Introduction / Context:
Minimum-phase systems are important in control and signal processing because they achieve the smallest possible phase shift for a given magnitude response. They are stable and causal with all zeros located in the left half-plane.
Given Data / Assumptions:
- Minimum-phase refers to transfer functions with all poles stable (LHP) and all zeros in the LHP.
- Focus is on zero locations since pole stability is assumed in any well-designed system.
Concept / Approach:
Minimum-phase condition means every zero contributes nonnegative phase lag consistent with stability. If a zero is in the right half-plane, the system becomes nonminimum-phase even if stable.
Step-by-Step Solution:
Check pole locations: for stability, poles in LHP are required but not sufficient for minimum phase.Zeros: all must also lie in LHP to avoid extra phase lag.Thus, correct characterization is 'all zeros in LHP'.
Verification / Alternative check:
Textbooks: A system with transfer function (s+1)/(s+2) is minimum-phase; (s−1)/(s+2) is stable but nonminimum-phase due to RHP zero.
Why Other Options Are Wrong:
'Poles in LHP' only ensures stability, not minimum phase.'Poles in RHP' is unstable.'All except one' permits RHP zeros, disqualifying minimum phase.'All poles and zeros LHP' is technically correct but the defining feature is zeros, not poles (already assumed stable).
Common Pitfalls:
Confusing stable with minimum-phase; forgetting that nonminimum-phase systems can be stable but with undesired phase properties.
Final Answer:
all zeros lie in the left half-plane
Discussion & Comments