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
20 Hz	−∞ dB	0.998692	diverges
25 Hz	−∞ dB	0.998365	diverges
31.5 Hz	−∞ dB	0.997940	holds · ≤ 30.330 dB
40 Hz	−∞ dB	0.997385	diverges
50 Hz	−∞ dB	0.996733	holds · ≤ 36.018 dB
60 Hz	−∞ dB	0.996081	holds · ≤ 0.933 dB
80 Hz	−∞ dB	0.994778	holds · ≤ 14.753 dB
100 Hz	−∞ dB	0.993476	holds · ≤ 23.861 dB
125 Hz	−∞ dB	0.991852	holds · ≤ 15.647 dB
160 Hz	−∞ dB	0.989583	holds · ≤ 2.165 dB
200 Hz	−∞ dB	0.986996	holds · ≤ 14.146 dB
250 Hz	−∞ dB	0.983772	holds · ≤ 7.850 dB
315 Hz	−∞ dB	0.979597	holds · ≤ 1.709 dB
400 Hz	−∞ dB	0.974166	holds · ≤ 34.830 dB
500 Hz	−∞ dB	0.967816	holds · ≤ 0.143 dB
630 Hz	−∞ dB	0.959628	holds · ≤ 0.243 dB
800 Hz	−∞ dB	0.949033	holds · ≤ 0.969 dB
1 kHz	−∞ dB	0.936734	holds · ≤ 0.199 dB
1.25 kHz	−∞ dB	0.921614	holds · ≤ 0.362 dB
1.6 kHz	−∞ dB	0.900925	holds · ≤ 0.781 dB
2 kHz	−∞ dB	0.877973	holds · ≤ 0.167 dB
2.5 kHz	−∞ dB	0.850335	holds · ≤ 0.225 dB
3.15 kHz	−∞ dB	0.816178	holds · ≤ 0.017 dB
4 kHz	−∞ dB	0.774597	holds · ≤ 0.001 dB
5 kHz	−∞ dB	0.730270	holds · ≤ 0.099 dB
6.3 kHz	−∞ dB	0.680357	holds · ≤ 0.000 dB
8 kHz	−∞ dB	0.629016	holds · ≤ 0.000 dB
10 kHz	−∞ dB	0.590467	holds · ≤ 0.011 dB
12.5 kHz	−∞ dB	0.578174	holds · ≤ 0.002 dB
16 kHz	−∞ dB	0.629016	holds · ≤ 0.000 dB
20 kHz	−∞ dB	0.774597	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.

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.