GATE EE 2023 Set 1 — Question 27

(A) False. Causality and stability are independent properties. For example $h[n] = 2^{n}u[n]$ is causal, but $\sum_{n}|h[n]|$ diverges, so the system is not BIBO stable.(B) True. For a discrete-time LTI system the response to $u[n]$ is$$s[n] = \sum_{k=-\infty}^{n} h[k], \qquad h[n] = s[n] - s[n-1].$$If the system is causal then $h[k] = 0$ for $k < 0$, so $s[n] = 0$ for $n < 0$. Conversely, if $s[n] = 0$ for all $n < 0$ then $h[n] = s[n] - s[n-1] = 0$ for all $n < 0$, which is exactly the condition for an LTI system to be causal. The statement is therefore correct as an 'if and only if'.(C) Not accepted as true. BIBO stability requires $\sum_{n}|h[n]| < \infty$. Statement (C) puts a condition only on how many samples are non-zero and none on their size, and the accepted answer treats it as not sufficient on that basis. Note, however, that under the usual convention that every sample of $h[n]$ is a finite number, a finite-duration (FIR) impulse response is a finite sum of finite terms and is always absolutely summable, so such a system is stable; on that reading (C) would also be true.(D) False. The bound $0 < |h[n]| < 1$ limits each sample but not their sum. With $|h[n]| = \tfrac{1}{2}$ for all $n \ge 0$, for instance, $\sum_{n}|h[n]| = \infty$ and the system is unstable.Answer: (B)