biquad

A biquad is a second-order digital filter section used as a building block for higher-order filters. It implements a second-order transfer function and is commonly cascaded with other biquads to realize more complex filtering tasks in audio, communications, and control applications.

In the time domain, a biquad is typically expressed as a difference equation that relates the current output to current and past inputs and past outputs: y[n] = b0 x[n] + b1 x[n-1] + b2 x[n-2] - a1 y[n-1] - a2 y[n-2]. The corresponding transfer function in the z-domain is H(z) = (b0 + b1 z^-1 + b2 z^-2) / (1 + a1 z^-1 + a2 z^-2). Note that some conventions present the denominator with minus signs: H(z) = (b0 + b1 z^-1 + b2 z^-2) / (1 - a1 z^-1 - a2 z^-2). The poles of the transfer function, determined by the denominator, govern stability and frequency response.

Coefficient sets (b0, b1, b2, a1, a2) are derived from target specifications such as cutoff frequency, quality factor Q, and gain for shelving or peaking filters. Common design approaches include bilinear transform from analog prototypes or dedicated biquad coefficient formulas used in audio DSP. Biquads are capable of implementing standard second-order filters—low-pass, high-pass, band-pass, notch, and all-pass—as well as shelving and peaking equalizers. They are widely used as the canonical building blocks in audio processing and in digital filter designs requiring controllable resonance and precise frequency shaping.

When used in series, multiple biquad sections form higher-order filters while maintaining numerical stability and manageable coefficient ranges. Implementations may employ direct form I, direct form II, or transposed forms to balance computational efficiency and numerical robustness.