GATE 2023 EC – Question 58
Let $x_1(t)=u(t+1.5)-u(t-1.5)$ and $x_2(t)$ is shown in the figure below. For $y(t)=x_1(t)*x_2(t)$, the $\int_{-\infty}^{\infty}y(t)\,dt$ is ________ (rounded off to the nearest integer).
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Correct answer: 15
Explanation
The area of a convolution is the product of the areas: $\int y\,dt=\int x_1\,dt\cdot\int x_2\,dt$. $x_1$ is a rectangle of width 3 and height 1, area 3. $x_2$ consists of an impulse of weight 1 at $t=-3$, a rectangle of height 1 from $-1$ to $1$ (area 2) and an impulse of weight 2 at $t=2$: total area $1+2+2=5$. So $\int y\,dt=3\times5=15$.