MinimumPhase

Minimum-phase is a term used in signal processing to describe a class of linear time-invariant systems or transfer functions that are stable, causal, and possess a favorable phase characteristic for a given magnitude response. In discrete-time systems, a system is minimum-phase if all its zeros lie inside the unit circle. In continuous-time systems, the zeros (and poles for stability) lie in the left half of the complex plane. An equivalent condition is that the inverse system is also stable and causal. For real-valued systems, zeros occur in complex-conjugate pairs.

A central property of minimum-phase systems is their phase behavior. For a fixed magnitude response, a minimum-phase system has the smallest possible phase, meaning its impulse response energy is as concentrated as possible near the start of the response. This relationship between magnitude and phase is formalized by the Hilbert transform. Because of this, the inverse of a minimum-phase system is stable and realizable, which is advantageous for deconvolution and real-time filtering.

Any stable, causal system can be factored into a minimum-phase component and an all-pass component that carries the remaining phase. If a system has zeros outside the unit circle (or in the right-half plane for continuous time), reflecting those zeros inside (and adjusting gain accordingly) yields an equivalent minimum-phase system with the same magnitude response.

Applications of minimum-phase design include audio processing, speech processing, system identification, and seismology, where predictable, causal, and invertible filters are desirable.