GATE 2022 EC – Question 46
The outputs of four systems ($S_1$, $S_2$, $S_3$, and $S_4$) corresponding to the input signal $\sin(t)$, for all time $t$, are as follows.
$S_1$: input $\sin(t)$, output $\sin(-t)$
$S_2$: input $\sin(t)$, output $\sin(t+1)$
$S_3$: input $\sin(t)$, output $\sin(2t)$
$S_4$: input $\sin(t)$, output $\sin^2(t)$
Based on the given information, which of the four systems is/are definitely NOT LTI (linear and time-invariant)?
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Show answer and explanation
Correct answer: (C) $S_3$; (D) $S_4$
Explanation
An LTI system cannot create frequencies that are not in its input. $S_3$ turns $\sin t$ into $\sin 2t$, so it cannot be LTI. $S_4$ produces $\sin^2 t=\frac{1-\cos2t}{2}$ with a DC term and a doubled frequency, so it is not LTI either. $S_1$ has $\sin(-t)=-\sin t$ (a gain of $-1$) and $S_2$ is a phase shift of 1 rad, both of which an LTI system can produce, so they are not definitely non-LTI.