# 800 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/800-hz
Page title: 800 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:** 800 Hz low-shelf filter — Q = 0.7071, gain +6 dB
- **Gain at f0 (800 Hz):** +3.00 dB — exact at every sample rate — the bilinear transform maps this value, not just the frequency
- **Pole radius at 48 kHz:** 0.9395820 — conjugate pair at ±3.58°, 0.0604 from the circle
- **−3 dB point:** 801.4 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 0.9395810 in Q1.14
- **Coefficients at 48 kHz:** b0 = 1.025990, b1 = -1.871870, b2 = 0.860461, a1 = -1.875507, a2 = 0.882814 — 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 800 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 | 1.160044 | -1.164492 | 0.412629 | -1.266295 | 0.470870 | 0.686200 | 20.0% |
| 16 kHz | 1.079109 | -1.599316 | 0.637712 | -1.628622 | 0.687514 | 0.829165 | 10.0% |
| 22.1 kHz | 1.057118 | -1.713650 | 0.721172 | -1.729772 | 0.762168 | 0.873023 | 7.3% |
| 32 kHz | 1.039159 | -1.805503 | 0.798216 | -1.813452 | 0.829426 | 0.910728 | 5.0% |
| 44.1 kHz | 1.028312 | -1.860232 | 0.849105 | -1.864518 | 0.873131 | 0.934415 | 3.6% |
| 48 kHz | 1.025990 | -1.871870 | 0.860461 | -1.875507 | 0.882814 | 0.939582 | 3.3% |
| 96 kHz | 1.012928 | -1.936775 | 0.927602 | -1.937711 | 0.939594 | 0.969326 | 1.7% |
| 192 kHz | 1.006446 | -1.968612 | 0.963120 | -1.968850 | 0.969328 | 0.984545 | 0.8% |

## 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 | — | 0.9395820 | 0.9395820 | yes | reference |
| float32 | — | 0.9395820 | 0.9395820 | yes | 0.0000 dB |
| 32-bit fixed | Q1.30 | 0.9395820 | 0.9395820 | yes | 0.0000 dB |
| 24-bit fixed | Q1.22 | 0.9395820 | 0.9395820 | yes | 0.0001 dB |
| 16-bit fixed | Q1.14 | 0.9395810 | 0.9395810 | yes | 0.0157 dB |

## Questions this page answers

### What are the biquad coefficients for a 800 Hz low-shelf filter at 48 kHz?

b0 = 1.025990, b1 = -1.871870, b2 = 0.860461, a1 = -1.875507, a2 = 0.882814, 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 800 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 0.9395810, against 0.9395820 exact, and the response drifts by at most 0.016 dB inside the band. 24-bit takes that to 0.0001 dB.

### Where is the real −3 dB point of a 800 Hz low-shelf filter?

801.4 Hz, which is 1.002× the 800 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 800 Hz low-shelf filter?

0.9395820 at 48 kHz, as a conjugate pair at ±3.58°. Pole radius rises toward 1 as the corner frequency falls relative to the sample rate — the same filter at 192 kHz sits at 0.9845445 and at 8 kHz at 0.6861999. 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 800 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/800-hz. Free to quote and cite with attribution and a link to the canonical page.