# Minimum phase

The analysis in terms of poles and zeroes 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 zeroes, can in practice be implemented as efficiently as any other.

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.

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

Performing frequency analysis for the discrete-time case will provide some insight. The time-domain equation is the following.

Applying the Z-transform gives the following relation in the z-domain.

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.

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

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.

For example, the two continuous-time LTI systems described by the transfer functions

For example, the continuous-time LTI system described by transfer function