Biquad Filter Cheat Sheet

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.