# 12.5 kHz high shelf Lifts or drops everything above f0 by a fixed amount and leaves the bottom flat. Solved at 5 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/12500-hz Page title: 12.5 kHz 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:** 12.5 kHz high-shelf filter — Q = 0.7071, gain +6 dB - **Gain at f0 (12.5 kHz):** +3.00 dB — exact at every sample rate — the bilinear transform maps this value, not just the frequency - **Pole radius at 48 kHz:** 0.4303221 — conjugate pair at ±108.78°, 0.570 from the circle - **−3 dB point:** 12486.4 Hz — 0.999× 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:** 0.000 ms - **16-bit fixed point:** holds — largest pole 0.4303262 in Q1.14 - **Coefficients at 48 kHz:** b0 = 1.394022, b1 = -0.174821, b2 = 0.243054, a1 = 0.277078, a2 = 0.185177 — 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 12.5 kHz are absent because the filter does not exist there — equivalently, the f₀/Nyquist column never reaches 100%. That is why this table is 5 rows and a 20 Hz page is 8. | Sample rate | b0 | b1 | b2 | a1 | a2 | Pole r | f0/Nyquist | |---|---|---|---|---|---|---|---| | 32 kHz | 1.175315 | 1.080406 | 0.382119 | 1.199380 | 0.438460 | 0.662163 | 78.1% | | 44.1 kHz | 1.353846 | 0.061882 | 0.232783 | 0.442164 | 0.206348 | 0.454256 | 56.7% | | 48 kHz | 1.394022 | -0.174821 | 0.243054 | 0.277078 | 0.185177 | 0.430322 | 52.1% | | 96 kHz | 1.649840 | -1.734897 | 0.617557 | -0.736700 | 0.269199 | 0.518844 | 26.0% | | 192 kHz | 1.808195 | -2.745062 | 1.109358 | -1.332770 | 0.505261 | 0.710817 | 13.0% | ## 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 | — | 0.4303221 | 0.4303221 | yes | reference | | float32 | — | 0.4303221 | 0.4303221 | yes | 0.0000 dB | | 32-bit fixed | Q1.30 | 0.4303221 | 0.4303221 | yes | 0.0000 dB | | 24-bit fixed | Q1.22 | 0.4303221 | 0.4303221 | yes | 0.0000 dB | | 16-bit fixed | Q1.14 | 0.4303262 | 0.4303262 | yes | 0.0002 dB | ## Questions this page answers ### What are the biquad coefficients for a 12.5 kHz high-shelf filter at 48 kHz? b0 = 1.394022, b1 = -0.174821, b2 = 0.243054, a1 = 0.277078, a2 = 0.185177, 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 12.5 kHz high-shelf filter stable in 16-bit fixed point? Yes. Rounding the five coefficients to a shared 1.14 scale leaves the largest pole at 0.4303262, against 0.4303221 exact, and the response drifts by at most 0.000 dB inside the band. 24-bit takes that to 0.0000 dB. ### Where is the real −3 dB point of a 12.5 kHz high-shelf filter? 12486.4 Hz, which is 0.999× the 12.5 kHz 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 12.5 kHz high-shelf filter? 0.4303221 at 48 kHz, as a conjugate pair at ±108.78°. Pole radius rises toward 1 as the corner frequency falls relative to the sample rate — the same filter at 192 kHz sits at 0.7108169 and at 8 kHz it is above Nyquist. 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 12.5 kHz 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/12500-hz. Free to quote and cite with attribution and a link to the canonical page.