Question

The continuous-time unit impulse signal is applied as an input to a continuous-time linear time-invariant system \( \mathcal{S} \). The output is observed to be the continuous-time unit step signal \( u(t) \). Which one of the following statements is true?

• A. Every bounded input signal applied to S results in a bounded output signal.
• B. t is possible to find a bounded input signal which when applied to S results in an unbounded output signal.
• C. On applying any input signal to S, the output signal is always bounded.
• D. On applying any input signal to S, the output signal is always unbounded.

Hint:

A system is BIBO stable only if its impulse response is absolutely integrable. If \( h(t) = u(t) \), the system is not BIBO stable.

Answer 1

The Correct Option is B

The impulse response \( h(t) \) of the system \( \mathcal{S} \) is the output observed when the input is a Dirac delta \( \delta(t) \). Given: \[ h(t) = u(t) \] Now consider the bounded-input bounded-output (BIBO) stability condition: A system is BIBO stable if and only if its impulse response is absolutely integrable: \[ \int_{-\infty}^{\infty} |h(t)| \, dt<\infty \] But for \( h(t) = u(t) \), we have: \[ \int_{0}^{\infty} 1 \, dt = \infty \] So the system is not BIBO stable. Hence, there exists at least one bounded input signal that can lead to an unbounded output signal.