# Minimum Phase Systems

In control theory and signal processing, a linear, time-invariant system is said to be minimum-phase if the system and its inverse are causal and stable.

The most general causal LTI transfer function can be uniquely factored into a series of an all-pass and a minimum phase system. The system function is then the product of the two parts, and in the time domain the response of the system is the convolution of the two-part responses. The difference between a minimum phase and a general transfer function is that a minimum phase system has all of the poles and zeroes of its transfer function in the left half of the s-plane representation (in discrete time, respectively, inside the unit circle of the z-plane). Since inverting a system function leads to poles turning to zeroes and vice versa, and poles on the right side (s-plane imaginary line) or outside (z-plane unit circle) of the complex plane lead to unstable systems, only the class of minimum phase systems is closed under inversion. Intuitively, the minimum phase part of a general causal system implements its amplitude response with minimum group delay, while its all pass part corrects its phase response alone to correspond with the original system function.

from Minimum Phase - Wikipedia

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