# 2.5 kHz 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/2500-hz
Page title: 2.5 kHz 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:** 2.5 kHz peaking EQ filter — Q = 1.0000, gain +6 dB
- **Gain at f0 (2.5 kHz):** +6.00 dB — exact at every sample rate — the bilinear transform maps this value, not just the frequency
- **Pole radius at 48 kHz:** 0.8920120 — conjugate pair at ±17.61°, 0.108 from the circle
- **−3 dB point:** 1549.8 Hz — 0.620× f0 at Q = 1.0000
- **Peak of the magnitude response:** +5.99 dB at 2.54 kHz — evaluated on the unit circle, not sketched
- **Group delay at f0:** 0.091 ms
- **16-bit fixed point:** holds — largest pole 0.8920288 in Q1.14
- **Coefficients at 48 kHz:** b0 = 1.101673, b1 = -1.700389, b2 = 0.694012, a1 = -1.700389, a2 = 0.795685 — 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 2.5 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 | 1.245269 | 0.576752 | 0.261858 | 0.576752 | 0.507127 | 0.712128 | 62.5% |
| 16 kHz | 1.226315 | -0.858476 | 0.318901 | -0.858476 | 0.545216 | 0.738387 | 31.3% |
| 22.1 kHz | 1.187006 | -1.229215 | 0.437202 | -1.229215 | 0.624208 | 0.790068 | 22.7% |
| 32 kHz | 1.142323 | -1.511612 | 0.571676 | -1.511612 | 0.713999 | 0.844985 | 15.6% |
| 44.1 kHz | 1.109350 | -1.668516 | 0.670908 | -1.668516 | 0.780259 | 0.883322 | 11.3% |
| 48 kHz | 1.101673 | -1.700389 | 0.694012 | -1.700389 | 0.795685 | 0.892012 | 10.4% |
| 96 kHz | 1.054259 | -1.865709 | 0.836707 | -1.865709 | 0.890966 | 0.943910 | 5.2% |
| 192 kHz | 1.027981 | -1.937271 | 0.915792 | -1.937271 | 0.943772 | 0.971479 | 2.6% |

## 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 | — | 0.8920120 | 0.8920120 | yes | reference |
| float32 | — | 0.8920120 | 0.8920120 | yes | 0.0000 dB |
| 32-bit fixed | Q1.30 | 0.8920120 | 0.8920120 | yes | 0.0000 dB |
| 24-bit fixed | Q1.22 | 0.8920119 | 0.8920119 | yes | 0.0000 dB |
| 16-bit fixed | Q1.14 | 0.8920288 | 0.8920288 | yes | 0.0021 dB |

## Questions this page answers

### What are the biquad coefficients for a 2.5 kHz peaking EQ filter at 48 kHz?

b0 = 1.101673, b1 = -1.700389, b2 = 0.694012, a1 = -1.700389, a2 = 0.795685, 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 2.5 kHz peaking EQ filter stable in 16-bit fixed point?

Yes. Rounding the five coefficients to a shared 1.14 scale leaves the largest pole at 0.8920288, against 0.8920120 exact, and the response drifts by at most 0.002 dB inside the band. 24-bit takes that to 0.0000 dB.

### Where is the real −3 dB point of a 2.5 kHz peaking EQ filter?

1549.8 Hz, which is 0.620× the 2.5 kHz 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 5810.3 Hz to 2652.3 Hz across the sweep while f0 never moves.

### How close to the unit circle are the poles of a 2.5 kHz peaking EQ filter?

0.8920120 at 48 kHz, as a conjugate pair at ±17.61°. Pole radius rises toward 1 as the corner frequency falls relative to the sample rate — the same filter at 192 kHz sits at 0.9714795 and at 8 kHz at 0.7121284. 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 2.5 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/peaking/2500-hz. Free to quote and cite with attribution and a link to the canonical page.