# 40 Hz peaking eq

Boosts or cuts a band around f0 and leaves both ends of the spectrum at unity. 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/peaking/40-hz
Page title: 40 Hz Peaking EQ 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:** 40 Hz peaking EQ filter — Q = 1.0000, gain +6 dB
- **Gain at f0 (40 Hz):** +6.00 dB — exact at every sample rate — the bilinear transform maps this value, not just the frequency
- **Pole radius at 48 kHz:** 0.9981483 — conjugate pair at ±0.28°, 0.00185 from the circle
- **−3 dB point:** 24.7 Hz — 0.617× f0 at Q = 1.0000
- **Peak of the magnitude response:** +5.99 dB at 41 Hz — evaluated on the unit circle, not sketched
- **Group delay at f0:** 5.603 ms
- **16-bit fixed point:** diverges — largest pole 1.0000000 in Q1.14
- **Coefficients at 48 kHz:** b0 = 1.001841, b1 = -1.996273, b2 = 0.994459, a1 = -1.996273, a2 = 0.996300 — 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 40 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 | 1.010944 | -1.977031 | 0.967063 | -1.977031 | 0.978007 | 0.988943 | 1.0% |
| 16 kHz | 1.005503 | -1.988696 | 0.983439 | -1.988696 | 0.988942 | 0.994455 | 0.5% |
| 22.1 kHz | 1.003999 | -1.991834 | 0.987964 | -1.991834 | 0.991963 | 0.995974 | 0.4% |
| 32 kHz | 1.002759 | -1.994394 | 0.991696 | -1.994394 | 0.994455 | 0.997224 | 0.3% |
| 44.1 kHz | 1.002004 | -1.995941 | 0.993970 | -1.995941 | 0.995974 | 0.997985 | 0.2% |
| 48 kHz | 1.001841 | -1.996273 | 0.994459 | -1.996273 | 0.996300 | 0.998148 | 0.2% |
| 96 kHz | 1.000921 | -1.998141 | 0.997227 | -1.998141 | 0.998148 | 0.999074 | 0.1% |
| 192 kHz | 1.000461 | -1.999072 | 0.998613 | -1.999072 | 0.999074 | 0.999537 | 0.0% |

## 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 | — | 0.9981483 | 0.9981483 | yes | reference |
| float32 | — | 0.9981483 | 0.9981483 | yes | 0.0198 dB |
| 32-bit fixed | Q1.30 | 0.9981483 | 0.9981483 | yes | 0.0001 dB |
| 24-bit fixed | Q1.22 | 0.9981483 | 0.9981483 | yes | 0.0769 dB |
| 16-bit fixed | Q1.14 | 1.0000000 (real) | 0.9981367 | no | n/a — diverges |

## Questions this page answers

### What are the biquad coefficients for a 40 Hz peaking EQ filter at 48 kHz?

b0 = 1.001841, b1 = -1.996273, b2 = 0.994459, a1 = -1.996273, a2 = 0.996300, with a0 normalised to 1 — the RBJ Audio EQ Cookbook form at Q = 1.0000 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 40 Hz peaking EQ filter stable in 16-bit fixed point?

No. Rounding the five coefficients to a shared 1.14 scale pushes the poles onto the real axis and the larger one out to 1.0000000 — at or outside the unit circle, which is a filter that diverges rather than one that is merely inaccurate. Note that √|a2| still reads 0.9981367 here, comfortably inside the circle: the shortcut is the geometric mean of the two real poles and it does not see this. Use 24-bit (largest pole 0.9981483) or float32.

### Where is the real −3 dB point of a 40 Hz peaking EQ filter?

24.7 Hz, which is 0.617× the 40 Hz corner. f0 and the −3 dB point are the same frequency only at Q = 1/√2; this page is designed at Q = 1.0000, and the Q sweep above shows the point moving from 96.8 Hz to 42.4 Hz across the sweep while f0 never moves.

### How close to the unit circle are the poles of a 40 Hz peaking EQ filter?

0.9981483 at 48 kHz, as a conjugate pair at ±0.28°. Pole radius rises toward 1 as the corner frequency falls relative to the sample rate — the same filter at 192 kHz sits at 0.9995368 and at 8 kHz at 0.9889426. 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 +6.00 dB at 40 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/peaking/40-hz. Free to quote and cite with attribution and a link to the canonical page.