# 80 Hz high shelf

Lifts or drops everything above f0 by a fixed amount and leaves the bottom 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/highshelf/80-hz
Page title: 80 Hz High 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:** 80 Hz high-shelf filter — Q = 0.7071, gain +6 dB
- **Gain at f0 (80 Hz):** +3.00 dB — exact at every sample rate — the bilinear transform maps this value, not just the frequency
- **Pole radius at 48 kHz:** 0.9912380 — conjugate pair at ±0.50°, 0.00876 from the circle
- **−3 dB point:** 79.9 Hz — 0.998× f0 at Q = 0.7071
- **Peak of the magnitude response:** +6.00 dB at 24.00 kHz — evaluated on the unit circle, not sketched
- **Group delay at f0:** -0.000 ms
- **16-bit fixed point:** holds — largest pole 0.9912335 in Q2.13
- **Coefficients at 48 kHz:** b0 = 1.990141, b1 = -3.955483, b2 = 1.965496, a1 = -1.982399, a2 = 0.982553 — 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 80 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 | 1.964748 | -3.782654 | 1.823199 | -1.894498 | 0.899791 | 0.948573 | 2.0% |
| 16 kHz | 1.979939 | -3.885870 | 1.907289 | -1.947210 | 0.948568 | 0.973944 | 1.0% |
| 22.1 kHz | 1.984131 | -3.914444 | 1.931033 | -1.961690 | 0.962410 | 0.981025 | 0.7% |
| 32 kHz | 1.987585 | -3.938021 | 1.950780 | -1.973600 | 0.973944 | 0.986886 | 0.5% |
| 44.1 kHz | 1.989688 | -3.952392 | 1.962885 | -1.980843 | 0.981025 | 0.990467 | 0.4% |
| 48 kHz | 1.990141 | -3.955483 | 1.965496 | -1.982399 | 0.982553 | 0.991238 | 0.3% |
| 96 kHz | 1.992700 | -3.972984 | 1.980323 | -1.991199 | 0.991238 | 0.995609 | 0.2% |
| 192 kHz | 1.993981 | -3.981750 | 1.987779 | -1.995600 | 0.995609 | 0.997802 | 0.1% |

## 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 | — | 0.9912380 | 0.9912380 | yes | reference |
| float32 | — | 0.9912380 | 0.9912380 | yes | 0.0033 dB |
| 32-bit fixed | Q2.29 | 0.9912380 | 0.9912380 | yes | 0.0000 dB |
| 24-bit fixed | Q2.21 | 0.9912381 | 0.9912381 | yes | 0.0001 dB |
| 16-bit fixed | Q2.13 | 0.9912335 | 0.9912335 | yes | 0.7316 dB |

## Questions this page answers

### What are the biquad coefficients for a 80 Hz high-shelf filter at 48 kHz?

b0 = 1.990141, b1 = -3.955483, b2 = 1.965496, a1 = -1.982399, a2 = 0.982553, 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 80 Hz high-shelf filter stable in 16-bit fixed point?

Yes. Rounding the five coefficients to a shared 2.13 scale leaves the largest pole at 0.9912335, against 0.9912380 exact, and the response drifts by at most 0.732 dB inside the band. 24-bit takes that to 0.0001 dB.

### Where is the real −3 dB point of a 80 Hz high-shelf filter?

79.9 Hz, which is 0.998× the 80 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 80 Hz high-shelf filter?

0.9912380 at 48 kHz, as a conjugate pair at ±0.50°. Pole radius rises toward 1 as the corner frequency falls relative to the sample rate — the same filter at 192 kHz sits at 0.9978023 and at 8 kHz at 0.9485730. 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 80 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/highshelf/80-hz. Free to quote and cite with attribution and a link to the canonical page.