# 10 kHz band-pass

Passes a band centred on f0 at unity gain and rejects everything either side. Solved at 6 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/bandpass/10000-hz
Page title: 10 kHz Band-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:** 10 kHz band-pass filter — Q = 1.0000
- **Gain at f0 (10 kHz):** 0.00 dB — exact at every sample rate — the bilinear transform maps this value, not just the frequency
- **Pole radius at 48 kHz:** 0.5904671 — conjugate pair at ±72.81°, 0.410 from the circle
- **−3 dB point:** 13640.5 Hz — 1.364× f0 at Q = 1.0000
- **Peak of the magnitude response:** 0.00 dB at 9.96 kHz — evaluated on the unit circle, not sketched
- **Group delay at f0:** 0.043 ms
- **16-bit fixed point:** holds — largest pole 0.5904772 in Q0.15
- **Coefficients at 48 kHz:** b0 = 0.325674, b1 = 0.000000, b2 = -0.325674, a1 = -0.349057, a2 = 0.348651 — 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 10 kHz are absent because the filter does not exist there — equivalently, the f₀/Nyquist column never reaches 100%. That is why this table is 6 rows and a 20 Hz page is 8.

| Sample rate | b0 | b1 | b2 | a1 | a2 | Pole r | f0/Nyquist |
|---|---|---|---|---|---|---|---|
| 22.1 kHz | 0.125851 | 0.000000 | -0.125851 | 1.674254 | 0.748297 | 0.865042 | 90.7% |
| 32 kHz | 0.315977 | 0.000000 | -0.315977 | 0.523528 | 0.368045 | 0.606667 | 62.5% |
| 44.1 kHz | 0.330959 | 0.000000 | -0.330959 | -0.194717 | 0.338081 | 0.581447 | 45.4% |
| 48 kHz | 0.325674 | 0.000000 | -0.325674 | -0.349057 | 0.348651 | 0.590467 | 41.7% |
| 96 kHz | 0.233353 | 0.000000 | -0.233353 | -1.216444 | 0.533295 | 0.730270 | 20.8% |
| 192 kHz | 0.138466 | 0.000000 | -0.138466 | -1.631626 | 0.723069 | 0.850335 | 10.4% |

## 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 | — | 0.5904671 | 0.5904671 | yes | reference |
| float32 | — | 0.5904671 | 0.5904671 | yes | 0.0000 dB |
| 32-bit fixed | Q0.31 | 0.5904671 | 0.5904671 | yes | 0.0000 dB |
| 24-bit fixed | Q0.23 | 0.5904671 | 0.5904671 | yes | 0.0000 dB |
| 16-bit fixed | Q0.15 | 0.5904772 | 0.5904772 | yes | 0.0004 dB |

## Questions this page answers

### What are the biquad coefficients for a 10 kHz band-pass filter at 48 kHz?

b0 = 0.325674, b1 = 0.000000, b2 = -0.325674, a1 = -0.349057, a2 = 0.348651, with a0 normalised to 1 — the RBJ Audio EQ Cookbook form at Q = 1.0000. 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 10 kHz band-pass filter stable in 16-bit fixed point?

Yes. Rounding the five coefficients to a shared 0.15 scale leaves the largest pole at 0.5904772, against 0.5904671 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 10 kHz band-pass filter?

13640.5 Hz, which is 1.364× the 10 kHz corner. f0 and the −3 dB point are the same frequency only at Q = 1/√2; this page is designed at Q = 1.0000, and the Q sweep above shows the point moving from 16437.2 Hz to 10375.0 Hz across the sweep while f0 never moves.

### How close to the unit circle are the poles of a 10 kHz band-pass filter?

0.5904671 at 48 kHz, as a conjugate pair at ±72.81°. Pole radius rises toward 1 as the corner frequency falls relative to the sample rate — the same filter at 192 kHz sits at 0.8503345 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 0.00 dB at 10 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/bandpass/10000-hz. Free to quote and cite with attribution and a link to the canonical page.