# 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.

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## Using Antenna Toolbox with Phased Array Systems

Intermediate
Example
Theory

When you create antenna arrays such as a uniform linear array (ULA), you can use antennas that are built into Phased Array System Toolbox™. Alternatively, you can use Antenna Toolbox™...

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## Control systems with non-minimum phase dynamics

8 min
Beginner
Video
Theory

This video describes control systems that have non-minimum phase dynamics, characterized by a zero of the input--output transfer function in the right-half-plane. Physically, these systems...

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## Control Systems in Practice, Part 6: What Are Non-Minimum Phase Systems?

14 min
Beginner
Video
Theory

We like to categorize transfer functions into groups and label them because it helps us understand how a particular system will behave simply by knowing the group that it’s part of. We gain...

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