Question:

Consider a continuous-time signal \[ x(t) = -t^2 \left\{ u(t+4) - u(t-4) \right\} \] where \( u(t) \) is the continuous-time unit step function. Let \( \delta(t) \) be the continuous-time unit impulse function. The value of \[ \int_{-\infty}^{\infty} x(t)\delta(t+3) \, dt \] is:

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To evaluate \( \int x(t)\delta(t+a)\,dt \), apply the sifting property: it equals \( x(-a) \). Ensure that the value lies within the domain where \( x(t) \) is defined and non-zero.
Updated On: Apr 16, 2025
  • \( -9 \)
  • \( 9 \)
  • \( 3 \)
  • \( -3 \)
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The Correct Option is A

Solution and Explanation

We use the sifting property of the Dirac delta function:
-∞ x ( t ) δ ( t + 3 ) dt = x ( -3 )
Now evaluate \( x(-3) \). The signal \( x(t) \) is defined as:
x ( t ) = - t 2 -4 < t < 4 < -9 in
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