Computed

400 Hz low shelf

Lifts or drops everything below f0 by a fixed amount and leaves the top flat. 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 low shelf 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. Shelves fix the slope at S = 1, so there is no Q family to draw; the faint traces are the gain sweep.

Gain at f0 (400 Hz)

+3.00 dB

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

Pole radius at 48 kHz

0.9693264

conjugate pair at ±1.79°, 0.0307 from the circle

−3 dB point

400.7 Hz

1.002× f0 at Q = 0.7071

16-bit fixed point

holds

largest pole 0.9693169 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 400 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	1.079109	-1.599316	0.637712	-1.628622	0.687514	0.829165	10.0%
16 kHz	1.039159	-1.805503	0.798216	-1.813452	0.829426	0.910728	5.0%
22.1 kHz	1.028312	-1.860232	0.849105	-1.864518	0.873131	0.934415	3.6%
32 kHz	1.019444	-1.904518	0.893398	-1.906594	0.910766	0.954341	2.5%
44.1 kHz	1.014078	-1.931099	0.921458	-1.932206	0.934430	0.966659	1.8%
48 kHz	1.012928	-1.936775	0.927602	-1.937711	0.939594	0.969326	1.7%
96 kHz	1.006446	-1.968612	0.963120	-1.968850	0.969328	0.984545	0.8%
192 kHz	1.003218	-1.984364	0.981387	-1.984424	0.984545	0.992242	0.4%

const float b0 = 1.01292827f, b1 = -1.93677478f, b2 = 0.92760199f;
const float a1 = -1.93771142f, a2 = 0.93959362f;   // 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	—	0.9693264	0.9693264	yes	reference
float32	—	0.9693264	0.9693264	yes	0.0002 dB
32-bit fixed	Q1.30	0.9693264	0.9693264	yes	0.0000 dB
24-bit fixed	Q1.22	0.9693263	0.9693263	yes	0.0008 dB
16-bit fixed	Q1.14	0.9693169	0.9693169	yes	0.0268 dB

What gain does at 400 Hz

A shelf reaches half its dB gain at f₀ — exactly half, at every sample rate — and its full gain on the far side.

Gain	At f₀	b0	a1	a2	Pole r
-12 dB	−6.00 dB	0.974264	-1.895516	0.900710	0.949057
-6 dB	−3.00 dB	0.987237	-1.912055	0.915763	0.956955
-3 dB	−1.50 dB	0.993621	-1.919319	0.922450	0.960442
+3 dB	+1.50 dB	1.006419	-1.932100	0.934331	0.966608
+6 dB	+3.00 dB	1.012928	-1.937711	0.939594	0.969326
+12 dB	+6.00 dB	1.026416	-1.947584	0.948923	0.974127

Questions this filter answers

What are the biquad coefficients for a 400 Hz low-shelf filter at 48 kHz?

b0 = 1.012928, b1 = -1.936775, b2 = 0.927602, a1 = -1.937711, a2 = 0.939594, with a0 normalised to 1 — the RBJ Audio EQ Cookbook form at Q = 0.7071 and +6 dB of gain. 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 400 Hz low-shelf filter stable in 16-bit fixed point?

Yes. Rounding the five coefficients to a shared 1.14 scale leaves the largest pole at 0.9693169, against 0.9693264 exact, and the response drifts by at most 0.027 dB inside the band. 24-bit takes that to 0.0008 dB.

Where is the real −3 dB point of a 400 Hz low-shelf filter?

400.7 Hz, which is 1.002× the 400 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 a shelf does not read Q at all — its slope is fixed at S = 1, matching Web Audio's BiquadFilterNode, so the gain sweep above is the family that moves this filter rather than a Q sweep.

How close to the unit circle are the poles of a 400 Hz low-shelf filter?

0.9693264 at 48 kHz, as a conjugate pair at ±1.79°. Pole radius rises toward 1 as the corner frequency falls relative to the sample rate — the same filter at 192 kHz sits at 0.9922423 and at 8 kHz at 0.8291649. 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 400 Hz corner as every other filter type.

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 +3.00 dB at 400 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.