Maximum 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.

In contradistinction, a linear, non-minimum phase transfer function can be modeled as minimum phase transfer function in series with an all-pass-filter, the characteristic issue of that series combination will be zeroes in the right-half-plane. A consequence of zeroes in the right-half-plane, is that the inverted function is not stable. The all pass filter (can also be transport delay) inserts 'excess phase', that is why the resulting function would be non-minimum phase.

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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