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.^[1][2]

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 H(z)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.

Inverse system[edit]

A system Hdisplaystyle mathbb H is invertible if we can uniquely determine its input from its output. I.e., we can find a system Hinvdisplaystyle mathbb H _inv such that if we apply Hdisplaystyle mathbb H followed by Hinvdisplaystyle mathbb H _inv, we obtain the identity system Idisplaystyle mathbb I . (See Inverse matrix for a finite-dimensional analog). I.e.,

HinvH=Idisplaystyle mathbb H _inv,mathbb H =mathbb I

Suppose that x~displaystyle tilde x is input to system Hdisplaystyle mathbb H and gives output y~displaystyle tilde y.

Hx~=y~displaystyle mathbb H ,tilde x=tilde y

Applying the inverse system Hinvdisplaystyle mathbb H _inv to y~displaystyle tilde y gives the following.

Hinvy~=HinvHx~=Ix~=x~displaystyle mathbb H _inv,tilde y=mathbb H _inv,mathbb H ,tilde x=mathbb I ,tilde x=tilde x

So we see that the inverse system Hinvdisplaystyle mathbb H _inv allows us to determine uniquely the input x~displaystyle tilde x from the output y~displaystyle tilde y.

Discrete-time example[edit]

Suppose that the system Hdisplaystyle mathbb H is a discrete-time, linear, time-invariant (LTI) system described by the impulse response h(n)displaystyle h(n) for n in Z. Additionally, suppose Hinvdisplaystyle mathbb H _inv has impulse response hinv(n)displaystyle h_inv(n). The cascade of two LTI systems is a convolution. In this case, the above relation is the following:

(h∗hinv)(n)=∑k=−∞∞h(k)hinv(n−k)=δ(n)displaystyle (h*h_inv)(n)=sum _k=-infty ^infty h(k),h_inv(n-k)=delta (n)

where δ(n)displaystyle delta (n) is the Kronecker delta or the identity system in the discrete-time case. Note that this inverse system Hinvdisplaystyle mathbb H _inv need not be unique.

Minimum phase system[edit]

When we impose the constraints of causality and stability, the inverse system is unique; and the system Hdisplaystyle mathbb H and its inverse Hinvdisplaystyle mathbb H _inv are called minimum-phase. The causality and stability constraints in the discrete-time case are the following (for time-invariant systems where h is the system's impulse response):

Causality[edit]

h(n)=0∀n<0displaystyle h(n)=0,,forall ,n<0

and

hinv(n)=0∀n<0displaystyle h_inv(n)=0,,forall ,n<0

Stability[edit]

∑n=−∞∞|h(n)|=‖h‖1<∞_1<infty

and

∑n=−∞∞|hinv(n)|=‖hinv‖1<∞_1<infty

See the article on stability for the analogous conditions for the continuous-time case.

Frequency analysis[edit]

Discrete-time frequency analysis[edit]

Performing frequency analysis for the discrete-time case will provide some insight. The time-domain equation is the following.

(h∗hinv)(n)=δ(n)displaystyle (h*h_inv)(n)=,!delta (n)

Applying the Z-transform gives the following relation in the z-domain.

H(z)Hinv(z)=1displaystyle H(z),H_inv(z)=1

From this relation, we realize that

Hinv(z)=1H(z)displaystyle H_inv(z)=frac 1H(z)

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

H(z)=A(z)D(z)displaystyle H(z)=frac A(z)D(z)

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

Hinv(z)=D(z)A(z)displaystyle H_inv(z)=frac D(z)A(z)

So, causality and stability for Hinv(z)displaystyle 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.

Continuous-time frequency analysis[edit]

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.

(h∗hinv)(t)=δ(t)displaystyle (h*h_inv)(t)=,!delta (t)

where δ(t)displaystyle delta (t) is the Dirac delta function. The Dirac delta function is the identity operator in the continuous-time case because of the sifting property with any signal x (t).

δ(t)∗x(t)=∫−∞∞δ(t−τ)x(τ)dτ=x(t)displaystyle delta (t)*x(t)=int _-infty ^infty delta (t-tau )x(tau )dtau =x(t)

Applying the Laplace transform gives the following relation in the s-plane.

H(s)Hinv(s)=1displaystyle H(s),H_inv(s)=1

From this relation, we realize that

Hinv(s)=1H(s)displaystyle H_inv(s)=frac 1H(s)

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

H(s)=A(s)D(s)displaystyle H(s)=frac A(s)D(s)

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 left-half s-plane. We also know that

Hinv(s)=D(s)A(s)displaystyle H_inv(s)=frac D(s)A(s)

So, causality and stability for Hinv(s)displaystyle H_inv(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.

Relationship of magnitude response to phase response[edit]

A minimum-phase system, whether discrete-time or continuous-time, 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 continuous-time case, let

H(jω) =def H(s)|s=jω displaystyle H(jomega ) stackrel mathrm def = H(s)_s=jomega

be the complex frequency response of system H(s). Then, only for a minimum-phase system, the phase response of H(s) is related to the gain by

arg⁡[H(jω)]=−Hlog⁡( right)rbrace

and, inversely,

log⁡(|H(jω)|)=log⁡(|H(j∞)|)+Harg⁡[H(jω)] right)=log left(.

Stated more compactly, let

H(jω)=|H(jω)|ejarg⁡[H(jω)] =def eα(ω)ejϕ(ω)=eα(ω)+jϕ(ω) H(jomega )

where α(ω)displaystyle alpha (omega ) and ϕ(ω)displaystyle phi (omega ) are real functions of a real variable. Then

ϕ(ω)=−Hα(ω) displaystyle phi (omega )=-mathcal Hlbrace alpha (omega )rbrace

and

α(ω)=α(∞)+Hϕ(ω) displaystyle alpha (omega )=alpha (infty )+mathcal Hlbrace phi (omega )rbrace .

The Hilbert transform operator is defined to be

Hx(t) =def x^(t)=1π∫−∞∞x(τ)t−τdτ displaystyle mathcal Hlbrace x(t)rbrace stackrel mathrm def = widehat x(t)=frac 1pi int _-infty ^infty frac x(tau )t-tau ,dtau .

An equivalent corresponding relationship is also true for discrete-time minimum-phase systems.

Minimum phase in the time domain[edit]

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.

∑n=m∞|h(n)|2∀m∈Z+^2,,,,,,,forall ,min mathbb Z ^+

Minimum phase as minimum group delay[edit]

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 adisplaystyle a of the transfer function H(z)displaystyle H(z). Let's place this zero adisplaystyle a inside the unit circle (|a|<1aright) and see how the group delay is affected.

a=|a|eiθa where θa=Arg(a)e^itheta _a,mbox where ,theta _a=mboxArg(a)

Since the zero adisplaystyle a contributes the factor 1−az−1displaystyle 1-az^-1 to the transfer function, the phase contributed by this term is the following.

ϕa(ω)=Arg(1−ae−iω)displaystyle phi _aleft(omega right)=mboxArgleft(1-ae^-iomega right)

=Arg(1−|a|eiθae−iω)aright

=Arg(1−|a|e−i(ω−θa))aright

=Arg(1−+isin(ω−θa))displaystyle =mboxArgleft(leftaright+ileftarightright)

=Arg(−1−cos⁡(ω−θa)+isin⁡(ω−θa))displaystyle =mboxArgleft(left^-1-cos(omega -theta _a)right+ileftsin(omega -theta _a)rightright)

ϕa(ω)displaystyle phi _a(omega ) contributes the following to the group delay.

−dϕa(ω)dω=sin2⁡(ω−θa)+cos2⁡(ω−θa)−|a|−1cos⁡(ω−θa)sin2⁡(ω−θa)+cos2⁡(ω−θa)+|a|−2−2|a|−1cos⁡(ω−θa)displaystyle -frac dphi _a(omega )domega =frac sin ^2(omega -theta _a)+cos ^2(omega -theta _a)-leftsin ^2(omega -theta _a)+cos ^2(omega -theta _a)+left

−dϕa(ω)dω=|a|−cos⁡(ω−θa)|a|+|a|−1−2cos⁡(ω−θa)displaystyle -frac dphi _a(omega )domega =frac left+left

The denominator and θadisplaystyle theta _a are invariant to reflecting the zero adisplaystyle a outside of the unit circle, i.e., replacing adisplaystyle a with (a−1)∗displaystyle (a^-1)^*. However, by reflecting adisplaystyle a outside of the unit circle, we increase the magnitude of |a|aright in the numerator. Thus, having adisplaystyle a inside the unit circle minimizes the group delay contributed by the factor 1−az−1displaystyle 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−1displaystyle 1-a_iz^-1 is additive. I.e., for a transfer function with Ndisplaystyle N zeros,

Arg(∏i=1N(1−aiz−1))=∑i=1NArg(1−aiz−1)displaystyle mboxArgleft(prod _i=1^Nleft(1-a_iz^-1right)right)=sum _i=1^NmboxArgleft(1-a_iz^-1right)

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.

Illustration of the calculus above. Top and bottom are filters with same gain response (on the left : the Nyquist diagrams, on the right : phase responses), but the filter on the top with a=0.8<1displaystyle a=0.8<1 has the smallest amplitude in phase response.

Non-minimum phase[edit]

Systems that are causal and stable whose inverses are causal and unstable are known as non-minimum-phase systems. A given non-minimum phase system will have a greater phase contribution than the minimum-phase system with the equivalent magnitude response.

Maximum phase[edit]

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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 – discuss]} 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

s+10s+5ands−10s+5displaystyle frac s+10s+5qquad textandqquad frac s-10s+5

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.

Mixed phase[edit]

A mixed-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.

For example, the continuous-time LTI system described by transfer function

(s+1)(s−5)(s+10)(s+2)(s+4)(s+6)displaystyle frac (s+1)(s-5)(s+10)(s+2)(s+4)(s+6)

is stable and causal; however, it has zeros on both the left- and right-hand sides of the complex plane. Hence, it is a mixed-phase system.

Linear phase[edit]

A linear-phase system has constant group delay. Non-trivial linear phase or nearly linear phase systems are also mixed phase.

See also[edit]

• 
  All-pass filter – A special non-minimum-phase case.


• 
  Kramers–Kronig relation – Minimum phase system in physics

References[edit]

Further reading[edit]