Question

Let \(x_1(t) = u(t+1.5) - u(t-1.5)\) and \(x_2(t)\) is shown in the figure below. For \(y(t) = x_1(t) \ast x_2(t)\), the \[ \int_{-\infty}^{\infty} y(t) \, dt \] is ___________ (rounded off to the nearest integer).

Hint:

N/A

Answer 1

Use $\displaystyle \int_{-\infty}^{\infty}(x_1 \ast x_2)(t)\,dt = \left( \int_{-\infty}^{\infty}x_1(t)\,dt \right) \left( \int_{-\infty}^{\infty}x_2(t)\,dt \right)$.

$x_1(t) = u(t+1.5) - u(t-1.5)$ is a unit–amplitude rectangular pulse of width 3, $\Rightarrow \displaystyle \int x_1 = 3$.

From the figure, $x_2(t) = \delta(t+3) + [u(t+1)-u(t)] + \delta(t-1) + 2\delta(t-2)$, so
$\displaystyle \int x_2 = 1 + (1) + (1) + 2 = 5$.

Hence, $\displaystyle \int y(t)\,dt = (3)(5) = 15$.

\[ \boxed{15} \]