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| In [[control theory]] and [[signal processing]], a [[LTI system theory|linear, time-invariant]] system is said to be '''minimum-phase''' if the system and its [[Inverse (mathematics)|inverse]] are [[causal system|causal]] and [[BIBO stability|stable]].<ref>{{cite book |author=Hassibi, Babak; Kailath, Thomas; Sayed, Ali H. |title=Linear estimation |publisher=Prentice Hall |location=Englewood Cliffs, N.J |year=2000 |pages=193 |isbn=0-13-022464-2}}</ref><ref>J. O. Smith III, ''[http://ccrma.stanford.edu/~jos/filters/Definition_Minimum_Phase_Filters.html Introduction to Digital Filters with Audio Applications]'' (September 2007 Edition).</ref>
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| For example, a discrete-time system with [[rational function|rational]] [[transfer function]] <math>H(z)</math> can only satisfy [[Causal#Engineering|causality]] and [[BIBO stability|stability]] requirements if all of its [[Pole (complex analysis)|pole]]s are inside the [[unit circle]]. However, we are free to choose whether the [[Zero (complex analysis)|zero]]s of the system are inside or outside the [[unit circle]]. A system 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.
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| == Inverse system ==
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| A system <math>\mathbb{H}</math> is invertible if we can uniquely determine its input from its output. I.e., we can find a system <math>\mathbb{H}_{inv}</math> such that if we apply <math>\mathbb{H}</math> followed by <math>\mathbb{H}_{inv}</math>, we obtain the identity system <math>\mathbb{I}</math>. (See [[Inverse matrix]] for a finite-dimensional analog). I.e.,
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| :<math>\mathbb{H}_{inv} \, \mathbb{H} = \mathbb{I}</math> | |
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| Suppose that <math>\tilde{x}</math> is input to system <math>\mathbb{H}</math> and gives output <math>\tilde{y}</math>.
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| :<math>\mathbb{H} \, \tilde{x} = \tilde{y}</math>
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| Applying the inverse system <math>\mathbb{H}_{inv}</math> to <math>\tilde{y}</math> gives the following.
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| :<math>\mathbb{H}_{inv} \, \tilde{y} = \mathbb{H}_{inv} \, \mathbb{H} \, \tilde{x} = \mathbb{I} \, \tilde{x} = \tilde{x}</math>
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| So we see that the inverse system <math>\mathbb{H}_{inv}</math> allows us to determine uniquely the input <math>\tilde{x}</math> from the output <math>\tilde{y}</math>.
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| === Discrete-time example ===
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| Suppose that the system <math>\mathbb{H}</math> is a discrete-time, [[LTI system theory|linear, time-invariant]] (LTI) system described by the [[impulse response]] <math>h(n) \, \forall \, n \, \in \mathbb{Z}</math>. Additionally, <math>\mathbb{H}_{inv}</math> has impulse response <math>h_{inv}(n) \, \forall \, n \, \in \mathbb{Z}</math>. The cascade of two LTI systems is a [[convolution]]. In this case, the above relation is the following:
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| :<math>(h * h_{inv}) (n) = \sum_{k=-\infty}^{\infty} h(k) \, h_{inv} (n-k) = \delta (n)</math>
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| where <math>\delta (n)</math> is the [[Kronecker delta]] or the [[identity matrix|identity]] system in the discrete-time case. Note that this inverse system <math>\mathbb{H}_{inv}</math> is not unique.
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| == Minimum phase system ==
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| When we impose the constraints of [[causal]]ity and [[BIBO stability|stability]], the inverse system is unique; and the system <math>\mathbb{H}</math> and its inverse <math>\mathbb{H}_{inv}</math> 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):
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| === Causality ===
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| :<math>h(n) = 0 \,\, \forall \, n < 0</math>
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| and
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| :<math>h_{inv} (n) = 0 \,\, \forall \, n < 0</math>
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| === Stability ===
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| :<math>\sum_{n = -\infty}^{\infty}{\left|h(n)\right|} = \| h \|_{1} < \infty</math>
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| and
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| :<math>\sum_{n = -\infty}^{\infty}{\left|h_{inv}(n)\right|} = \| h_{inv} \|_{1} < \infty</math>
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| See the article on [[BIBO stability|stability]] for the analogous conditions for the continuous-time case.
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| == Frequency analysis ==
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| === Discrete-time frequency analysis ===
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| Performing frequency analysis for the discrete-time case will provide some insight. The time-domain equation is the following.
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| :<math>(h * h_{inv}) (n) = \,\! \delta (n)</math>
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| Applying the [[Z-transform]] gives the following relation in the z-domain.
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| :<math>H(z) \, H_{inv}(z) = 1</math>
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| From this relation, we realize that
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| :<math>H_{inv}(z) = \frac{1}{H(z)}</math>
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| For simplicity, we consider only the case of a [[rational function|rational]] [[transfer function]] ''H'' (''z''). Causality and stability imply that all [[pole (complex analysis)|poles]] of ''H'' (''z'') must be strictly inside the [[unit circle]] (See [[BIBO stability#Discrete signals|stability]]). Suppose
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| :<math>H(z) = \frac{A(z)}{D(z)}</math>
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| where ''A'' (''z'') and ''D'' (''z'') are [[polynomial]] in ''z''. Causality and stability imply that the [[zero (complex analysis)|poles]] – the [[Root of a function|root]]s of ''D'' (''z'') – must be strictly inside the [[unit circle]]. We also know that
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| :<math>H_{inv}(z) = \frac{D(z)}{A(z)}</math>
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| So, causality and stability for <math>H_{inv}(z)</math> imply that its [[pole (complex analysis)|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.
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| === Continuous-time frequency analysis ===
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| 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.
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| :<math>(h * h_{inv}) (t) = \,\! \delta (t)</math> | |
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| where <math>\delta(t)</math> 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'').
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| :<math>\delta(t) * x(t) = \int_{-\infty}^{\infty} \delta(t - \tau) x(\tau) d \tau = x(t)</math>
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| Applying the [[Laplace transform]] gives the following relation in the [[s-plane]].
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| :<math>H(s) \, H_{inv}(s) = 1</math>
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| From this relation, we realize that
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| :<math>H_{inv}(s) = \frac{1}{H(s)}</math>
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| Again, for simplicity, we consider only the case of a [[rational function|rational]] [[transfer function]] ''H''(''s''). Causality and stability imply that all [[pole (complex analysis)|poles]] of ''H'' (''s'') must be strictly inside the left-half [[s-plane]] (See [[BIBO stability#Continuous signals|stability]]). Suppose
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| :<math>H(s) = \frac{A(s)}{D(s)}</math>
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| where ''A'' (''s'') and ''D'' (''s'') are [[polynomial]] in ''s''. Causality and stability imply that the [[pole (complex analysis)|poles]] – the [[Root of a function|root]]s of ''D'' (''s'') – must be inside the left-half [[s-plane]]. We also know that
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| :<math>H_{inv}(s) = \frac{D(s)}{A(s)}</math>
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| So, causality and stability for <math>H_{inv}(s)</math> imply that its [[pole (complex analysis)|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]].
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| === Relationship of magnitude response to phase response ===
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| 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 [[neper]]s which is proportional to [[Decibel|dB]]) is related to the phase angle of the frequency response (measured in [[radian]]s) by the [[Hilbert transform]]. That is, in the continuous-time case, let
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| :<math>H(j \omega) \ \stackrel{\mathrm{def}}{=}\ H(s) \Big|_{s = j \omega} \ </math>
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| 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
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| :<math> \arg \left[ H(j \omega) \right] = -\mathcal{H} \lbrace \log \left( |H(j \omega)| \right) \rbrace \ </math>
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| and, inversely,
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| :<math> \log \left( |H(j \omega)| \right) = \log \left( |H(j \infty)| \right) + \mathcal{H} \lbrace \arg \left[H(j \omega) \right] \rbrace \ </math>.
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| Stated more compactly, let
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| :<math>H(j \omega) = |H(j \omega)| e^{j \arg \left[H(j \omega) \right]} \ \stackrel{\mathrm{def}}{=}\ e^{\alpha(\omega)} e^{j \phi(\omega)} = e^{\alpha(\omega) + j \phi(\omega)} \ </math>
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| where <math>\alpha(\omega)</math> and <math>\phi(\omega)</math> are real functions of a real variable. Then
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| :<math> \phi(\omega) = -\mathcal{H} \lbrace \alpha(\omega) \rbrace \ </math>
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| and
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| :<math> \alpha(\omega) = \alpha(\infty) + \mathcal{H} \lbrace \phi(\omega) \rbrace \ </math>.
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| The Hilbert transform operator is defined to be
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| :<math>\mathcal{H} \lbrace x(t) \rbrace \ \stackrel{\mathrm{def}}{=}\ \widehat{x}(t) = \frac{1}{\pi}\int_{-\infty}^{\infty}\frac{x(\tau)}{t-\tau}\, d\tau \ </math> .
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| An equivalent corresponding relationship is also true for discrete-time minimum-phase systems.
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| == Minimum phase in the time domain ==
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| For all [[causal]] and [[BIBO stability|stable]] systems that have the same [[frequency response|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]].
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| :<math> \sum_{n = m}^{\infty} \left| h(n) \right|^2 \,\,\,\,\,\,\, \forall \, m \in \mathbb{Z}^{+}</math>
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| == Minimum phase as minimum group delay ==
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| For all [[causal]] and [[BIBO stability|stable]] systems that have the same [[frequency response|magnitude response]], the minimum phase system has the minimum [[group delay]]. The following proof illustrates this idea of minimum [[group delay]].
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| Suppose we consider one [[Zero (complex analysis)|zero]] <math>a</math> of the [[transfer function]] <math>H(z)</math>. Let's place this [[Zero (complex analysis)|zero]] <math>a</math> inside the [[unit circle]] (<math>\left| a \right| < 1</math>) and see how the [[group delay]] is affected.
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| :<math>a = \left| a \right| e^{i \theta_a} \, \mbox{ where } \, \theta_a = \mbox{Arg}(a)</math>
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| Since the [[Zero (complex analysis)|zero]] <math>a</math> contributes the factor <math>1 - a z^{-1}</math> to the [[transfer function]], the phase contributed by this term is the following.
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| :<math>\phi_a \left(\omega \right) = \mbox{Arg} \left(1 - a e^{-i \omega} \right)</math>
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| :<math>= \mbox{Arg} \left(1 - \left| a \right| e^{i \theta_a} e^{-i \omega} \right)</math>
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| :<math>= \mbox{Arg} \left(1 - \left| a \right| e^{-i (\omega - \theta_a)} \right)</math>
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| :<math>= \mbox{Arg} \left( \left\{ 1 - \left| a \right| cos( \omega - \theta_a ) \right\} + i \left\{ \left| a \right| sin( \omega - \theta_a ) \right\}\right)</math>
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| :<math>= \mbox{Arg} \left( \left\{ \left| a \right|^{-1} - \cos( \omega - \theta_a ) \right\} + i \left\{ \sin( \omega - \theta_a ) \right\} \right)</math>
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| <math>\phi_a (\omega)</math> contributes the following to the [[group delay]].
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| :<math>-\frac{d \phi_a (\omega)}{d \omega} =
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| \frac{ \sin^2( \omega - \theta_a ) + \cos^2( \omega - \theta_a ) - \left| a \right|^{-1} \cos( \omega - \theta_a )
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| }{
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| \sin^2( \omega - \theta_a ) + \cos^2( \omega - \theta_a ) + \left| a \right|^{-2} - 2 \left| a \right|^{-1} \cos( \omega - \theta_a )
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| }</math>
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| :<math> -\frac{d \phi_a (\omega)}{d \omega} =
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| \frac{ \left| a \right| - \cos( \omega - \theta_a )
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| }{
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| \left| a \right| + \left| a \right|^{-1} - 2 \cos( \omega - \theta_a )
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| }</math>
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| The denominator and <math>\theta_a</math> are invariant to reflecting the [[Zero (complex analysis)|zero]] <math>a</math> outside of the [[unit circle]], i.e., replacing <math>a</math> with <math>(a^{-1})^{*}</math>. However, by reflecting <math>a</math> outside of the unit circle, we increase the magnitude of <math>\left| a \right|</math> in the numerator. Thus, having <math>a</math> inside the [[unit circle]] minimizes the [[group delay]] contributed by the factor <math>1 - a z^{-1}</math>. We can extend this result to the general case of more than one [[Zero (complex analysis)|zero]] since the phase of the multiplicative factors of the form <math>1 - a_i z^{-1}</math> is additive. I.e., for a [[transfer function]] with <math>N</math> [[Zero (complex analysis)|zero]]s,
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| :<math>\mbox{Arg}\left( \prod_{i = 1}^N \left( 1 - a_i z^{-1} \right) \right) = \sum_{i = 1}^N \mbox{Arg}\left( 1 - a_i z^{-1} \right) </math>
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| So, a minimum phase system with all [[Zero (complex analysis)|zero]]s inside the [[unit circle]] minimizes the [[group delay]] since the [[group delay]] of each individual [[Zero (complex analysis)|zero]] is minimized.
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| == Non-minimum phase ==
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| 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.
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| === Maximum phase ===
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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. That is,
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| * The zeros of the discrete-time system are outside the [[unit circle]].
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| * The zeros of the continuous-time system are in the right-hand side of the [[complex plane]].
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| 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.
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| For example, the two continuous-time LTI systems described by the transfer functions
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| :<math>\frac{s + 10}{s + 5} \qquad \text{and} \qquad \frac{s - 10}{s + 5}</math>
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| 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.
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| === Mixed phase ===
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| A ''mixed-phase'' system has some of its [[Zero (complex analysis)|zero]]s 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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| For example, the continuous-time LTI system described by transfer function
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| :<math>\frac{ (s + 1)(s - 5)(s + 10) }{ (s+2)(s+4)(s+6) }</math>
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| 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.
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| === Linear phase ===
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| A [[linear phase|linear-phase]] system has constant [[group delay]]. Non-trivial linear phase or nearly linear phase systems are also mixed phase.
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| ==See also==
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| * [[All-pass filter]]{{spaced ndash}} A special non-minimum-phase case.
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| * [[Kramers–Kronig relation]]{{spaced ndash}} Minimum phase system in physics
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| ==References==
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| {{reflist}}
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| ==Further reading==
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| {{refbegin}}
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| *Dimitris G. Manolakis, Vinay K. Ingle, Stephen M. Kogon : ''Statistical and Adaptive Signal Processing'', pp. 54–56, McGraw-Hill, ISBN 0-07-040051-2
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| *Boaz Porat : ''A Course in Digital Signal Processing'', pp. 261–263, John Wiley and Sons, ISBN 0-471-14961-6
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| {{refend}}
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| [[Category:Digital signal processing]]
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| [[Category:Control theory]]
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