Butterworth

The Butterworth filter is a type of signal processing filter designed to have a maximally flat magnitude response in the passband, avoiding ripple. It was proposed by the British engineer Stephen Butterworth in the 1930s and is widely used in analog and digital filter design.

For an analog low-pass Butterworth filter of order n, the squared magnitude response is |H(jω)|^2 = 1 / [1 + (ω/ωc)^{2n}], where ωc is the cutoff frequency. The passband remains flat, the −3 dB point lies at ωc, and the roll-off steepness increases with the order n. The design achieves a monotonic, ripple-free response in the passband, but the phase response is nonlinear, which can affect transient behavior.

Butterworth filters are typically realized as cascaded second-order sections, enabling stable, practical implementations. They can be designed for low-pass, high-pass, band-pass, and band-stop configurations. In digital signal processing, analog prototypes are transformed via methods such as bilinear transform or matched-z to produce stable infinite impulse response (IIR) filters.

Due to their flat passband characteristics, Butterworth filters are commonly used in audio processing, communications, and instrumentation where a smooth transition to the stopband is desirable. They are often considered alongside other filter families when trade-offs between passband flatness, transition sharpness, and phase response are evaluated.

Beyond signal processing, Butterworth is also a surname and may refer to other topics named after the designer, including institutions, publications, and brands.