GATE 2025 EC – Question 17
Consider the discrete-time system below with input $x[n]$ and output $y[n]$. In the figure, $h_1[n]$ and $h_2[n]$ denote the impulse responses of LTI Subsystems 1 and 2, respectively. Also, $\delta[n]$ is the unit impulse, and $b>0$.
Assuming $h_2[n]\ne\delta[n]$, the overall system (denoted by the dashed box) is ________.
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Correct answer: (D) nonlinear and time variant
Explanation
$y[n]=h_2[n]*\big(h_1[n]*x[n]+b\delta[n]\big)-b\delta[n]=(h_2*h_1*x)[n]+b\big(h_2[n]-\delta[n]\big)$. The second term does not depend on $x$ and is non-zero because $h_2\ne\delta$, so zero input gives non-zero output: the system is not linear (it is affine). The offset is fixed in time and does not shift when the input is shifted, so the system is also time variant.