 Minimum phase

In control theory and signal processing, a linear, timeinvariant system is said to be minimumphase if the system and its inverse are causal and stable.^{[1]}^{[2]}
For example, a discretetime 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 minimumphase if all its zeros are also inside the unit circle. Insight is given below as to why this system is called minimumphase.
Contents
Inverse system
A system is invertible if we can uniquely determine its input from its output. I.e., we can find a system such that if we apply followed by , we obtain the identity system . (See Inverse matrix for a finitedimensional 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 .
Discretetime example
Suppose that the system is a discretetime, linear, timeinvariant (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:
where δ(n) is the Kronecker delta or the identity system in the discretetime case. Note that this inverse system is not unique.
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 minimumphase. The causality and stability constraints in the discretetime case are the following (for timeinvariant systems where h is the system's impulse response):
Causality
and
Stability
and
See the article on stability for the analogous conditions for the continuoustime case.
Frequency analysis
Discretetime frequency analysis
Performing frequency analysis for the discretetime case will provide some insight. The timedomain equation is the following.
Applying the Ztransform gives the following relation in the zdomain.
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 (See stability). Suppose
where A (z) and D (z) are polynomial in z. Causality and stability imply that the poles – the roots of D (z) – must be strictly inside the unit circle. We also know that
So, causality and stability for H_{inv}(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.
Continuoustime frequency analysis
Analysis for the continuoustime case proceeds in a similar manner except that we use the Laplace transform for frequency analysis. The timedomain equation is the following.
where δ(t) is the Dirac delta function. The Dirac delta function is the identity operator in the continuoustime case because of the sifting property with any signal x (t).
Applying the Laplace transform gives the following relation in the splane.
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 lefthalf splane (See stability). Suppose
where A (s) and D (s) are polynomial in s. Causality and stability imply that the poles – the roots of D (s) – must be inside the lefthalf splane. We also know that
So, causality and stability for H_{inv}(s) imply that its poles – the roots of A (s) – must be strictly inside the lefthalf splane. These two constraints imply that both the zeros and the poles of a minimum phase system must be strictly inside the lefthalf splane.
Relationship of magnitude response to phase response
A minimumphase system, whether discretetime or continuoustime, has an additional useful property that the natural logarithm of the magnitude of the frequency response (the "gain" measured in nepers which is proportional to dB) is related to the phase angle of the frequency response (measured in radians) by the Hilbert transform. That is, in the continuoustime case, let
be the complex frequency response of system H(s). Then, only for a minimumphase system, the phase response of H(s) is related to the gain by
and, inversely,
 .
Stated more compactly, let
where α(ω) and ϕ(ω) are real functions of a real variable. Then
and
 .
The Hilbert transform operator is defined to be
 .
An equivalent corresponding relationship is also true for discretetime minimumphase systems.
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 start 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. The following proof illustrates this idea of minimum group delay.
Suppose we consider one zero a of the transfer function H(z). Let's place this zero a inside the unit circle () and see how the group delay is affected.
Since the zero a contributes the factor 1 − az ^{− 1} to the transfer function, the phase contributed by this term is the following.
ϕ_{a}(ω) contributes the following to the group delay.
The denominator and θ_{a} 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 − a_{i}z ^{− 1} is additive. I.e., for a transfer function with N zeros,
So, a minimum phase system with all zeros inside the unit circle minimizes the group delay since the group delay of each individual zero is minimized.
Nonminimum phase
Systems that are causal and stable whose inverses are causal and unstable are known as nonminimumphase systems. A given nonminimum phase system will have a greater phase contribution than the minimumphase system with the equivalent magnitude response.
Maximum phase
A maximumphase system is the opposite of a minimum phase system. A causal and stable LTI system is a maximumphase system if its inverse is causal and unstable. That is,
 The zeros of the discretetime system are outside the unit circle.
 The zeros of the continuoustime system are in the righthand side of the complex plane.
Such a system is called a maximumphase system because it has the maximum group delay of the set of systems that have the same magnitude response. In this set of equalmagnituderesponse systems, the maximum phase system will have maximum energy delay.
For example, the two continuoustime LTI systems described by the transfer functions
have equivalent magnitude responses; however, the first system has a much larger contribution to the phase shift. Hence, in this set, the second system is the minimumphase system and the first system is the maximumphase system.
Mixed phase
A mixedphase 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.
For example, the continuoustime LTI system described by transfer function
is stable and causal; however, it has zeros on both the left and righthand sides of the complex plane. Hence, it is a mixedphase system.
Linear phase
A linearphase system has constant group delay. Nontrivial linear phase or nearly linear phase systems are also mixed phase.
See also
 Allpass filter – A special nonminimumphase case.
 Kramers–Kronig_relation – Minimum phase system in physics
References
 ^ Hassibi, Babak; Kailath, Thomas; Sayed, Ali H. (2000). Linear estimation. Englewood Cliffs, N.J: Prentice Hall. pp. 193. ISBN 0130224642.
 ^ J. O. Smith III, Introduction to Digital Filters with Audio Applications (September 2007 Edition).
Further reading
 Dimitris G. Manolakis, Vinay K. Ingle, Stephen M. Kogon : Statistical and Adaptive Signal Processing, pp. 5456, McGrawHill, ISBN 0070400512
 Boaz Porat : A Course in Digital Signal Processing, pp. 261263, John Wiley and Sons, ISBN 0471149616
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