Editing Minimum phase (section)

=== Maximum phase ===
{{unreferenced section|date=September 2014}}
A ''maximum-phase'' system is the opposite of a minimum phase system. A causal and stable LTI system is a ''maximum-phase'' system if its inverse is causal and unstable.{{dubious|date=September 2014}} That is,
* The zeros of the discrete-time system are outside the [[unit circle]].
* The zeros of the continuous-time system are in the right-hand side of the [[complex plane]].
Such a system is called a ''maximum-phase system'' because it has the maximum [[group delay]] of the set of systems that have the same magnitude response. In this set of equal-magnitude-response systems, the maximum phase system will have maximum energy delay.

For example, the two continuous-time LTI systems described by the transfer functions
<math display="block">\frac{s + 10}{s + 5} \qquad \text{and} \qquad \frac{s - 10}{s + 5}</math>

have equivalent magnitude responses; however, the second system has a much larger contribution to the phase shift. Hence, in this set, the second system is the maximum-phase system and the first system is the minimum-phase system. These systems are also famously known as nonminimum-phase systems that raise many stability concerns in control. One recent solution to these systems is moving the RHP zeros to the LHP using the PFCD method.<ref>{{Cite book|title=Analytical Statistical Study of Linear Parallel Feedforward Compensators for Nonminimum Phase Systems|last=Noury|first=K. |date=2019|doi = 10.1115/DSCC2019-9126 |chapter = Analytical Statistical Study of Linear Parallel Feedforward Compensators for Nonminimum-Phase Systems|isbn = 978-0-7918-5914-8|s2cid=214446227 }}</ref>