GATE EC 2024 Set 1 — Question 24

Given the transfer function of a causal discrete-time LTI system:$$H(z) = \frac{2z^2+3}{(z+\frac{1}{3})(z-\frac{1}{3})} = \frac{2z^2+3}{z^2 - \frac{1}{9}}$$1. Stability:
The poles of the system are the roots of the denominator: $(z+\frac{1}{3})(z-\frac{1}{3}) = 0 \Rightarrow z = \pm \frac{1}{3}$.
For a causal LTI system to be stable, all its poles must lie inside the unit circle ($|z| < 1$). Since $|\pm \frac{1}{3}| = \frac{1}{3} < 1$, the system is stable. Thus, statement (A) is true.2. Minimum Phase System:
A system is minimum phase if all its poles and zeros lie inside the unit circle.
The zeros are the roots of the numerator: $2z^2 + 3 = 0 \Rightarrow z^2 = -\frac{3}{2} \Rightarrow z = \pm j\sqrt{1.5}$.
The magnitude of the zeros is $|z| = \sqrt{1.5} \approx 1.225$, which is greater than 1. Since the zeros lie outside the unit circle, the system is not a minimum phase system. Thus, statement (B) is false.3. Initial Value Theorem:
The initial value of the impulse response $h[0]$ is given by:$$h[0] = \lim_{z \to \infty} H(z) = \lim_{z \to \infty} \frac{2z^2+3}{z^2 - \frac{1}{9}} = \lim_{z \to \infty} \frac{2 + \frac{3}{z^2}}{1 - \frac{1}{9z^2}} = 2$$Thus, statement (C) is true.4. Final Value Theorem:
For a stable system, the final value of the impulse response $h[n]$ as $n \to \infty$ is given by:$$h[\infty] = \lim_{z \to 1} (z-1)H(z) = \lim_{z \to 1} (z-1) \frac{2z^2+3}{z^2 - \frac{1}{9}} = (1-1) \cdot \frac{2(1)^2+3}{1^2 - \frac{1}{9}} = 0 \cdot \frac{5}{8/9} = 0$$Alternatively, since all poles are strictly inside the unit circle, the impulse response $h[n]$ must decay to zero as $n \to \infty$. Thus, statement (D) is true.Therefore, the correct statements are (A), (C), and (D).