# 200 Hz 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/200-hz
Page title: 200 Hz 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:** 200 Hz low-shelf filter — Q = 0.7071, gain +6 dB
- **Gain at f0 (200 Hz):** +3.00 dB — exact at every sample rate — the bilinear transform maps this value, not just the frequency
- **Pole radius at 48 kHz:** 0.9845445 — conjugate pair at ±0.89°, 0.0155 from the circle
- **−3 dB point:** 200.4 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 0.9845300 in Q1.14
- **Coefficients at 48 kHz:** b0 = 1.006446, b1 = -1.968612, b2 = 0.963120, a1 = -1.968850, a2 = 0.969328 — 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 200 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 | 1.039159 | -1.805503 | 0.798216 | -1.813452 | 0.829426 | 0.910728 | 5.0% |
| 16 kHz | 1.019444 | -1.904518 | 0.893398 | -1.906594 | 0.910766 | 0.954341 | 2.5% |
| 22.1 kHz | 1.014078 | -1.931099 | 0.921458 | -1.932206 | 0.934430 | 0.966659 | 1.8% |
| 32 kHz | 1.009683 | -1.952748 | 0.945194 | -1.953279 | 0.954346 | 0.976906 | 1.3% |
| 44.1 kHz | 1.007017 | -1.965814 | 0.959925 | -1.966096 | 0.966661 | 0.983189 | 0.9% |
| 48 kHz | 1.006446 | -1.968612 | 0.963120 | -1.968850 | 0.969328 | 0.984545 | 0.8% |
| 96 kHz | 1.003218 | -1.984364 | 0.981387 | -1.984424 | 0.984545 | 0.992242 | 0.4% |
| 192 kHz | 1.001608 | -1.992197 | 0.990650 | -1.992212 | 0.992242 | 0.996114 | 0.2% |

## 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 | — | 0.9845445 | 0.9845445 | yes | reference |
| float32 | — | 0.9845445 | 0.9845445 | yes | 0.0005 dB |
| 32-bit fixed | Q1.30 | 0.9845445 | 0.9845445 | yes | 0.0000 dB |
| 24-bit fixed | Q1.22 | 0.9845445 | 0.9845445 | yes | 0.0011 dB |
| 16-bit fixed | Q1.14 | 0.9845300 | 0.9845300 | yes | 1.1759 dB |

## Questions this page answers

### What are the biquad coefficients for a 200 Hz low-shelf filter at 48 kHz?

b0 = 1.006446, b1 = -1.968612, b2 = 0.963120, a1 = -1.968850, a2 = 0.969328, 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 200 Hz 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 0.9845300, against 0.9845445 exact, and the response drifts by at most 1.176 dB inside the band. 24-bit takes that to 0.0011 dB.

### Where is the real −3 dB point of a 200 Hz low-shelf filter?

200.4 Hz, which is 1.002× the 200 Hz 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 200 Hz low-shelf filter?

0.9845445 at 48 kHz, as a conjugate pair at ±0.89°. Pole radius rises toward 1 as the corner frequency falls relative to the sample rate — the same filter at 192 kHz sits at 0.9961136 and at 8 kHz at 0.9107281. 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 200 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/lowshelf/200-hz. Free to quote and cite with attribution and a link to the canonical page.