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GATE 2025 EE – Question 42

Signals and Systems · Linear time invariant and causal systems · 2 marks · Multiple choice

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

  1. Every bounded input signal applied to $\boldsymbol S$ results in a bounded output signal.
  2. It is possible to find a bounded input signal which when applied to $\boldsymbol S$ results in an unbounded output signal.
  3. On applying any input signal to $\boldsymbol S$, the output signal is always bounded.
  4. On applying any input signal to $\boldsymbol S$, the output signal is always unbounded.

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Correct answer: (B) It is possible to find a bounded input signal which when applied to $\boldsymbol S$ results in an unbounded output signal.

Explanation

The impulse response is $h(t)=u(t)$, so $\int_{-\infty}^\infty|h(t)|dt=\infty$: the system (an integrator) is not BIBO stable. A bounded input such as $x(t)=u(t)$ gives $y(t)=t\,u(t)$, which is unbounded. Not every input gives an unbounded output (e.g. an input of zero gives zero), so A, C and D are false and B is true.