Question

For an LTI system with impulse response \(h(t)=e^{-2t}u(t)\), the system is

• A. Unstable
• B. Non-causal
• C. Stable but non-causal
• D. Stable and causal

Hint:

For continuous-time LTI systems, \[ \boxed{ \begin{aligned} \text{Causal} &\iff h(t)=0,\; t \text{Stable} &\iff \int_{-\infty}^{\infty}|h(t)|dt<\infty. \end{aligned} } \]

Answer 1

The Correct Option is D

Step 1: Check causality. Since \[ h(t)=e^{-2t}u(t), \] the impulse response is zero for \[ t<0. \] Therefore, the system is causal.

Step 2: Check stability. For an LTI system to be BIBO stable, \[ \int_{-\infty}^{\infty}|h(t)|\,dt<\infty. \] Here, \[ \int_{0}^{\infty}e^{-2t}\,dt = \frac12, \] which is finite. Hence, the system is stable. Therefore, \[ \boxed{\text{Stable and causal}} \] is the correct answer. \[ \boxed{(D)} \]