GATE 2022 EE Q.48: solution and answer

Official answer: BSource: Official answer key, GATE 2022 EE (IIT Kharagpur)

1. Linearity (superposition test): apply $x(t)=a{x}_1(t)+b{x}_2(t)$. The system reads the input at the instant $e^t$, so $y(t)=a{x}_1(e^t)+b{x}_2(e^t)=a{y}_1(t)+b{y}_2(t)$. Additivity and homogeneity both hold. Only the time axis is warped; the amplitude is not touched. So the system is linear.

2. Causality test: a system is causal if $y(t)$ depends only on $x(\tau )$ for $\tau \le t$. Here $y(t)$ depends on the input at the single instant $\tau =e^t$.

3. Compare $e^t$ with $t$ for every real $t$. For $t<0$: $e^t>0>t$. For $t\ge 0$: every term of $e^t=1+t+\frac{t^2}{2!}+\cdots$ is non-negative, so $e^t\ge 1+t>t$. So $e^t>t$ always, and the output at every instant needs an input value from a later time.

4. Check at one instant: $y(0)=x(e^0)=x(1)$, so the output at $t=0$ needs the input at $t=1$. Likewise $y(1)=x(e)\approx x(2.718)$. One counter-example is enough to prove the system is non-causal.

5. Conclusion: the system is linear and non-causal, which is option (B).