The GATE Grind

GATE 2023 EE – Question 31

Signals and Systems · Linear time invariant and causal systems · 1 mark · Numerical answer

For the signals $x(t)$ and $y(t)$ shown in the figure, $z(t)=x(t)*y(t)$ is maximum at $t=T_1$. Then $T_1$ in seconds is ____________ (Round off to the nearest integer).

Diagram for GATE 2023 EE question 31

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Correct answer: 4

Explanation

$x(t)$ is a unit-height rectangle of width 2 (from $-1$ to $1$), so $z(t)=\int_{t-1}^{t+1}y(u)\,du$ and $\dfrac{dz}{dt}=y(t+1)-y(t-1)$. $y$ is a ramp rising linearly from $-1$ at $t=1$ to $2$ at $t=5$, and zero elsewhere. For $2\le t<4$ both ends lie on the ramp and $y(t+1)-y(t-1)=1.5>0$, so $z$ increases. For $t>4$ the upper end has passed the drop at 5, so $y(t+1)=0$ and $\dfrac{dz}{dt}=-y(t-1)<0$ (the lower end is beyond the zero crossing at 2.33). So $z$ peaks where $t+1=5$, i.e. $t=4$.