# 12.5 kHz high-pass

Rejects everything below the corner at 12 dB/octave and passes what is above it. 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/highpass/12500-hz
Page title: 12.5 kHz High-pass 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-pass filter — Q = 0.7071
- **Gain at f0 (12.5 kHz):** −3.01 dB — exact at every sample rate — the bilinear transform maps this value, not just the frequency
- **Pole radius at 48 kHz:** 0.4154670 — conjugate pair at ±95.30°, 0.585 from the circle
- **−3 dB point:** 12500.0 Hz — 1.000× f0 at Q = 0.7071
- **Peak of the magnitude response:** 0.00 dB at 24.00 kHz — evaluated on the unit circle, not sketched
- **Group delay at f0:** 0.030 ms
- **16-bit fixed point:** holds — largest pole 0.4154605 in Q0.15
- **Coefficients at 48 kHz:** b0 = 0.273980, b1 = -0.547960, b2 = 0.273980, a1 = 0.076693, a2 = 0.172613 — 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 | 0.078349 | -0.156698 | 0.078349 | 1.067264 | 0.380659 | 0.616976 | 78.1% |
| 44.1 kHz | 0.233925 | -0.467850 | 0.233925 | 0.246648 | 0.182348 | 0.427022 | 56.7% |
| 48 kHz | 0.273980 | -0.547960 | 0.273980 | 0.076693 | 0.172613 | 0.415467 | 52.1% |
| 96 kHz | 0.555241 | -1.110482 | 0.555241 | -0.901782 | 0.319181 | 0.564961 | 26.0% |
| 192 kHz | 0.748290 | -1.496580 | 0.748290 | -1.432186 | 0.560975 | 0.748983 | 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 | — | 0.4154670 | 0.4154670 | yes | reference |
| float32 | — | 0.4154670 | 0.4154670 | yes | 0.0000 dB |
| 32-bit fixed | Q0.31 | 0.4154670 | 0.4154670 | yes | 0.0000 dB |
| 24-bit fixed | Q0.23 | 0.4154669 | 0.4154669 | yes | 0.0001 dB |
| 16-bit fixed | Q0.15 | 0.4154605 | 0.4154605 | yes | 0.0003 dB |

## Questions this page answers

### What are the biquad coefficients for a 12.5 kHz high-pass filter at 48 kHz?

b0 = 0.273980, b1 = -0.547960, b2 = 0.273980, a1 = 0.076693, a2 = 0.172613, 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 12.5 kHz high-pass filter stable in 16-bit fixed point?

Yes. Rounding the five coefficients to a shared 0.15 scale leaves the largest pole at 0.4154605, against 0.4154670 exact, and the response drifts by at most 0.000 dB inside the band. 24-bit takes that to 0.0001 dB.

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

12500.0 Hz, which is 1.000× 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 the Q sweep above shows the point moving from 15711.7 Hz to 12921.4 Hz across the sweep while f0 never moves.

### How close to the unit circle are the poles of a 12.5 kHz high-pass filter?

0.4154670 at 48 kHz, as a conjugate pair at ±95.30°. Pole radius rises toward 1 as the corner frequency falls relative to the sample rate — the same filter at 192 kHz sits at 0.7489825 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 −3.01 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/highpass/12500-hz. Free to quote and cite with attribution and a link to the canonical page.