Computed

20 Hz high-pass

Rejects everything below the corner at 12 dB/octave and passes what is above it. Solved at 8 sample rates, with the poles, the real −3 dB point and the word length it stops working at.

Open the interactive designer →All high-pass frequencies

Magnitude response at 48 kHz

Every vertex is 20·log₁₀|H(e^jω)| evaluated on the unit circle — not a sketch of the filter's shape. The faint traces behind it are the same corner at every Q in the sweep below.

Gain at f0 (20 Hz)

−3.01 dB

exact at every sample rate — the bilinear transform maps this value, not just the frequency

Pole radius at 48 kHz

0.9981505

conjugate pair at ±0.11°, 0.00185 from the circle

−3 dB point

20.0 Hz

1.000× f0 at Q = 0.7071

16-bit fixed point

diverges

largest pole 1.0000000 in Q1.14

Coefficients, at every sample rate

The cookbook computes w₀ = 2πf₀/F_s, so the same filter is a different set of numbers at every rate. Rates whose Nyquist limit is at or below 20 Hz are absent because the filter does not exist there — equivalently, the f₀/Nyquist column never reaches 100%. That is why this table is 8 rows and a 20 Hz page is 8.

Sample rate	b0	b1	b2	a1	a2	Pole r	f₀/Nyquist
8 kHz	0.988954	-1.977908	0.988954	-1.977786	0.978031	0.988954	0.5%
16 kHz	0.994462	-1.988924	0.994462	-1.988893	0.988954	0.994462	0.3%
22.1 kHz	0.995978	-1.991957	0.995978	-1.991940	0.991973	0.995978	0.2%
32 kHz	0.997227	-1.994454	0.997227	-1.994446	0.994462	0.997227	0.1%
44.1 kHz	0.997987	-1.995974	0.997987	-1.995970	0.995978	0.997987	0.1%
48 kHz	0.998151	-1.996301	0.998151	-1.996298	0.996304	0.998151	0.1%
96 kHz	0.999075	-1.998150	0.999075	-1.998149	0.998151	0.999075	0.0%
192 kHz	0.999537	-1.999075	0.999537	-1.999074	0.999075	0.999537	0.0%

const float b0 = 0.99815051f, b1 = -1.99630102f, b2 = 0.99815051f;
const float a1 = -1.99629760f, a2 = 0.99630444f;   // a0 == 1

What word length this filter survives

The fixed-point rows round all five coefficients to one shared scale, which is what a q15/q31 biquad section does with its post-shift.Pole radius is solved from the quadratic, not taken as √|a₂| — once rounding pushes the poles onto the real axis those two disagree, and the convenient one reports a comfortable margin on a filter that has already left the unit circle. Error is the worst deviation from float64 across frequencies where the response is within 40 dB of its own peak; below that it is measuring the −200 dB floor.

Word format	Q format	Largest pole	√\|a₂\| says	Stable	Worst error in band
float64	—	0.9981505	0.9981505	yes	reference
float32	—	0.9981505	0.9981505	yes	0.0119 dB
32-bit fixed	Q1.30	0.9981505	0.9981505	yes	0.0050 dB
24-bit fixed	Q1.22	0.9981505	0.9981505	yes	1.3692 dB
16-bit fixed	Q1.14	1.0000000 (real)	0.9981367	no	n/a — diverges

Read the two pole columns against each other on the 16-bit row.√|a₂| reports 0.9981367 — well inside the unit circle — while the larger real pole is actually at 1.0000000. The shortcut is the geometric mean of the two poles and is exact only while they are a conjugate pair. Evaluated rather than inferred, the quantised filter has +122 dB of gain near DC in that format — a filter that diverges, not one that is merely inaccurate.

What Q does at 20 Hz

f₀ does not move with Q — the −3 dB point does. They are the same frequency only at Q = 1/√2, which is why a Butterworth corner is the one people quote and why every other Q surprises somebody.

Q	At f₀	Peak	−3 dB	Pole r	Group delay at f₀
0.5	−6.02 dB	0.00 dB @ 24.00 kHz	31.1 Hz	0.997385	7.959 ms
0.7071	−3.01 dB	0.00 dB @ 23.23 kHz	20.0 Hz	0.998151	11.253 ms
1	0.00 dB	+1.25 dB @ 28 Hz	17.1 Hz	0.998692	15.907 ms
2	+6.02 dB	+6.29 dB @ 21 Hz	32.0 Hz	0.999346	31.735 ms
4	+12.04 dB	+12.10 dB @ 20 Hz	23.6 Hz	0.999673	62.852 ms
10	+20.00 dB	+19.77 dB @ 20 Hz	21.2 Hz	0.999869	147.620 ms

Questions this filter answers

What are the biquad coefficients for a 20 Hz high-pass filter at 48 kHz?

b0 = 0.998151, b1 = -1.996301, b2 = 0.998151, a1 = -1.996298, a2 = 0.996304, with a0 normalised to 1 — the RBJ Audio EQ Cookbook form at Q = 0.7071. Every other sample rate in the table above gives different numbers for the same filter, because w0 = 2πf0/Fs and every cosine and sine downstream of it moves.

Is a 20 Hz high-pass filter stable in 16-bit fixed point?

No. Rounding the five coefficients to a shared 1.14 scale pushes the poles onto the real axis and the larger one out to 1.0000000 — at or outside the unit circle, which is a filter that diverges rather than one that is merely inaccurate. Note that √|a2| still reads 0.9981367 here, comfortably inside the circle: the shortcut is the geometric mean of the two real poles and it does not see this. Use 24-bit (largest pole 0.9981505) or float32.

Where is the real −3 dB point of a 20 Hz high-pass filter?

20.0 Hz, which is 1.000× the 20 Hz corner. f0 and the −3 dB point are the same frequency only at Q = 1/√2; this page is designed at Q = 0.7071, and the Q sweep above shows the point moving from 31.1 Hz to 21.2 Hz across the sweep while f0 never moves.

How close to the unit circle are the poles of a 20 Hz high-pass filter?

0.9981505 at 48 kHz, as a conjugate pair at ±0.11°. Pole radius rises toward 1 as the corner frequency falls relative to the sample rate — the same filter at 192 kHz sits at 0.9995373 and at 8 kHz at 0.9889542. That distance is the whole story of the fixed-point table: a pole a few parts in 10⁵ from the circle has nowhere to be rounded to.

The neighbouring corners

One third-octave either side, and the same 20 Hz corner as every other filter type.

25 Hzhigh-pass

Method and limits. Coefficients follow the RBJ Audio EQ Cookbook, the bilinear transform of the analog prototype prewarped so the corner lands exactly on f0 — which is why −3.01 dB at 20 Hz holds at every sample rate in the table rather than only at low f0/Fs. Shelves fix the slope at S = 1, matching Web Audio's BiquadFilterNode, so Q is not read for those two types. Pole and zero radii are the roots of the quadratic, not sqrt(|a2|). The fixed-point rows model a single shared coefficient scale and no other quantisation: they say nothing about signal-path headroom, limit cycles or the accumulator width your implementation uses, all of which can make a filter that passes this table still misbehave. Nothing on this page is fetched or estimated — it is solved from the type and the frequency in the URL.