Minimum phase 548131 215935003 2008-05-30T10:17:09Z Prozac1980 2120668 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 are [[causal system|causal]] and [[BIBO stability|stable]]. 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 inside the unit circle. Insight is given below as to why this system is called minimum-phase. == Inverse system == 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., :<math>\mathbb{H} \, \mathbb{H}_{inv} = \mathbb{I}</math> Suppose that <math>\tilde{x}</math> is input to system <math>\mathbb{H}</math> and gives output <math>\tilde{y}</math>. :<math>\mathbb{H} \, \tilde{x} = \tilde{y}</math> Applying the inverse system <math>\mathbb{H}_{inv}</math> to <math>\tilde{y}</math> gives the following. :<math>\mathbb{H}_{inv} \, \tilde{y} = \mathbb{H}_{inv} \, \mathbb{H} \, \tilde{x} = \mathbb{I} \, \tilde{x} = \tilde{x}</math> 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>. === Discrete-time example === 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: :<math>(h * h_{inv}) (n) = \sum_{k=-\infty}^{\infty} h(k) \, h_{inv} (n-k) = \delta (n)</math> 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. == Minimum phase system == 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): === Causality === :<math>h(n) = h_{inv} (n) = 0 \,\, \forall \, n < 0</math> === Stability === :<math>\sum_{n = -\infty}^{\infty}{\left|h(n)\right|} = \| h \|_{1} < \infty</math> :<math>\sum_{n = -\infty}^{\infty}{\left|h_{inv}(n)\right|} = \| h_{inv} \|_{1} < \infty</math> See the article on [[BIBO stability|stability]] for the analogous conditions for the continuous-time case. == Frequency analysis == === Discrete-time frequency analysis === [[Image:min-phase-discrete-time.png|thumb|300px|right|A minimum-phase system in the [[complex plane]].]] Performing frequency analysis for the discrete-time case will provide some insight. The time-domain equation is the following. :<math>(h * h_{inv}) (n) = \,\! \delta (n)</math> Applying the [[Z-transform]] gives the following relation in the z-domain. :<math>H(z) \, H_{inv}(z) = 1</math> From this relation, we realize that :<math>H_{inv}(z) = \frac{1}{H(z)}</math> 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]] in the [[complex plane]] (See [[BIBO stability#Discrete signals|stability]]). Suppose :<math>H(z) = \frac{A(z)}{D(z)}</math> where ''A'' (''z'') and ''D'' (''z'') are [[polynomial]] in ''z''. Causality and stability imply that the [[pole (complex analysis)|poles]] -- the [[Root (mathematics)|root]]s of ''D'' (''z'') -- must be strictly inside the [[unit circle]]. We also know that :<math>H_{inv}(z) = \frac{D(z)}{A(z)}</math> So, causality and stability for <math>H_{inv}(z)</math> imply that its [[pole (complex analysis)|poles]] -- the [[Root (mathematics)|root]]s 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 === [[Image:min-phase-continuous-time.png|thumb|300px|right|A minimum-phase system in the [[s plane]].]] 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. :<math>(h * h_{inv}) (t) = \,\! \delta (t)</math> 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''). :<math>\delta(t) * x(t) = \int_{-\infty}^{\infty} \delta(t - \tau) x(\tau) d \tau = x(t)</math> Applying the [[Laplace transform]] gives the following relation in the [[s-plane]]. :<math>H(s) \, H_{inv}(s) = 1</math> From this relation, we realize that :<math>H_{inv}(s) = \frac{1}{H(s)}</math> 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 :<math>H(s) = \frac{A(s)}{D(s)}</math> where ''A'' (''s'') and ''D'' (''s'') are [[polynomial]] in ''s''. Causality and stability imply that the [[pole (complex analysis)|poles]] -- the [[Root (mathematics)|root]]s of ''D'' (''s'') -- must be inside the left-half [[s-plane]]. We also know that :<math>H_{inv}(s) = \frac{D(s)}{A(s)}</math> So, causality and stability for <math>H_{inv}(s)</math> imply that its [[pole (complex analysis)|poles]] -- the [[Root (mathematics)|root]]s 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 === 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 :<math>H(j \omega) \ \stackrel{\mathrm{def}}{=}\ H(s) \Big|_{s = j \omega} \ </math> 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 :<math> \arg \left[ H(j \omega) \right] = -\mathcal{H} \lbrace \log \left( |H(j \omega)| \right) \rbrace \ </math> and, inversely, :<math> \log \left( |H(j \omega)| \right) = \log \left( |H(j \infty)| \right) + \mathcal{H} \lbrace \arg \left[H(j \omega) \right] \rbrace \ </math>. Stated more compactly, let :<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> where <math>\alpha(\omega)</math> and <math>\phi(\omega)</math> are real functions of a real variable. Then :<math> \phi(\omega) = -\mathcal{H} \lbrace \alpha(\omega) \rbrace \ </math> and :<math> \alpha(\omega) = \alpha(\infty) + \mathcal{H} \lbrace \phi(\omega) \rbrace \ </math>. The Hilbert transform operator is defined to be :<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> . An equivalent corresponding relationship is also true for discrete-time minimum-phase systems. == Minimum phase in the time domain == 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]]. :<math> \sum_{n = m}^{\infty} \left| h(n) \right|^2 \,\,\,\,\,\,\, \forall \, m \in \mathbb{Z}^{+}</math> == Minimum phase as minimum group delay == 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]]. 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]]. 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. :<math>a = \left| a \right| e^{i \theta_a} \, \mbox{ where } \, \theta_a = \mbox{Arg}(a)</math> 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. :<math>\phi_a \left(\omega \right) = \mbox{Arg} \left(1 - a e^{-i \omega} \right)</math> :<math>= \mbox{Arg} \left(1 - \left| a \right| e^{i \theta_a} e^{-i \omega} \right)</math> :<math>= \mbox{Arg} \left(1 - \left| a \right| e^{-i (\omega - \theta_a)} \right)</math> :<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> :<math>= \mbox{Arg} \left( \left\{ \left| a \right|^{-1} - \cos( \omega - \theta_a ) \right\} + i \left\{ \sin( \omega - \theta_a ) \right\} \right)</math> <math>\phi_a (\omega)</math> contributes the following to the [[group delay]]. :<math>-\frac{d \phi_a (\omega)}{d \omega} = \frac{ \sin^2( \omega - \theta_a ) + \cos^2( \omega - \theta_a ) - \left| a \right|^{-1} \cos( \omega - \theta_a ) }{ \sin^2( \omega - \theta_a ) + \cos^2( \omega - \theta_a ) + \left| a \right|^{-2} - 2 \left| a \right|^{-1} \cos( \omega - \theta_a ) }</math> :<math> -\frac{d \phi_a (\omega)}{d \omega} = \frac{ \left| a \right| - \cos( \omega - \theta_a ) }{ \left| a \right| + \left| a \right|^{-1} - 2 \cos( \omega - \theta_a ) }</math> 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, :<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> 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. == Maximum phase == A maximum phase system is the opposite of a minimum phase system. It is a causal and stable system that has all of its [[Zero (complex analysis)|zero]]s outside the [[unit circle]] and all of its poles inside the [[unit circle]]. Of all causal and stable systems with a specified magnitude response, a maximum phase system has maximal [[group delay]] and its [[impulse response]] has maximum energy delay. == Mixed phase == A '''mixed phase''' or '''non minimum-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. A [[linear phase]] system has constant group delay. Non-trivial linear phase or nearly linear phase systems are also mixed phase. == References == *Dimitris G. Manolakis, Vinay K. Ingle, Stephen M. Kogon : ''Statistical and Adaptive Signal Processing'', pp. 54-56, McGraw-Hill, ISBN 0-07-040051-2 *Boaz Porat : ''A Course in Digital Signal Processing'', pp. 261-263, John Wiley and Sons, ISBN 0-471-14961-6 [[Category:Digital signal processing]] [[Category:Control theory]]