Form
| Direct-form transfer | H(z) = (b₀ + b₁ z⁻¹ + b₂ z⁻²) / (1 + a₁ z⁻¹ + a₂ z⁻²) |
|---|---|
| Difference equation | y[n] = b₀x[n] + b₁x[n−1] + b₂x[n−2] − a₁y[n−1] − a₂y[n−2] |
| ω₀ | = 2π · f₀ / fs |
| α | = sin(ω₀) / (2 · Q) |
Low-pass (RBJ biquad)
| b₀ | = (1 − cos ω₀) / 2 |
|---|---|
| b₁ | = 1 − cos ω₀ |
| b₂ | = (1 − cos ω₀) / 2 |
| a₀ | = 1 + α |
| a₁ | = −2 · cos ω₀ |
| a₂ | = 1 − α |
High-pass
| b₀ | = (1 + cos ω₀) / 2 |
|---|---|
| b₁ | = −(1 + cos ω₀) |
| b₂ | = (1 + cos ω₀) / 2 |
| a₀/a₁/a₂ | Same as LP |
Peaking EQ (gain A)
| A | = 10^(dB_gain / 40) |
|---|---|
| b₀ | = 1 + α · A |
| b₁ | = −2 · cos ω₀ |
| b₂ | = 1 − α · A |
| a₀ | = 1 + α / A |
| a₁ | = −2 · cos ω₀ |
| a₂ | = 1 − α / A |
Normalization
- Divide all coefficients by a₀ so the leading denominator coefficient is 1.
- Q typically 0.707 for maximally flat (Butterworth Q).
- Higher Q → sharper peak; lower Q → gentler.
Notes
- Reference: Robert Bristow-Johnson's "Cookbook formulae for audio EQ biquad filter coefficients".
- Cascade biquads for higher-order filters (two biquads = 4th order).
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