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]^[3]

For example, a discrete-time 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 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

A system $\mathbb{H}$ is invertible if we can uniquely determine its input from its output. I.e., we can find a system $\mathbb{H}_{inv}$ such that if we apply $\mathbb{H}$ followed by $\mathbb{H}_{inv}$ , we obtain the identity system $\mathbb{I}$ . (See Inverse matrix for a finite-dimensional analog). I.e.,

$\mathbb{H} \, \mathbb{H}_{inv} = \mathbb{I}$

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 $\mathbb{H}_{inv}$ to $\tilde{y}$ gives the following.

$\mathbb{H}_{inv} \, \tilde{y} = \mathbb{H}_{inv} \, \mathbb{H} \, \tilde{x} = \mathbb{I} \, \tilde{x} = \tilde{x}$

So we see that the inverse system $\mathbb{H}_{inv}$ allows us to determine uniquely the input $\tilde{x}$ from the output $\tilde{y}$ .

Discrete-time example

Suppose that the system $\mathbb{H}$ is a discrete-time, linear, time-invariant (LTI) system described by the impulse response $h(n) \, \forall \, n \, \in \mathbb{Z}$ . Additionally, $\mathbb{H}_{inv}$ has impulse response $h_{inv}(n) \, \forall \, n \, \in \mathbb{Z}$ . The cascade of two LTI systems is a convolution. In this case, the above relation is the following:

$(h * h_{inv}) (n) = \sum_{k=-\infty}^{\infty} h(k) \, h_{inv} (n-k) = \delta (n)$

where $δ(n)$ is the Kronecker delta or the identity system in the discrete-time case. Note that this inverse system $\mathbb{H}_{inv}$ is not unique.

Non-minimum phase

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

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,

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

$\frac{s + 10}{s + 5} \qquad \text{and} \qquad \frac{s - 10}{s + 5}$

have equivalent magnitude responses; however, the second system has a much larger contribution to the phase shift. Hence, in this set, the first system is the minimum-phase system and the second system is the maximum-phase system.

Mixed phase

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

$\frac{ (s + 1)(s - 5)(s + 10) }{ (s+2)(s+4)(s+6) }$

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

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

Group delay and phase delay:

All signal components are delayed when passing through a device such as an amplifier or a loudspeaker. The signal delay can be (and often is) different for different frequencies. The delay variation means that signals consisting of different frequency components suffer delay (or time) distortion. A small delay variation is usually not a problem, but larger delays can cause trouble such as poor fidelity and intersymbol interference. High speed modems use adaptive equalizers to compensate for group delay.

In physics, and in particular in optics, the term group delay has the following meanings:

1. The rate of change of the total phase shift with respect to angular frequency,

$\tau_g = -\frac{d\phi}{d\omega}$

through a device or transmission medium, where $\phi \$ is the total phase shift in radians, and $\omega \$ is the angular frequency in radians per unit time, equal to $2 \pi f \$ , where $f \$ is the frequency (hertz if group delay is measured in seconds).

2. In an optical fiber, the transit time required for optical power, traveling at a given mode's group velocity, to travel a given distance.

Note: For optical fiber dispersion measurement purposes, the quantity of interest is group delay per unit length, which is the reciprocal of the group velocity of a particular mode. The measured group delay of a signal through an optical fiber exhibits a wavelength dependence due to the various dispersion mechanisms present in the fiber.

It is often desirable for the group delay to be constant across all frequencies; otherwise there is temporal smearing of the signal. Because group delay is $\tau_g(\omega) = -\frac{d\phi}{d\omega}$ , as defined in (1), it therefore follows that a constant group delay can be achieved if the transfer function of the device or medium has a linear phase response (i.e., $\phi(\omega) = \phi(0) - \tau_g \omega \$ where the group delay $\tau_g \$ is a constant). The degree of nonlinearity of the phase indicates the deviation of the group delay from a constant.

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Thursday, December 30, 2010

Mininimun Phase Filter