GATE 2023 EC Q.58: solution and answer

Official answer: 15Source: Official answer key, GATE 2023 EC (IIT Kanpur)

1. Area property of convolution: for $y(t)=x_1(t)\ast x_2(t)$, $${\int }_{-\infty }^{\infty }y(t)dt=\left({\int }_{-\infty }^{\infty }{x}_1(t)dt\right)\left({\int }_{-\infty }^{\infty }{x}_2(t)dt\right)$$ This is $Y(0)=X_1(0)X_2(0)$ in the Fourier domain, since $X(0)={\int }_{-\infty }^{\infty }x(t)dt$.

2. $x_1(t)=u(t+1.5)-u(t-1.5)$ is a unit-height pulse on $-1.5<t<1.5$, so its area is $1\times 3=3$.

3. Read $x_2(t)$ from the figure: $x_2(t)=\delta (t+3)+\left[u(t+1)-u(t-1)\right]+2\delta (t-2)$.

4. Area of $x_2$: the impulse at $t=-3$ contributes its strength $1$, the pulse contributes $1\times 2=2$, and the impulse at $t=2$ contributes its strength $2$. Total $=1+2+2=5$.

5. So $${\int }_{-\infty }^{\infty }y(t)dt=3\times 5=15$$