# 4 kHz high shelf

Lifts or drops everything above f0 by a fixed amount and leaves the bottom flat. Solved at 7 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/4000-hz
Page title: 4 kHz 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:** 4 kHz high-shelf filter — Q = 0.7071, gain +6 dB
- **Gain at f0 (4 kHz):** +3.00 dB — exact at every sample rate — the bilinear transform maps this value, not just the frequency
- **Pole radius at 48 kHz:** 0.6477253 — conjugate pair at ±26.62°, 0.352 from the circle
- **−3 dB point:** 3993.2 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 0.6477312 in Q2.13
- **Coefficients at 48 kHz:** b0 = 1.761096, b1 = -2.440864, b2 = 0.941186, a1 = -1.158131, a2 = 0.419548 — 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 4 kHz are absent because the filter does not exist there — equivalently, the f₀/Nyquist column never reaches 100%. That is why this table is 7 rows and a 20 Hz page is 8.

| Sample rate | b0 | b1 | b2 | a1 | a2 | Pole r | f0/Nyquist |
|---|---|---|---|---|---|---|---|
| 16 kHz | 1.412538 | -0.284719 | 0.252511 | 0.201566 | 0.178764 | 0.422805 | 50.0% |
| 22.1 kHz | 1.541779 | -1.065171 | 0.396617 | -0.316062 | 0.189288 | 0.435072 | 36.3% |
| 32 kHz | 1.661560 | -1.808464 | 0.646972 | -0.781659 | 0.281727 | 0.530779 | 25.0% |
| 44.1 kHz | 1.742695 | -2.322911 | 0.880675 | -1.089277 | 0.389736 | 0.624289 | 18.1% |
| 48 kHz | 1.761096 | -2.440864 | 0.941186 | -1.158131 | 0.419548 | 0.647725 | 16.7% |
| 96 kHz | 1.872195 | -3.163970 | 1.370386 | -1.566136 | 0.644746 | 0.802961 | 8.3% |
| 192 kHz | 1.932341 | -3.564119 | 1.653523 | -1.780867 | 0.802613 | 0.895887 | 4.2% |

## 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 | — | 0.6477253 | 0.6477253 | yes | reference |
| float32 | — | 0.6477253 | 0.6477253 | yes | 0.0000 dB |
| 32-bit fixed | Q2.29 | 0.6477253 | 0.6477253 | yes | 0.0000 dB |
| 24-bit fixed | Q2.21 | 0.6477253 | 0.6477253 | yes | 0.0000 dB |
| 16-bit fixed | Q2.13 | 0.6477312 | 0.6477312 | yes | 0.0041 dB |

## Questions this page answers

### What are the biquad coefficients for a 4 kHz high-shelf filter at 48 kHz?

b0 = 1.761096, b1 = -2.440864, b2 = 0.941186, a1 = -1.158131, a2 = 0.419548, 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 4 kHz 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 0.6477312, against 0.6477253 exact, and the response drifts by at most 0.004 dB inside the band. 24-bit takes that to 0.0000 dB.

### Where is the real −3 dB point of a 4 kHz high-shelf filter?

3993.2 Hz, which is 0.998× the 4 kHz 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 4 kHz high-shelf filter?

0.6477253 at 48 kHz, as a conjugate pair at ±26.62°. Pole radius rises toward 1 as the corner frequency falls relative to the sample rate — the same filter at 192 kHz sits at 0.8958865 and at 8 kHz it is above Nyquist. 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 4 kHz 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/4000-hz. Free to quote and cite with attribution and a link to the canonical page.