Introduction / Context:
In digital signal processing, classification of systems into minimum-phase, maximum-phase, or mixed-phase depends on the locations of zeros of the system's transfer function relative to the unit circle in the z-plane. This question tests knowledge of system classification based on pole-zero locations.
Given Data / Assumptions:
- Transfer function: H(z) = 6 + z^(-1) + z^(-2).
- No poles other than at z = 0 from negative powers of z.
- Classification depends only on zero locations.
Concept / Approach:
For a system to be minimum-phase, all its zeros must lie inside the unit circle. For maximum-phase, all zeros must lie outside. For mixed-phase, some zeros lie inside and some outside. The magnitude response is the same for all three, but the phase response differs.
Step-by-Step Solution:
Rewrite H(z) with common denominator: H(z) = (6z^2 + z + 1)/z^2.Denominator corresponds to poles at z = 0 (stable, inside unit circle).Zeros are roots of 6z^2 + z + 1 = 0.Discriminant = 1 − 24 = −23 < 0 → complex conjugate roots.Magnitude of roots = sqrt(1/6) ≈ 0.408 < 1.Thus, all zeros are inside unit circle → minimum-phase system.
Verification / Alternative check:
Check |z| of roots explicitly: roots lie at (−1 ± j√23)/12 → magnitude less than 1.
Why Other Options Are Wrong:
Maximum phase: requires zeros outside unit circle, not the case here.Mixed phase: requires mix of inside and outside, not the case.None/Linear phase: irrelevant here; linear phase is a property of FIR symmetry, not applicable.
Common Pitfalls:
Forgetting to multiply by z^2 and solve numerator polynomial properly, or misclassifying because magnitude check is skipped.
Final Answer:
Minimum phase
Discussion & Comments