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GATE 2025 EE – Question 40

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

Let continuous-time signals $x_1(t)$ and $x_2(t)$ be

$$x_1(t)=\begin{cases}1,&t\in[0,1]\\2-t,&t\in[1,2]\\0,&\text{otherwise}\end{cases}\quad\text{and}\quad x_2(t)=\begin{cases}t,&t\in[0,1]\\2-t,&t\in[1,2]\\0,&\text{otherwise}\end{cases}$$

Consider the convolution $y(t)=x_1(t)*x_2(t)$. Then $\int_{-\infty}^{\infty}y(t)\,dt$ is

  1. 1.5
  2. 2.5
  3. 3.5
  4. 4

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Correct answer: (A) 1.5

Explanation

The integral of a convolution is the product of the integrals: $\int y\,dt=\int x_1\,dt\cdot\int x_2\,dt$. $\int x_1dt=1\cdot1+\tfrac12=1.5$ (a unit-height step on $[0,1]$, then a ramp down on $[1,2]$). $\int x_2dt=1$ (a triangle of height 1 and base 2). So the result is $1.5\times1=1.5$.