# 500 Hz band-pass

Passes a band centred on f0 at unity gain and rejects everything either side. 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/bandpass/500-hz
Page title: 500 Hz 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:** 500 Hz band-pass filter — Q = 1.0000
- **Gain at f0 (500 Hz):** 0.00 dB — exact at every sample rate — the bilinear transform maps this value, not just the frequency
- **Pole radius at 48 kHz:** 0.9678161 — conjugate pair at ±3.25°, 0.0322 from the circle
- **−3 dB point:** 808.6 Hz — 1.617× f0 at Q = 1.0000
- **Peak of the magnitude response:** 0.00 dB at 498 Hz — evaluated on the unit circle, not sketched
- **Group delay at f0:** 0.637 ms
- **16-bit fixed point:** holds — largest pole 0.9678045 in Q1.14
- **Coefficients at 48 kHz:** b0 = 0.031666, b1 = 0.000000, b2 = -0.031666, a1 = -1.932521, a2 = 0.936668 — 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 500 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 | 0.160610 | 0.000000 | -0.160610 | -1.550990 | 0.678779 | 0.823881 | 12.5% |
| 16 kHz | 0.088876 | 0.000000 | -0.088876 | -1.787234 | 0.822248 | 0.906779 | 6.3% |
| 22.1 kHz | 0.066291 | 0.000000 | -0.066291 | -1.848497 | 0.867419 | 0.931353 | 4.5% |
| 32 kHz | 0.046719 | 0.000000 | -0.046719 | -1.897381 | 0.906562 | 0.952136 | 3.1% |
| 44.1 kHz | 0.034366 | 0.000000 | -0.034366 | -1.926370 | 0.931268 | 0.965022 | 2.3% |
| 48 kHz | 0.031666 | 0.000000 | -0.031666 | -1.932521 | 0.936668 | 0.967816 | 2.1% |
| 96 kHz | 0.016096 | 0.000000 | -0.016096 | -1.966754 | 0.967808 | 0.983772 | 1.0% |
| 192 kHz | 0.008114 | 0.000000 | -0.008114 | -1.983505 | 0.983771 | 0.991852 | 0.5% |

## 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 | — | 0.9678161 | 0.9678161 | yes | reference |
| float32 | — | 0.9678161 | 0.9678161 | yes | 0.0000 dB |
| 32-bit fixed | Q1.30 | 0.9678161 | 0.9678161 | yes | 0.0000 dB |
| 24-bit fixed | Q1.22 | 0.9678161 | 0.9678161 | yes | 0.0000 dB |
| 16-bit fixed | Q1.14 | 0.9678045 | 0.9678045 | yes | 0.0053 dB |

## Questions this page answers

### What are the biquad coefficients for a 500 Hz band-pass filter at 48 kHz?

b0 = 0.031666, b1 = 0.000000, b2 = -0.031666, a1 = -1.932521, a2 = 0.936668, 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 500 Hz band-pass filter stable in 16-bit fixed point?

Yes. Rounding the five coefficients to a shared 1.14 scale leaves the largest pole at 0.9678045, against 0.9678161 exact, and the response drifts by at most 0.005 dB inside the band. 24-bit takes that to 0.0000 dB.

### Where is the real −3 dB point of a 500 Hz band-pass filter?

808.6 Hz, which is 1.617× the 500 Hz 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 1205.0 Hz to 525.8 Hz across the sweep while f0 never moves.

### How close to the unit circle are the poles of a 500 Hz band-pass filter?

0.9678161 at 48 kHz, as a conjugate pair at ±3.25°. Pole radius rises toward 1 as the corner frequency falls relative to the sample rate — the same filter at 192 kHz sits at 0.9918523 and at 8 kHz at 0.8238807. 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 500 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/bandpass/500-hz. Free to quote and cite with attribution and a link to the canonical page.