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ISRO EC 2023 Q.19: solution and answer

ISRO 2023 · ECQ.19 · 1 mark · MCQ

An LTI system is minimum phase if

  1. (A)

    All poles are inside unit circle

  2. (B)

    All zeros are inside unit circle

  3. (C)

    All poles are outside unit circle

  4. (D)

    All poles and zeros are inside unit circle

See the question in the paper
ISRO EC 2023 question 19Tap to open full size

Answer and solution

ANSWER

Option (d): all poles and zeros are inside the unit circle.

Official answer: DSource: ISRO's revised and final answer key of 16 February 2024

  1. A causal discrete-time LTI system with transfer function is stable exactly when all its poles lie inside the unit circle . That condition is about stability alone, not about minimum phase.

  2. A system is minimum phase when both and its inverse are causal and stable.

  3. is causal and stable when all poles of are inside .

  4. The poles of are the zeros of . So is causal and stable when all zeros of are inside .

  5. Both conditions together require all poles and all zeros inside the unit circle. Option (a) alone ignores the inverse system, and option (b) alone ignores the stability of the system itself. Option (c) places the poles outside the circle, so the system would be unstable.

  6. Hence the correct option is (d).

KEY IDEA

A minimum phase system and its inverse are both causal and stable, which puts every pole and every zero of inside the unit circle.

WATCH OUT FOR

Choosing option (a) because stability needs poles inside the unit circle, and forgetting that the inverse system must also be stable, which constrains the zeros.

From the ISRO EC 2023 paper. Every question of this paper · The full paper (PDF, ISRO) · All 5 ISRO z-transform questions, with notes

More ISRO z-transform questions