Questions: Minimum Phase Systems and Factorization

Question 1

A discrete-time channel has transfer function H(z) = (z – 2)/(z – 0.5). An engineer wants to design a stable causal equalizer E(z) such that E(z)·H(z) = 1. What problem arises?

• A. No problem — any rational system has a rational inverse that is also stable and causal
• B. The inverse E(z) = (z – 0.5)/(z – 2) has a pole at z = 2, outside the unit circle, making the equalizer unstable
• C. The channel pole at z = 0.5 prevents inversion because poles cannot be canceled
• D. Equalization is only possible if the channel has no poles at all

Answer: (B)

Inverting H(z) = (z–2)/(z–0.5) gives E(z) = (z–0.5)/(z–2), which places a pole at z = 2 — outside the unit circle, making E(z) unstable. The zero at z = 2 in H(z) becomes a pole in E(z). This is exactly why minimum-phase is the key criterion for stable invertibility: only when all zeros lie inside the unit circle will the inverse have all poles inside the unit circle too, preserving stability. H(z) here is non-minimum-phase, so no stable causal inverse exists.

Question 2

Two causal stable systems A and B have exactly the same magnitude response |H(e^jω)| for all frequencies, but system A is minimum-phase while system B is not. What must be true about their transfer functions?

• A. Their transfer functions are identical — magnitude response uniquely determines a causal system
• B. System B has the same minimum-phase factor as A but an additional all-pass component that adds phase lag without changing magnitude
• C. System B must be unstable, since a non-minimum-phase system sharing the same magnitude as a stable system is impossible
• D. The two systems differ in pole locations but have identical zeros

Answer: (B)

By the min-phase factorization theorem, any stable causal system factors as H = H_min · H_ap, where H_min is minimum-phase and H_ap is all-pass (|H_ap| = 1 everywhere). If A and B share the same magnitude response, they share the same minimum-phase factor H_min. B differs only in having a non-trivial all-pass component that moves some zeros outside the unit circle (to conjugate-reciprocal positions), keeping magnitude unchanged but adding phase lag. Both remain stable (poles inside unit circle).

Question 3

A minimum-phase system introduces less phase lag at every frequency than any other stable, causal system with the same magnitude response.

• T. True
• F. False

Answer: True

This is the defining property of minimum-phase systems — the name means 'minimum group delay.' An all-pass factor has unit magnitude everywhere but introduces additional phase lag. Any non-minimum-phase system with the same magnitude has a non-trivial all-pass component H_ap, contributing extra phase lag beyond what H_min alone introduces. The minimum-phase system has no all-pass component (H_ap = 1), making its phase response the smallest possible for that magnitude profile.

Question 4

Moving a zero from inside to outside the unit circle (to its conjugate-reciprocal position) changes both the magnitude and phase responses of the system.

• T. True
• F. False

Answer: False

Moving a zero z₀ (inside the unit circle) to its conjugate reciprocal 1/z₀* (outside the unit circle) changes only the phase response — the magnitude response |H(e^jω)| is unchanged. This is exactly the all-pass substitution: the factor (z – 1/z₀*)/(1 – z₀*z⁻¹) has unit magnitude everywhere. This is why two systems can share an identical magnitude spectrum yet have different phase responses — they have the same minimum-phase factor but differ in their all-pass components.

Question 5

Why are minimum-phase systems called 'stably invertible,' and what goes wrong when you attempt to invert a non-minimum-phase system with a stable causal filter?

Model answer: A minimum-phase system has all zeros inside the unit circle. Its causal inverse 1/H_min(z) has poles exactly where H_min has zeros — all inside the unit circle — so the inverse is also stable and causal. For a non-minimum-phase system, at least one zero lies outside the unit circle; inverting it places a pole outside the unit circle, making the equalizer unstable. You cannot perfectly equalize a non-minimum-phase channel with a stable causal filter; you must either accept instability, use non-causal processing, or apply regularized approximations.

This has direct practical consequences: channel equalization, echo cancellation, and seismic deconvolution are feasible with a stable causal inverse only when the system is minimum-phase. The first diagnostic question in any equalization problem is therefore: are all zeros inside the unit circle? If yes, perfect causal inversion is achievable. If no, the engineer must choose between approximation methods — there is no exact stable causal solution.