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GATE 2024 EC – Question 24

Networks, Signals and Systems · LTI Systems · 1 mark · Multiple select

For a causal discrete-time LTI system with transfer function

$$H(z)=\frac{2z^2+3}{\left(z+\frac13\right)\left(z-\frac13\right)},$$

which of the following statements is/are true?

  1. The system is stable.
  2. The system is a minimum phase system.
  3. The initial value of the impulse response is 2.
  4. The final value of the impulse response is 0.

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Show answer and explanation

Correct answer: (A) The system is stable.; (C) The initial value of the impulse response is 2.; (D) The final value of the impulse response is 0.

Explanation

(A) The poles are $\pm\frac13$, inside the unit circle, so the causal system is stable. (B) The zeros satisfy $2z^2+3=0$, i.e. $z=\pm j\sqrt{1.5}$, which lie outside the unit circle, so it is not minimum phase. (C) $h[0]=\lim_{z\to\infty}H(z)=2$. (D) All poles are inside the unit circle, so the impulse response decays and $h[\infty]=0$.