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 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.

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}_\text{inv}</math> such that if we apply <math>\mathbb{H}</math> followed by <math>\mathbb{H}_\text{inv}</math>, we obtain the identity system <math>\mathbb{I}</math>. (See Inverse matrix for a finite-dimensional analog). That is,

\mathbb{H}_\text{inv} \mathbb{H} = \mathbb{I}.

</math>

Suppose that <math>\tilde{x}</math> is input to system <math>\mathbb{H}</math> and gives output <math>\tilde{y}</math>:

\mathbb{H} \tilde{x} = \tilde{y}.

</math>

Applying the inverse system <math>\mathbb{H}_\text{inv}</math> to <math>\tilde{y}</math> gives

\mathbb{H}_\text{inv} \tilde{y} = \mathbb{H}_\text{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, linear, time-invariant (LTI) system described by the impulse response <math>h(n)</math> for in . Additionally, suppose <math>\mathbb{H}_\text{inv}</math> has impulse response <math>h_\text{inv}(n)</math>. The cascade of two LTI systems is a convolution. In this case, the above relation is the following:

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

</math>

where <math>\delta(n)</math> is the Kronecker delta, or the identity system in the discrete-time case. (Changing the order of <math>h_\text{inv}</math> and <math>h</math> is allowed because of commutativity of the convolution operation.) Note that this inverse system <math>\mathbb{H}_\text{inv}</math> need not be unique.

Minimum-phase system

When we impose the constraints of causality and stability, the inverse system is unique; and the system <math>\mathbb{H}</math> and its inverse <math>\mathbb{H}_\text{inv}</math> are called minimum-phase. The causality and stability constraints in the discrete-time case are the following (for time-invariant systems where is the system's impulse response, and <math>\|{\cdot}\|_1</math> is the ℓ<sup>1</sup> norm):

Causality

h(n) = 0\ \forall n < 0

</math>

and

h_\text{inv}(n) = 0\ \forall n < 0.

</math>

Stability

\sum_{n=-\infty}^\infty |h(n)| = \|h\|_1 < \infty

</math>

and

\sum_{n=-\infty}^\infty |h_\text{inv}(n)| = \|h_\text{inv}\|_1 < \infty.

</math>

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

Converting an IIR filter to a minimum-phase filter

Source:

Any stable causal infinite impulse response (IIR) filter <math>H(z)</math> can be expressed as a cascade of a minimum-phase filter <math>H_{mp}(z)</math> and an all-pass filter <math>H_{ap}(z)</math>.

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

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.

Linear phase

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

Inverse system

Discrete-time example

Minimum-phase system

Causality

Stability

Converting an IIR filter to a minimum-phase filter

Mixed phase

Linear phase

See also

References

Further reading