GATE 2023 IN – Question 21

Signals and Systems · Signals and linear time-invariant systems · 1 mark · Multiple choice

A system has $Y(s)/X(s)=(s-\pi)/(s+\pi)$. If $u(t)$ is the unit step, which input gives steady-state output $y(t)=\sin(\pi t)$?

$x(t)=\sin(\pi t)u(t)$
$x(t)=\sin(\pi t+\pi/2)u(t)$
$x(t)=\sin(\pi t-\pi/2)u(t)$
$x(t)=\cos(\pi t+\pi/4)u(t)$

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Correct answer: (C) $x(t)=\sin(\pi t-\pi/2)u(t)$

Explanation

At the desired frequency, $H(j\pi)=(j-1)/(j+1)=j$. The magnitude is one and phase is $+\pi/2$. Input phase must therefore be $-\pi/2$ to produce the required sine in steady state. Answer: **C**. The stable pole makes the transient decay.