.949033	holds · ≤ 0.969 dB	0.950331647 · -1.890251261 · 0.950331647 · -1.890251261 · 0.900663293
1 kHz	−∞ dB	0.936734	holds · ≤ 0.199 dB	0.938735232 · -1.861408445 · 0.938735232 · -1.861408445 · 0.877470465
1.25 kHz	−∞ dB	0.921614	holds · ≤ 0.362 dB	0.924686387 · -1.824671315 · 0.924686387 · -1.824671315 · 0.849372773
1.6 kHz	−∞ dB	0.900925	holds · ≤ 0.781 dB	0.905833330 · -1.772077397 · 0.905833330 · -1.772077397 · 0.811666661
2 kHz	−∞ dB	0.877973	holds · ≤ 0.167 dB	0.885418424 · -1.710497046 · 0.885418424 · -1.710497046 · 0.770836849
2.5 kHz	−∞ dB	0.850335	holds · ≤ 0.225 dB	0.861534419 · -1.631625797 · 0.861534419 · -1.631625797 · 0.723068837
3.15 kHz	−∞ dB	0.816178	holds · ≤ 0.017 dB	0.833073403 · -1.526503638 · 0.833073403 · -1.526503638 · 0.666146806
4 kHz	−∞ dB	0.774597	holds · ≤ 0.001 dB	0.800000000 · -1.385640646 · 0.800000000 · -1.385640646 · 0.600000000
5 kHz	−∞ dB	0.730270	holds · ≤ 0.099 dB	0.766647336 · -1.216444450 · 0.766647336 · -1.216444450 · 0.533294672
6.3 kHz	−∞ dB	0.680357	holds · ≤ 0.000 dB	0.731442613 · -0.993007581 · 0.731442613 · -0.993007581 · 0.462885225
8 kHz	−∞ dB	0.629016	holds · ≤ 0.000 dB	0.697830521 · -0.697830521 · 0.697830521 · -0.697830521 · 0.395661041
10 kHz	−∞ dB	0.590467	holds · ≤ 0.011 dB	0.674325697 · -0.349056666 · 0.674325697 · -0.349056666 · 0.348651394
12.5 kHz	−∞ dB	0.578174	holds · ≤ 0.002 dB	0.667142801 · 0.087266454 · 0.667142801 · 0.087266454 · 0.334285603
16 kHz	−∞ dB	0.629016	holds · ≤ 0.000 dB	0.697830521 · 0.697830521 · 0.697830521 · 0.697830521 · 0.395661041
20 kHz	−∞ dB	0.774597	holds · ≤ 0.001 dB	0.800000000 · 1.385640646 · 0.800000000 · 1.385640646 · 0.600000000

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.

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 = 1.0000, 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.