Minimum phase

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. The most general causal LTI transfer function can be uniquely factored into a series of an all-pass and a minimum phase system. The system function is then the product of the two parts, and in the time domain the response of the system is the convolution of the two part responses. The difference between a minimum-phase and a general transfer function is that a minimum-phase system has all of the poles and zeros of its transfer function in the left half of the s-plane representation (in discrete time, respectively, inside the unit circle of the z plane). Since inverting a system function leads to poles turning to zeros and conversely, and poles on the right side (s-plane imaginary line) or outside (z-plane unit circle) of the complex plane lead to unstable systems, only the class of minimum-phase systems is closed under inversion. Intuitively, the minimum-phase part of a general causal system implements its amplitude response with minimal group delay, while its all-pass part corrects its phase response alone to correspond with the original system function. The analysis in terms of poles and zeros is exact only in the case of transfer functions which can be expressed as ratios of polynomials.…

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. The most general causal LTI transfer function can be uniquely factored into a series of an all-pass and a minimum phase system. The system function is then the product of the two parts, and in the time domain the response of the system is the convolution of the two part responses. The difference between a minimum-phase and a general transfer function is that a minimum-phase system has all of the poles and zeros of its transfer function in the left half of the s-plane representation (in discrete time, respectively, inside the unit circle of the z plane). Since inverting a system function leads to poles turning to zeros and conversely, and poles on the right side (s-plane imaginary line) or outside (z-plane unit circle) of the complex plane lead to unstable systems, only the class of minimum-phase systems is closed under inversion. Intuitively, the minimum-phase part of a general causal system implements its amplitude response with minimal group delay, while its all-pass part corrects its phase response alone to correspond with the original system function. The analysis in terms of poles and zeros is exact only in the case of transfer functions which can be expressed as ratios of polynomials. In the continuous-time case, such systems translate into networks of conventional, idealized LCR networks. In discrete time, they conveniently translate into approximations thereof, using addition, multiplication, and unit delay. It can be shown that in both cases, system functions of rational form with increasing order can be used to efficiently approximate any other system function; thus even system functions lacking a rational form, and so possessing an infinitude of poles and/or zeros, can in practice be implemented as efficiently as any other. In the context of causal, stable systems, we would in theory be free to choose whether the zeros of the system function are outside of the stable range (to the right or outside) if the closure condition wasn't an issue. However, inversion is of great practical importance, just as theoretically perfect factorizations are in their own right. (Cf. the spectral symmetric/antisymmetric decomposition as another important example, leading e.g. to Hilbert transform techniques.) Many physical systems also naturally tend towards minimum-phase response, and sometimes have to be inverted using other physical systems obeying the same constraint. Insight is given below as to why this system is called minimum-phase, and why the basic idea applies even when the system function cannot be cast into a rational form that could be implemented. \mathbb{H}_\text{inv} \mathbb{H} = \mathbb{I}. Suppose that \tilde{x} is input to system \mathbb{H} and gives output \tilde{y} : \mathbb{H} \tilde{x} = \tilde{y}. Applying the inverse system \mathbb{H}_\text{inv} to \tilde{y} gives \mathbb{H}_\text{inv} \tilde{y} = \mathbb{H}_\text{inv} \mathbb{H} \tilde{x} = \mathbb{I} \tilde{x} = \tilde{x}. So we see that the inverse system \mathbb{H}_{inv} allows us to determine uniquely the input \tilde{x} from the output \tilde{y} . (h_\text{inv} * h)(n) = (h * h_\text{inv})(n) = \sum_{k=-\infty}^\infty h(k) h_\text{inv}(n - k) = \delta(n), where \delta(n) is the Kronecker delta, or the identity system in the discrete-time case. (Changing the order of h_\text{inv} and h is allowed because of commutativity of the convolution operation.) Note that this inverse system \mathbb{H}_\text{inv} need not be unique. h(n) = 0\ \forall n and h_\text{inv}(n) = 0\ \forall n \sum_{n=-\infty}^\infty |h(n)| = \|h\|_1 and \sum_{n=-\infty}^\infty |h_\text{inv}(n)| = \|h_\text{inv}\|_1 See the article on stability for the analogous conditions for the continuous-time case. (h * h_\text{inv})(n) = \delta(n). Applying the Z-transform gives the following relation in the z domain: H(z) H_\text{inv}(z) = 1. From this relation, we realize that H_\text{inv}(z) = \frac{1}{H(z)}. For simplicity, we consider only the case of a rational transfer function . Causality and stability imply that all poles of must be strictly inside the unit circle (see stability). Suppose H(z) = \frac{A(z)}{D(z)}, where and are polynomial in . Causality and stability imply that the poles the roots of must be strictly inside the unit circle. We also know that H_\text{inv}(z) = \frac{D(z)}{A(z)}, so causality and stability for H_\text{inv}(z) imply that its poles the roots of 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. (h * h_\text{inv})(t) = \delta(t), where \delta(t) is the Dirac delta function the identity operator in the continuous-time case because of the sifting property with any signal : (\delta * x)(t) = \int_{-\infty}^\infty \delta(t - \tau) x(\tau) \,d\tau = x(t). Applying the Laplace transform gives the following relation in the s-plane: H(s) H_\text{inv}(s) = 1, from which we realize that H_\text{inv}(s) = \frac{1}{H(s)}. Again, for simplicity, we consider only the case of a rational transfer function . Causality and stability imply that all poles of must be strictly inside the left-half s-plane (see stability). Suppose H(s) = \frac{A(s)}{D(s)}, where and are polynomial in . Causality and stability imply that the poles the roots of must be inside the left-half s-plane. We also know that H_\text{inv}(s) = \frac{D(s)}{A(s)}, so causality and stability for H_\text{inv}(s) imply that its poles the roots of 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. H(j\omega)\ \stackrel{\text{def}}{=}\ H(s)\Big|_{s=j\omega} be the complex frequency response of system . Then, only for a minimum-phase system, the phase response of is related to the gain by \arg[H(j\omega)] = -\mathcal{H}\big\{\log\big(|H(j\omega)|\big)\big\}, where \mathcal{H} denotes the Hilbert transform, and, inversely, \log\big(|H(j\omega)|\big) = \log\big(|H(j\infty)|\big) + \mathcal{H}\big\{\arg[H(j\omega)]\big\}. Stated more compactly, let H(j\omega) = |H(j\omega)| e^{j\arg[H(j\omega)]}\ \stackrel{\text{def}}{=}\ e^{\alpha(\omega)} e^{j\phi(\omega)} = e^{\alpha(\omega) + j\phi(\omega)}, where \alpha(\omega) and \phi(\omega) are real functions of a real variable. Then \phi(\omega) = -\mathcal{H}\{\alpha(\omega)\} and \alpha(\omega) = \alpha(\infty) + \mathcal{H}\{\phi(\omega)\}. The Hilbert transform operator is defined to be \mathcal{H}\{x(t)\}\ \stackrel{\text{def}}{=}\ \hat{x}(t) = \frac{1}{\pi} \int_{-\infty}^\infty \frac{x(\tau)}{t - \tau} \,d\tau. An equivalent corresponding relationship is also true for discrete-time minimum-phase systems. \sum_{n=m}^\infty |h(n)|^2 \quad \forall m \in \mathbb{Z}^+. a = \left| a \right| e^{i \theta_a} \, \text{ where } \, \theta_a = \operatorname{Arg}(a) Since the zero a contributes the factor 1 - a z^{-1} to the transfer function, the phase contributed by this term is the following. \begin{align} \phi_a \left(\omega \right) &= \operatorname{Arg} \left(1 - a e^{-i \omega} \right)\\ &= \operatorname{Arg} \left(1 - \left| a \right| e^{i \theta_a} e^{-i \omega} \right)\\ &= \operatorname{Arg} \left(1 - \left| a \right| e^{-i (\omega - \theta_a)} \right)\\ &= \operatorname{Arg} \left( \left\{ 1 - \left| a \right| \cos( \omega - \theta_a ) \right\} + i \left\{ \left| a \right| \sin( \omega - \theta_a ) \right\}\right)\\ &= \operatorname{Arg} \left( \left\{ \left| a \right|^{-1} - \cos( \omega - \theta_a ) \right\} + i \left\{ \sin( \omega - \theta_a ) \right\} \right) \end{align} \phi_a (\omega) contributes the following to the group delay. \begin{align} -\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 ) } \\ &= \frac{ \left| a \right| - \cos( \omega - \theta_a ) }{ \left| a \right| + \left| a \right|^{-1} - 2 \cos( \omega - \theta_a ) } \end{align} The denominator and \theta_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 \left| a \right| in the numerator. Thus, having a inside the unit circle minimizes the group delay contributed by the factor 1 - a z^{-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, \operatorname{Arg}\left( \prod_{i = 1}^N \left( 1 - a_i z^{-1} \right) \right) = \sum_{i = 1}^N \operatorname{Arg}\left( 1 - a_i z^{-1} \right) 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. s, on the right : phase responses), but the filter on the top with a = 0.8 has the smallest amplitude in phase response. \frac{s + 10}{s + 5} \qquad \text{and} \qquad \frac{s - 10}{s + 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. These systems are also famously known as nonminimum-phase systems that raise many stability concerns in control. One recent solution to these systems is moving the RHP zeros to the LHP using the PFCD method. \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. To control the transfer functions that include these systems some methods such as internal model controller (IMC), generalized Smith's predictor (GSP) and parallel feedforward control with derivative (PFCD) are proposed. ==See also== All-pass filter A special non-minimum-phase case. Kramers–Kronig relation Minimum phase system in physics ==References== ==Further reading== Dimitris G. Manolakis, Vinay K. Ingle, Stephen M. Kogon : Statistical and Adaptive Signal Processing, pp. 54–56, McGraw-Hill, Boaz Porat : A Course in Digital Signal Processing, pp. 261–263, John Wiley and Sons,

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