# 12.5 kHz low shelf

Lifts or drops everything below f0 by a fixed amount and leaves the top 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/lowshelf/12500-hz
Page title: 12.5 kHz Low 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 low-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:** 0.4175577 — conjugate pair at ±81.36°, 0.582 from the circle
- **−3 dB point:** 12513.6 Hz — 1.001× f0 at Q = 0.7071
- **Peak of the magnitude response:** +6.00 dB at 10 Hz — evaluated on the unit circle, not sketched
- **Group delay at f0:** -0.000 ms
- **16-bit fixed point:** holds — largest pole 0.4175853 in Q1.14
- **Coefficients at 48 kHz:** b0 = 1.431299, b1 = 0.396581, b2 = 0.265044, a1 = -0.125408, a2 = 0.174354 — 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 | 1.697641 | 2.036116 | 0.744347 | 0.919248 | 0.325120 | 0.570193 | 78.1% |
| 44.1 kHz | 1.473773 | 0.651649 | 0.304110 | 0.045709 | 0.171942 | 0.414659 | 56.7% |
| 48 kHz | 1.431299 | 0.396581 | 0.265044 | -0.125408 | 0.174354 | 0.417558 | 52.1% |
| 96 kHz | 1.209367 | -0.890940 | 0.325560 | -1.051555 | 0.374313 | 0.611811 | 26.0% |
| 192 kHz | 1.103455 | -1.470652 | 0.557532 | -1.518123 | 0.613516 | 0.783273 | 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 | — | 0.4175577 | 0.4175577 | yes | reference |
| float32 | — | 0.4175577 | 0.4175577 | yes | 0.0000 dB |
| 32-bit fixed | Q1.30 | 0.4175577 | 0.4175577 | yes | 0.0000 dB |
| 24-bit fixed | Q1.22 | 0.4175576 | 0.4175576 | yes | 0.0000 dB |
| 16-bit fixed | Q1.14 | 0.4175853 | 0.4175853 | yes | 0.0008 dB |

## Questions this page answers

### What are the biquad coefficients for a 12.5 kHz low-shelf filter at 48 kHz?

b0 = 1.431299, b1 = 0.396581, b2 = 0.265044, a1 = -0.125408, a2 = 0.174354, 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 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 0.4175853, against 0.4175577 exact, and the response drifts by at most 0.001 dB inside the band. 24-bit takes that to 0.0000 dB.

### Where is the real −3 dB point of a 12.5 kHz low-shelf filter?

12513.6 Hz, which is 1.001× 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 low-shelf filter?

0.4175577 at 48 kHz, as a conjugate pair at ±81.36°. Pole radius rises toward 1 as the corner frequency falls relative to the sample rate — the same filter at 192 kHz sits at 0.7832729 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.

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Source: MakerPortal — https://makerportal.ai/lab/biquad/lowshelf/12500-hz. Free to quote and cite with attribution and a link to the canonical page.