# 630 Hz low shelf

Lifts or drops everything below f0 by a fixed amount and leaves the top flat. 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/lowshelf/630-hz
Page title: 630 Hz Low shelf 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:** 630 Hz low-shelf filter — Q = 0.7071, gain +6 dB
- **Gain at f0 (630 Hz):** +3.00 dB — exact at every sample rate — the bilinear transform maps this value, not just the frequency
- **Pole radius at 48 kHz:** 0.9521124 — conjugate pair at ±2.81°, 0.0479 from the circle
- **−3 dB point:** 631.1 Hz — 1.002× f0 at Q = 0.7071
- **Peak of the magnitude response:** +6.00 dB at 10 Hz — evaluated on the unit circle, not sketched
- **Group delay at f0:** 0.000 ms
- **16-bit fixed point:** holds — largest pole 0.9520999 in Q1.14
- **Coefficients at 48 kHz:** b0 = 1.020424, b1 = -1.899645, b2 = 0.888377, a1 = -1.901929, a2 = 0.906518 — 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 630 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 | 1.125535 | -1.352286 | 0.494600 | -1.419089 | 0.553332 | 0.743863 | 15.8% |
| 16 kHz | 1.062065 | -1.688101 | 0.701407 | -1.706899 | 0.744674 | 0.862945 | 7.9% |
| 22.1 kHz | 1.044830 | -1.776658 | 0.772956 | -1.786916 | 0.807528 | 0.898626 | 5.7% |
| 32 kHz | 1.030753 | -1.847970 | 0.837349 | -1.852991 | 0.863081 | 0.929022 | 3.9% |
| 44.1 kHz | 1.022246 | -1.890572 | 0.879129 | -1.893267 | 0.898681 | 0.947988 | 2.9% |
| 48 kHz | 1.020424 | -1.899645 | 0.888377 | -1.901929 | 0.906518 | 0.952112 | 2.6% |
| 96 kHz | 1.010169 | -1.950359 | 0.942534 | -1.950943 | 0.952118 | 0.975765 | 1.3% |
| 192 kHz | 1.005073 | -1.975321 | 0.970841 | -1.975469 | 0.975766 | 0.987809 | 0.7% |

## 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 | — | 0.9521124 | 0.9521124 | yes | reference |
| float32 | — | 0.9521123 | 0.9521123 | yes | 0.0001 dB |
| 32-bit fixed | Q1.30 | 0.9521124 | 0.9521124 | yes | 0.0000 dB |
| 24-bit fixed | Q1.22 | 0.9521124 | 0.9521124 | yes | 0.0002 dB |
| 16-bit fixed | Q1.14 | 0.9520999 | 0.9520999 | yes | 0.0206 dB |

## Questions this page answers

### What are the biquad coefficients for a 630 Hz low-shelf filter at 48 kHz?

b0 = 1.020424, b1 = -1.899645, b2 = 0.888377, a1 = -1.901929, a2 = 0.906518, with a0 normalised to 1 — the RBJ Audio EQ Cookbook form at Q = 0.7071 and +6 dB of gain. 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 630 Hz low-shelf filter stable in 16-bit fixed point?

Yes. Rounding the five coefficients to a shared 1.14 scale leaves the largest pole at 0.9520999, against 0.9521124 exact, and the response drifts by at most 0.021 dB inside the band. 24-bit takes that to 0.0002 dB.

### Where is the real −3 dB point of a 630 Hz low-shelf filter?

631.1 Hz, which is 1.002× the 630 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 a shelf does not read Q at all — its slope is fixed at S = 1, matching Web Audio's BiquadFilterNode, so the gain sweep above is the family that moves this filter rather than a Q sweep.

### How close to the unit circle are the poles of a 630 Hz low-shelf filter?

0.9521124 at 48 kHz, as a conjugate pair at ±2.81°. Pole radius rises toward 1 as the corner frequency falls relative to the sample rate — the same filter at 192 kHz sits at 0.9878088 and at 8 kHz at 0.7438627. 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 +3.00 dB at 630 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/lowshelf/630-hz. Free to quote and cite with attribution and a link to the canonical page.