# 250 Hz high-pass

Rejects everything below the corner at 12 dB/octave and passes what is above it. 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/highpass/250-hz
Page title: 250 Hz 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:** 250 Hz high-pass filter — Q = 0.7071
- **Gain at f0 (250 Hz):** −3.01 dB — exact at every sample rate — the bilinear transform maps this value, not just the frequency
- **Pole radius at 48 kHz:** 0.9771257 — conjugate pair at ±1.33°, 0.0229 from the circle
- **−3 dB point:** 250.0 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:** 0.900 ms
- **16-bit fixed point:** holds — largest pole 0.9771248 in Q1.14
- **Coefficients at 48 kHz:** b0 = 0.977126, b1 = -1.954251, b2 = 0.977126, a1 = -1.953728, a2 = 0.954775 — 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 250 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 | 0.870331 | -1.740662 | 0.870331 | -1.723776 | 0.757547 | 0.870372 | 6.3% |
| 16 kHz | 0.932932 | -1.865864 | 0.932932 | -1.861361 | 0.870367 | 0.932935 | 3.1% |
| 22.1 kHz | 0.950874 | -1.901748 | 0.950874 | -1.899333 | 0.904163 | 0.950875 | 2.3% |
| 32 kHz | 0.965885 | -1.931771 | 0.965885 | -1.930606 | 0.932935 | 0.965885 | 1.6% |
| 44.1 kHz | 0.975128 | -1.950256 | 0.975128 | -1.949637 | 0.950875 | 0.975128 | 1.1% |
| 48 kHz | 0.977126 | -1.954251 | 0.977126 | -1.953728 | 0.954775 | 0.977126 | 1.0% |
| 96 kHz | 0.988497 | -1.976993 | 0.988497 | -1.976861 | 0.977126 | 0.988497 | 0.5% |
| 192 kHz | 0.994232 | -1.988463 | 0.994232 | -1.988430 | 0.988497 | 0.994232 | 0.3% |

## 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 | — | 0.9771257 | 0.9771257 | yes | reference |
| float32 | — | 0.9771257 | 0.9771257 | yes | 0.0001 dB |
| 32-bit fixed | Q1.30 | 0.9771257 | 0.9771257 | yes | 0.0007 dB |
| 24-bit fixed | Q1.22 | 0.9771257 | 0.9771257 | yes | 0.0004 dB |
| 16-bit fixed | Q1.14 | 0.9771248 | 0.9771248 | yes | 0.0742 dB |

## Questions this page answers

### What are the biquad coefficients for a 250 Hz high-pass filter at 48 kHz?

b0 = 0.977126, b1 = -1.954251, b2 = 0.977126, a1 = -1.953728, a2 = 0.954775, 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 250 Hz high-pass filter stable in 16-bit fixed point?

Yes. Rounding the five coefficients to a shared 1.14 scale leaves the largest pole at 0.9771248, against 0.9771257 exact, and the response drifts by at most 0.074 dB inside the band. 24-bit takes that to 0.0004 dB.

### Where is the real −3 dB point of a 250 Hz high-pass filter?

250.0 Hz, which is 1.000× the 250 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 the Q sweep above shows the point moving from 388.4 Hz to 264.3 Hz across the sweep while f0 never moves.

### How close to the unit circle are the poles of a 250 Hz high-pass filter?

0.9771257 at 48 kHz, as a conjugate pair at ±1.33°. Pole radius rises toward 1 as the corner frequency falls relative to the sample rate — the same filter at 192 kHz sits at 0.9942317 and at 8 kHz at 0.8703717. 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 250 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.

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Source: MakerPortal — https://makerportal.ai/lab/biquad/highpass/250-hz. Free to quote and cite with attribution and a link to the canonical page.