# 31.5 Hz all-pass

Leaves every magnitude alone and shifts phase, passing through −180° at f0. 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/allpass/31-5-hz
Page title: 31.5 Hz All-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:** 31.5 Hz all-pass filter — Q = 1.0000
- **Gain at f0 (31.5 Hz):** 0.00 dB — exact at every sample rate — the bilinear transform maps this value, not just the frequency
- **Pole radius at 48 kHz:** 0.9979405 — conjugate pair at ±0.20°, 0.00206 from the circle
- **−3 dB point:** none — the magnitude never falls 3 dB below its own peak inside the band
- **Peak of the magnitude response:** 0.00 dB at 18 Hz — evaluated on the unit circle, not sketched
- **Group delay at f0:** 20.206 ms
- **16-bit fixed point:** holds — largest pole 0.9979532 in Q1.14
- **Coefficients at 48 kHz:** b0 = 0.995885, b1 = -1.995868, b2 = 1.000000, a1 = -1.995868, a2 = 0.995885 — 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 31.5 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 | 0.975565 | -1.974960 | 1.000000 | -1.974960 | 0.975565 | 0.987707 | 0.8% |
| 16 kHz | 0.987706 | -1.987554 | 1.000000 | -1.987554 | 0.987706 | 0.993834 | 0.4% |
| 22.1 kHz | 0.991064 | -1.990984 | 1.000000 | -1.990984 | 0.991064 | 0.995522 | 0.3% |
| 32 kHz | 0.993834 | -1.993796 | 1.000000 | -1.993796 | 0.993834 | 0.996912 | 0.2% |
| 44.1 kHz | 0.995522 | -1.995502 | 1.000000 | -1.995502 | 0.995522 | 0.997759 | 0.1% |
| 48 kHz | 0.995885 | -1.995868 | 1.000000 | -1.995868 | 0.995885 | 0.997940 | 0.1% |
| 96 kHz | 0.997940 | -1.997936 | 1.000000 | -1.997936 | 0.997940 | 0.998970 | 0.1% |
| 192 kHz | 0.998970 | -1.998969 | 1.000000 | -1.998969 | 0.998970 | 0.999485 | 0.0% |

## 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 | — | 0.9979405 | 0.9979405 | yes | reference |
| float32 | — | 0.9979404 | 0.9979404 | yes | 0.0000 dB |
| 32-bit fixed | Q1.30 | 0.9979405 | 0.9979405 | yes | 0.0000 dB |
| 24-bit fixed | Q1.22 | 0.9979404 | 0.9979404 | yes | 0.0000 dB |
| 16-bit fixed | Q1.14 | 0.9979532 | 0.9979532 | yes | 0.0000 dB |

## Questions this page answers

### What are the biquad coefficients for a 31.5 Hz all-pass filter at 48 kHz?

b0 = 0.995885, b1 = -1.995868, b2 = 1.000000, a1 = -1.995868, a2 = 0.995885, 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 31.5 Hz all-pass filter stable in 16-bit fixed point?

Yes. Rounding the five coefficients to a shared 1.14 scale leaves the largest pole at 0.9979532, against 0.9979405 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 31.5 Hz all-pass filter?

This type has no −3 dB edge to find: its magnitude response is unity at every frequency, and what f0 marks is the point where the phase passes through −180°. At f0 the response measures 0.00 dB.

### How close to the unit circle are the poles of a 31.5 Hz all-pass filter?

0.9979405 at 48 kHz, as a conjugate pair at ±0.20°. Pole radius rises toward 1 as the corner frequency falls relative to the sample rate — the same filter at 192 kHz sits at 0.9994847 and at 8 kHz at 0.9877068. 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 31.5 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/allpass/31-5-hz. Free to quote and cite with attribution and a link to the canonical page.