Minimum phase

[1][2] The most general causal LTI transfer function can be uniquely factored into a series of an all-pass and a minimum phase system.

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 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.

the spectral symmetric/antisymmetric decomposition as another important example, leading e.g. to Hilbert transform techniques.)

is a discrete-time, linear, time-invariant (LTI) system described by the impulse response

is the Kronecker delta, or the identity system in the discrete-time case.

Performing frequency analysis for the discrete-time case will provide some insight.

where A(z) and D(z) are polynomial in z. Causality and stability imply that the poles – the roots of D(z) – must be strictly 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.

is the Dirac delta function – the identity operator in the continuous-time case because of the sifting property with any signal x(t):

where A(s) and D(s) are polynomial in s. Causality and stability imply that the poles – the roots of D(s) – must be inside the left-half s-plane.

imply that its 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.

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 nepers, which is proportional to dB) is related to the phase angle of the frequency response (measured in radians) by the Hilbert transform.

Then, only for a minimum-phase system, the phase response of H(s) is related to the gain by

An equivalent corresponding relationship is also true for discrete-time minimum-phase systems.

i.e., it minimizes the following function, which we can think of as the delay of energy in the impulse response:

The following proof illustrates this idea of minimum group delay.

inside the unit circle minimizes the group delay contributed by the factor

We can extend this result to the general case of more than one zero since the phase of the multiplicative factors of the form

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.

have equivalent magnitude responses; however, the second system has a much larger contribution to the phase shift.

One recent solution to these systems is moving the RHP zeros to the LHP using the PFCD method.

is stable and causal; however, it has zeros on both the left- and right-hand sides of the complex plane.