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E.g., a discrete-time system with rational transfer function 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 is minimum-phase if all its zeros are inside the unit circle. Insight is given below as to why this system is called minimum-phase.
But, first, we define exactly what we mean by inverse system.
A system is invertible if we can uniquely determine its output from its input. I.e., we can find a system such that if we apply followed by , we obtain the identity system . (See Inverse matrix for a finite-dimensional analog). I.e.,
Suppose that is input to system and gives output .
Applying the inverse system to gives the following.
So we see that the inverse system allows us to determine uniquely the input from the output .
Suppose that the system is a discrete-time, linear, time-invariant (LTI) system described by the impulse response . Additionally, has impulse response . The cascade of two LTI systems is a convolution. In this case, the above relation is the following:
Minimum phase system
When we impose the constraints of causality and stability, the inverse system is unique; and the system and its inverse are called minimum-phase. The causality and stability constraints in the discrete-time case are the following:
See the article on stability for the analogous conditions for the continuous-time case.
Discrete-time frequency analysis
Performing frequency analysis for the discrete-time case will provide some insight. The time-domain equation is the following.
Applying the Z-transform gives the following relation in the z-domain.
From this relation, we realize that
For simplicity, we consider only the case of a rational transfer function H (z). Causality and stability imply that all poles of H (z) must be strictly inside the unit circle in the complex plane (See stability). Suppose
So, causality and stability for Hinv(z) imply that its poles -- the roots of A (z) -- must be inside the unit circle. These two constraints imply that both the zeros and the poles of a minimum phase system must be strictly inside the unit circle.
Continuous-time frequency analysis
Analysis for the continuous-time case proceeds in a similar manner except that we use the Laplace transform for frequency analysis. The time-domain equation is the following.
From this relation, we realize that
Again, for simplicity, we consider only the case of a rational transfer function H(s). Causality and stability imply that all poles of H (s) must be strictly inside the left-half s-plane (See stability). Suppose
So, causality and stability for Hinv(s) imply that its poles -- the roots of A (s) -- must be strictly inside the left-half s-plane. These two constraints imply that both the zeros and the poles of a minimum phase system must be strictly inside the left-half s-plane.
Minimum phase in the time domain
For all causal and stable systems that have the same magnitude response, the minimum phase system has its energy concentrated near the beginning of the impulse response. I.e., it minimizes the following function which we can think of as the delay of energy in the impulse response.
Minimum phase as minimum group delay
For all causal and stable systems that have the same magnitude response, the minimum phase system has the minimum group delay. So, the proper term should be a minimum group delay system --- it's just that minimum phase has been assigned in the literature so the name stuck. The following proof illustrates this idea of minimum group delay.
φa(ω) contributes the following to the group delay.
The denominator and θa are invariant to reflecting the zero a outside of the unit circle, i.e., replacing a with (a - 1) * . However, by reflecting a outside of the unit circle, we increase the magnitude of in the numerator. Thus, having a inside the unit circle minimizes the group delay contributed by the factor 1 - az - 1. We can extend this result to the general case of more than one zero since the phase of the multiplicative factors of the form 1 - aiz - 1 is additive. I.e., for a transfer function with N zeros,
A maximum phase system is the opposite of a minimum phase system in several respects.
- It has all of its zeros outside the unit circle. It follows from the proof above that a system which has all its zeros outside the unit circle has maximal group delay; thus, the name maximum phase follows.
- If a maximum phase system is anticausal and stable (all its poles and zeros are outside the unit circle), then the inverse system is also anticausal and stable.
- A maximum phase system has the property that its impulse response has the maximum delay of energy.
A mixed phase or nonminimum-phase system has some of its zeros inside the unit circle and has others outside the unit circle. Thus, its group delay is neither minimum or maximum but somewhere between the group delay of the minimum and maximum phase equivalent system.
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