# 1.25 kHz low-pass

Passes everything below the corner and rolls off above it at 12 dB/octave. 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/lowpass/1250-hz
Page title: 1.25 kHz Low-pass 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:** 1.25 kHz low-pass filter — Q = 0.7071
- **Gain at f0 (1.25 kHz):** −3.01 dB — exact at every sample rate — the bilinear transform maps this value, not just the frequency
- **Pole radius at 48 kHz:** 0.8907442 — conjugate pair at ±6.66°, 0.109 from the circle
- **−3 dB point:** 1250.0 Hz — 1.000× f0 at Q = 0.7071
- **Peak of the magnitude response:** 0.00 dB at 10 Hz — evaluated on the unit circle, not sketched
- **Group delay at f0:** 0.181 ms
- **16-bit fixed point:** holds — largest pole 0.8907278 in Q1.14
- **Coefficients at 48 kHz:** b0 = 0.005989, b1 = 0.011977, b2 = 0.005989, a1 = -1.769471, a2 = 0.793425 — 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 1.25 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 | 0.139939 | 0.279879 | 0.139939 | -0.699738 | 0.259495 | 0.509407 | 31.3% |
| 16 kHz | 0.044280 | 0.088559 | 0.044280 | -1.322887 | 0.500006 | 0.707111 | 15.6% |
| 22.1 kHz | 0.025176 | 0.050352 | 0.025176 | -1.503695 | 0.604400 | 0.777432 | 11.3% |
| 32 kHz | 0.012787 | 0.025575 | 0.012787 | -1.655608 | 0.706757 | 0.840688 | 7.8% |
| 44.1 kHz | 0.007028 | 0.014056 | 0.007028 | -1.749243 | 0.777355 | 0.881678 | 5.7% |
| 48 kHz | 0.005989 | 0.011977 | 0.005989 | -1.769471 | 0.793425 | 0.890744 | 5.2% |
| 96 kHz | 0.001581 | 0.003162 | 0.001581 | -1.884418 | 0.890742 | 0.943792 | 2.6% |
| 192 kHz | 0.000407 | 0.000813 | 0.000407 | -1.942165 | 0.943791 | 0.971489 | 1.3% |

## 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 | — | 0.8907442 | 0.8907442 | yes | reference |
| float32 | — | 0.8907442 | 0.8907442 | yes | 0.0000 dB |
| 32-bit fixed | Q1.30 | 0.8907442 | 0.8907442 | yes | 0.0000 dB |
| 24-bit fixed | Q1.22 | 0.8907443 | 0.8907443 | yes | 0.0001 dB |
| 16-bit fixed | Q1.14 | 0.8907278 | 0.8907278 | yes | 0.0133 dB |

## Questions this page answers

### What are the biquad coefficients for a 1.25 kHz low-pass filter at 48 kHz?

b0 = 0.005989, b1 = 0.011977, b2 = 0.005989, a1 = -1.769471, a2 = 0.793425, with a0 normalised to 1 — the RBJ Audio EQ Cookbook form at Q = 0.7071. 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 1.25 kHz low-pass filter stable in 16-bit fixed point?

Yes. Rounding the five coefficients to a shared 1.14 scale leaves the largest pole at 0.8907278, against 0.8907442 exact, and the response drifts by at most 0.013 dB inside the band. 24-bit takes that to 0.0001 dB.

### Where is the real −3 dB point of a 1.25 kHz low-pass filter?

1250.0 Hz, which is 1.000× the 1.25 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 the Q sweep above shows the point moving from 805.6 Hz to 1308.5 Hz across the sweep while f0 never moves.

### How close to the unit circle are the poles of a 1.25 kHz low-pass filter?

0.8907442 at 48 kHz, as a conjugate pair at ±6.66°. Pole radius rises toward 1 as the corner frequency falls relative to the sample rate — the same filter at 192 kHz sits at 0.9714893 and at 8 kHz at 0.5094067. 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 −3.01 dB at 1.25 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/lowpass/1250-hz. Free to quote and cite with attribution and a link to the canonical page.