MinimumPhaseSystem

A minimum phase system is a concept in control theory and signal processing. It describes a causal and stable linear time-invariant (LTI) system where all its finite-duration impulse responses have their energy concentrated at the beginning of the response. More formally, a minimum phase system is characterized by its transfer function having all its zeros and poles in the open left-half of the complex plane. This means that for a rational transfer function, the real parts of all the roots of the numerator (zeros) and the denominator (poles) are negative.

The property of having zeros only in the open left-half plane is crucial. If a system has zeros in the right-half plane, it is called a non-minimum phase system. Systems with zeros on the imaginary axis are also considered non-minimum phase. The location of the poles, however, does not determine whether a system is minimum phase; it only dictates stability. A stable system requires all poles to be in the open left-half plane.

Minimum phase systems have several desirable properties. For instance, their inverse systems are also minimum phase and stable. This makes them easier to control and equalize. In practical applications, such as audio processing or communication systems, systems are often designed to be minimum phase to ensure predictable behavior and to avoid potential issues associated with non-minimum phase characteristics like phase distortion. Determining if a system is minimum phase involves examining the roots of its characteristic equation.