# 3.15 kHz 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/3150-hz
Page title: 3.15 kHz 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:** 3.15 kHz low-shelf filter — Q = 0.7071, gain +6 dB
- **Gain at f0 (3.15 kHz):** +3.00 dB — exact at every sample rate — the bilinear transform maps this value, not just the frequency
- **Pole radius at 48 kHz:** 0.7817335 — conjugate pair at ±14.40°, 0.218 from the circle
- **−3 dB point:** 3155.5 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 0.7817186 in Q1.14
- **Coefficients at 48 kHz:** b0 = 1.104296, b1 = -1.466169, b2 = 0.554971, a1 = -1.514329, a2 = 0.611107 — 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 3.15 kHz 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 | 1.705030 | 2.082957 | 0.765572 | 0.947271 | 0.334915 | 0.578718 | 78.8% |
| 16 kHz | 1.320211 | -0.258128 | 0.235445 | -0.581740 | 0.232044 | 0.481710 | 39.4% |
| 22.1 kHz | 1.230140 | -0.774127 | 0.298270 | -0.962250 | 0.340287 | 0.583341 | 28.6% |
| 32 kHz | 1.157502 | -1.178435 | 0.418037 | -1.277475 | 0.476499 | 0.690289 | 19.7% |
| 44.1 kHz | 1.113680 | -1.416026 | 0.527299 | -1.472131 | 0.584874 | 0.764771 | 14.3% |
| 48 kHz | 1.104296 | -1.466169 | 0.554971 | -1.514329 | 0.611107 | 0.781734 | 13.1% |
| 96 kHz | 1.051582 | -1.742118 | 0.744015 | -1.755456 | 0.782259 | 0.884454 | 6.6% |
| 192 kHz | 1.025580 | -1.873921 | 0.862483 | -1.877448 | 0.884536 | 0.940498 | 3.3% |

## 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 | — | 0.7817335 | 0.7817335 | yes | reference |
| float32 | — | 0.7817335 | 0.7817335 | yes | 0.0000 dB |
| 32-bit fixed | Q1.30 | 0.7817335 | 0.7817335 | yes | 0.0000 dB |
| 24-bit fixed | Q1.22 | 0.7817336 | 0.7817336 | yes | 0.0000 dB |
| 16-bit fixed | Q1.14 | 0.7817186 | 0.7817186 | yes | 0.0041 dB |

## Questions this page answers

### What are the biquad coefficients for a 3.15 kHz low-shelf filter at 48 kHz?

b0 = 1.104296, b1 = -1.466169, b2 = 0.554971, a1 = -1.514329, a2 = 0.611107, 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 3.15 kHz 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 0.7817186, against 0.7817335 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 3.15 kHz low-shelf filter?

3155.5 Hz, which is 1.002× the 3.15 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 3.15 kHz low-shelf filter?

0.7817335 at 48 kHz, as a conjugate pair at ±14.40°. Pole radius rises toward 1 as the corner frequency falls relative to the sample rate — the same filter at 192 kHz sits at 0.9404978 and at 8 kHz at 0.5787184. 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 3.15 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/lowshelf/3150-hz. Free to quote and cite with attribution and a link to the canonical page.