Active Crossover — High-Pass Biquad (IIR)

Digital Butterworth HPF coefficients:

ω₀ = 2π × f_c / f_s
α = sin(ω₀) / (2 × Q)

b₀ = (1 + cos(ω₀)) / 2
b₁ = −(1 + cos(ω₀))
b₂ = (1 + cos(ω₀)) / 2
a₀ = 1 + α
a₁ = −2 × cos(ω₀)
a₂ = 1 − α

Normalize: divide b₀, b₁, b₂, a₁, a₂ by a₀

Worked example: 80 Hz HPF, Q = 0.707 (Butterworth), 48 kHz sample rate:

ω₀ = 2π × 80 / 48000 = 0.01047 rad
α = sin(0.01047) / (2 × 0.707) = 0.01047 / 1.414 = 0.00740

b₀ = (1 + cos(0.01047)) / 2 = (1 + 0.99995) / 2 = 0.99997
b₁ = −(1 + 0.99995) = −1.99995
b₂ = 0.99997
a₀ = 1 + 0.00740 = 1.00740
a₁ = −2 × 0.99995 = −1.99990
a₂ = 1 − 0.00740 = 0.99260

After normalization (÷ by 1.00740):
b₀ = 0.9927    b₁ = −1.9854    b₂ = 0.9927
a₁ = −1.9852   a₂ = 0.9854

These coefficients are entered directly into DSP programming interfaces.