Question:medium
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?
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?
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A system is BIBO stable only if its impulse response is absolutely integrable. If \( h(t) = u(t) \), the system is not BIBO stable.
Updated On: Feb 3, 2026
Every bounded input signal applied to S results in a bounded output signal.
t is possible to find a bounded input signal which when applied to S results in an unbounded output signal.
On applying any input signal to S, the output signal is always bounded.
On applying any input signal to S, the output signal is always unbounded.
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The Correct Option is B
Solution and Explanation
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.
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