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 zeroes 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 zeroes and vice versa, 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 minimum 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 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. In the context of causal, stable systems, we would in theory be free to choose whether the zeroes 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 formula_1 is invertible if we can uniquely determine its input from its output. I.e., we can find a system formula_2 such that if we apply formula_1 followed by formula_2, we obtain the identity system formula_5. (See Inverse matrix for a  finite-dimensional analog). I.e., Suppose that formula_7 is input to system formula_1 and gives output formula_9. Applying the inverse system formula_2 to formula_9 gives the following. So we see that the inverse system formula_2 allows us to determine uniquely the input formula_7 from the output formula_9. Discrete-time example. Suppose that the system formula_1 is a discrete-time, linear, time-invariant (LTI) system described by the impulse response formula_18 for "n" in "Z". Additionally, suppose formula_2 has impulse response formula_20. The cascade of two LTI systems is a convolution. In this case, the above relation is the following: where formula_22  is the Kronecker delta or the identity system in the discrete-time case. (Changing the order of formula_23 and formula_24 is allowed because of commutativity of the convolution operation.) Note that this inverse system formula_2 need not be unique. Minimum phase system. When we impose the constraints of causality and stability, the inverse system is unique; and the system formula_1 and its inverse formula_2 are called minimum-phase. The causality and stability constraints in the discrete-time case are the following (for time-invariant systems where h is the system's impulse response): Causality. and Stability. and See the article  on stability for the analogous conditions for the continuous-time case. Frequency analysis. Discrete-time frequency analysis. 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. From this relation, we realize that For simplicity, we consider only the case of a rational transfer function "H"("z"). Causality and stability imply that all poles of "H"("z") must be strictly inside the unit circle (See stability). Suppose 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. We also know that So, causality and stability for formula_37 imply that its poles – the roots of "A"("z") – 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. Continuous-time frequency analysis. 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. where formula_39 is the Dirac delta function. The Dirac  delta function is the identity operator in the continuous-time case because of the sifting property with any signal "x"("t"). Applying the Laplace transform gives the following relation in the s-plane. From this relation, we realize that Again, for simplicity, we consider only the case of a rational transfer function "H"("s"). Causality and stability imply that all poles of "H"("s") must be strictly inside the left-half s-plane (See stability). Suppose 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 . We also know that So, causality and stability for formula_45 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. Relationship of magnitude response to phase response. 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. That is, in the continuous-time case, let be the complex frequency response of system "H"("s"). Then, only for a minimum-phase system, the phase response of "H"("s") is related to the gain by where formula_48 denotes the Hilbert transform, and, inversely, Stated more compactly, let where formula_51 and formula_52 are real functions of a real variable. Then and The Hilbert transform operator is defined to be An equivalent corresponding relationship is also true for discrete-time minimum-phase systems. Minimum phase in the time domain. For all  causal and stable systems that have the same magnitude response, the minimum phase system has its energy concentrated near the start of the impulse response. i.e., it minimizes the following function which we can think of as the delay of energy in the impulse response. Minimum phase as minimum group delay. For all causal and stable systems that have the same magnitude response, the minimum phase system has the minimum group delay. The following proof illustrates this idea of minimum group delay. Suppose we consider one zero formula_57 of the transfer function formula_58. Let's place this zero formula_57 inside  the unit circle (formula_60) and see how the group delay is affected. Since the zero formula_57 contributes the factor formula_63 to the transfer function, the phase contributed by this term is the following. formula_69 contributes the following to the group delay. The denominator and formula_72 are invariant to reflecting the zero formula_57 outside of the unit circle, i.e., replacing formula_57 with formula_75. However, by reflecting formula_57 outside of the unit circle, we increase the magnitude of formula_77 in the numerator. Thus, having formula_57 inside the unit circle minimizes the group delay contributed by the factor formula_63. We can extend this  result to the general case of more than one zero since the phase of the multiplicative factors of the form formula_80 is additive. I.e., for a transfer function with formula_81 zeros, 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. 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, 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 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. 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.