# 200 Hz high-pass

Rejects everything below the corner at 12 dB/octave and passes what is above it. 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/highpass/200-hz
Page title: 200 Hz High-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:** 200 Hz high-pass filter — Q = 0.7071
- **Gain at f0 (200 Hz):** −3.01 dB — exact at every sample rate — the bilinear transform maps this value, not just the frequency
- **Pole radius at 48 kHz:** 0.9816583 — conjugate pair at ±1.06°, 0.0183 from the circle
- **−3 dB point:** 200.0 Hz — 1.000× f0 at Q = 0.7071
- **Peak of the magnitude response:** 0.00 dB at 24.00 kHz — evaluated on the unit circle, not sketched
- **Group delay at f0:** 1.126 ms
- **16-bit fixed point:** holds — largest pole 0.9816430 in Q1.14
- **Coefficients at 48 kHz:** b0 = 0.981658, b1 = -1.963317, b2 = 0.981658, a1 = -1.962980, a2 = 0.963653 — 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 | 0.894859 | -1.789717 | 0.894859 | -1.778632 | 0.800803 | 0.894876 | 5.0% |
| 16 kHz | 0.945977 | -1.891954 | 0.945977 | -1.889033 | 0.894874 | 0.945978 | 2.5% |
| 22.1 kHz | 0.960503 | -1.921005 | 0.960503 | -1.919445 | 0.922566 | 0.960503 | 1.8% |
| 32 kHz | 0.972614 | -1.945228 | 0.972614 | -1.944478 | 0.945978 | 0.972614 | 1.3% |
| 44.1 kHz | 0.980052 | -1.960105 | 0.980052 | -1.959707 | 0.960503 | 0.980053 | 0.9% |
| 48 kHz | 0.981658 | -1.963317 | 0.981658 | -1.962980 | 0.963653 | 0.981658 | 0.8% |
| 96 kHz | 0.990787 | -1.981573 | 0.990787 | -1.981489 | 0.981658 | 0.990787 | 0.4% |
| 192 kHz | 0.995383 | -1.990765 | 0.995383 | -1.990744 | 0.990787 | 0.995383 | 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 | — | 0.9816583 | 0.9816583 | yes | reference |
| float32 | — | 0.9816583 | 0.9816583 | yes | 0.0005 dB |
| 32-bit fixed | Q1.30 | 0.9816583 | 0.9816583 | yes | 0.0011 dB |
| 24-bit fixed | Q1.22 | 0.9816583 | 0.9816583 | yes | 0.0021 dB |
| 16-bit fixed | Q1.14 | 0.9816430 | 0.9816430 | yes | 19.7173 dB |

## Questions this page answers

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

b0 = 0.981658, b1 = -1.963317, b2 = 0.981658, a1 = -1.962980, a2 = 0.963653, 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 200 Hz high-pass filter stable in 16-bit fixed point?

Yes. Rounding the five coefficients to a shared 1.14 scale leaves the largest pole at 0.9816430, against 0.9816583 exact, and the response drifts by at most 19.717 dB inside the band. 24-bit takes that to 0.0021 dB.

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

200.0 Hz, which is 1.000× 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 the Q sweep above shows the point moving from 310.7 Hz to 211.4 Hz across the sweep while f0 never moves.

### How close to the unit circle are the poles of a 200 Hz high-pass filter?

0.9816583 at 48 kHz, as a conjugate pair at ±1.06°. Pole radius rises toward 1 as the corner frequency falls relative to the sample rate — the same filter at 192 kHz sits at 0.9953827 and at 8 kHz at 0.8948758. 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 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.

---

Source: MakerPortal — https://makerportal.ai/lab/biquad/highpass/200-hz. Free to quote and cite with attribution and a link to the canonical page.