The GATE Grind

GATE 2023 EE – Question 27

Signals and Systems · Linear time invariant and causal systems · 1 mark · Multiple select

Which of the following statement(s) is/are true?

  1. If an LTI system is causal, it is stable
  2. A discrete time LTI system is causal if and only if its response to a step input $u[n]$ is 0 for $n<0$
  3. If a discrete time LTI system has an impulse response $h[n]$ of finite duration, the system is stable
  4. If the impulse response $0<|h[n]|<1$ for all $n$, then the LTI system is stable.

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (B) A discrete time LTI system is causal if and only if its response to a step input $u[n]$ is 0 for $n<0$; (C) If a discrete time LTI system has an impulse response $h[n]$ of finite duration, the system is stable

Explanation

Causality does not imply stability (an integrator is causal but unstable), so A is false. A discrete LTI system is causal exactly when $h[n]=0$ for $n<0$, which is equivalent to the step response being zero for $n<0$: B is true. A finite-duration $h[n]$ is absolutely summable, so the system is stable: C is true. $|h[n]|<1$ for all $n$ does not guarantee $\sum|h[n]|<\infty$ (e.g. $h[n]=1/2\cdot$ constant), so D is false.