Minimum phase

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.

For example, a discrete-time system with rational transfer function ${\displaystyle H(z)}$ can only satisfy causality and stability requirements if all of its poles are inside the unit circle. However, we are free to choose whether the zeros of the system are inside or outside the unit circle. A system with rational transfer function is minimum-phase if all its zeros are also inside the unit circle. Insight is given below as to why this system is called minimum-phase.

A system ${\displaystyle \mathbb {H} }$ is invertible if we can uniquely determine its input from its output. I.e., we can find a system ${\displaystyle \mathbb {H} _{inv}}$ such that if we apply ${\displaystyle \mathbb {H} }$ followed by ${\displaystyle \mathbb {H} _{inv}}$, we obtain the identity system ${\displaystyle \mathbb {I} }$. (See Inverse matrix for a finite-dimensional analog). I.e.,

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