# 20 Hz high shelf Lifts or drops everything above f0 by a fixed amount and leaves the bottom flat. Solved at 8 sample rates, with the poles, the real −3 dB point and the word length it stops working at. Canonical page: https://makerportal.ai/lab/biquad/highshelf/20-hz Page title: 20 Hz High shelf Biquad Coefficients — RBJ cookbook This markdown document and the HTML page above are rendered from the same solved values at build time, by the same functions. Nothing here is written by a language model and nothing is fetched at request time. ## Key figures - **Filter:** 20 Hz high-shelf filter — Q = 0.7071, gain +6 dB - **Gain at f0 (20 Hz):** +3.00 dB — exact at every sample rate — the bilinear transform maps this value, not just the frequency - **Pole radius at 48 kHz:** 0.9978023 — conjugate pair at ±0.13°, 0.00220 from the circle - **−3 dB point:** 20.0 Hz — 0.998× f0 at Q = 0.7071 - **Peak of the magnitude response:** +6.00 dB at 24.00 kHz — evaluated on the unit circle, not sketched - **Group delay at f0:** -0.003 ms - **16-bit fixed point:** diverges — largest pole 1.0000000 in Q2.13 - **Coefficients at 48 kHz:** b0 = 1.993981, b1 = -3.981750, b2 = 1.987779, a1 = -1.995600, a2 = 0.995609 — a0 normalised to 1 ## Coefficients, at every sample rate The cookbook computes w0 = 2πf0/Fs, 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 | 1.987585 | -3.938021 | 1.950780 | -1.973600 | 0.973944 | 0.986886 | 0.5% | | 16 kHz | 1.991420 | -3.964229 | 1.972896 | -1.986799 | 0.986886 | 0.993421 | 0.3% | | 22.1 kHz | 1.992473 | -3.971435 | 1.979007 | -1.990421 | 0.990467 | 0.995222 | 0.2% | | 32 kHz | 1.993340 | -3.977366 | 1.984047 | -1.993400 | 0.993421 | 0.996705 | 0.1% | | 44.1 kHz | 1.993867 | -3.980974 | 1.987118 | -1.995211 | 0.995222 | 0.997608 | 0.1% | | 48 kHz | 1.993981 | -3.981750 | 1.987779 | -1.995600 | 0.995609 | 0.997802 | 0.1% | | 96 kHz | 1.994621 | -3.986136 | 1.991517 | -1.997800 | 0.997802 | 0.998901 | 0.0% | | 192 kHz | 1.994942 | -3.988330 | 1.993389 | -1.998900 | 0.998901 | 0.999450 | 0.0% | ## What word length this filter survives All five coefficients are rounded 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 sqrt(|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. | Word format | Q format | Largest pole | sqrt of abs(a2) says | Stable | Worst error in band | |---|---|---|---|---|---| | float64 | — | 0.9978023 | 0.9978023 | yes | reference | | float32 | — | 0.9978023 | 0.9978023 | yes | 0.0266 dB | | 32-bit fixed | Q2.29 | 0.9978023 | 0.9978023 | yes | 0.0002 dB | | 24-bit fixed | Q2.21 | 0.9978022 | 0.9978022 | yes | 0.0441 dB | | 16-bit fixed | Q2.13 | 1.0000000 (real) | 0.9978003 | no | n/a — diverges | ## Questions this page answers ### What are the biquad coefficients for a 20 Hz high-shelf filter at 48 kHz? b0 = 1.993981, b1 = -3.981750, b2 = 1.987779, a1 = -1.995600, a2 = 0.995609, 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 20 Hz high-shelf filter stable in 16-bit fixed point? No. Rounding the five coefficients to a shared 2.13 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 0.9978003 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 0.9978022) or float32. ### Where is the real −3 dB point of a 20 Hz high-shelf filter? 20.0 Hz, which is 0.998× 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 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 20 Hz high-shelf filter? 0.9978023 at 48 kHz, as a conjugate pair at ±0.13°. Pole radius rises toward 1 as the corner frequency falls relative to the sample rate — the same filter at 192 kHz sits at 0.9994501 and at 8 kHz at 0.9868859. 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. ## 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 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. ## Related tool [Biquad Filter Designer](https://makerportal.ai/lab/biquad-filter-designer) — Interactive RBJ-cookbook biquad designer — pick a type, corner, Q and gain and hear the filter while reading its coefficients, poles and magnitude response. --- Source: MakerPortal — https://makerportal.ai/lab/biquad/highshelf/20-hz. Free to quote and cite with attribution and a link to the canonical page.