minimalphase

Minimum-phase is a term used in signal processing to describe systems or sequences whose phase response is as small as possible for a given magnitude response. In discrete-time, a causal, stable LTI system is minimum-phase if all zeros of its transfer function H(z) lie inside the unit circle (|z| < 1). Equivalently, the inverse system 1/H(z) is also causal and stable. In continuous-time, the zeros of the transfer function must lie in the left half-plane (Re(s) < 0), with the same requirement that the inverse be stable and causal.

A key consequence is that, among all systems with the same magnitude response, a minimum-phase system has the smallest possible phase lag and the smallest group delay. If only the magnitude response is known, the minimum-phase phase can be reconstructed from the magnitude via the Hilbert transform of the log-magnitude, a process central to spectral factorization and cepstral methods. The impulse response of a minimum-phase system tends to be front-loaded, with energy concentrated toward the beginning.

Non-minimum-phase systems have zeros outside the unit circle (discrete-time) or in the right-half plane (continuous-time), which leads to larger phase for a given magnitude and often to an unstable or noncausal inverse. In practice, the concept is important for deconvolution, equalization, and filter design, where a stable inverse is desirable and the phase behavior for a given magnitude response must be controlled.