The GATE Grind

GATE 2025 EE – Question 20

Signals and Systems · Linear time invariant and causal systems · 1 mark · Multiple choice

Consider a discrete-time linear time-invariant (LTI) system $\boldsymbol S$, where $y[n]=\boldsymbol S\{x[n]\}$.

Let

$$\boldsymbol S\{\delta[n]\}=\begin{cases}1,&n\in\{0,1,2\}\\0,&\text{otherwise}\end{cases}$$

where $\delta[n]$ is the discrete-time unit impulse function. For an input signal $x[n]$, the output $y[n]$ is

  1. $x[n]+x[n-1]+x[n-2]$
  2. $x[n-1]+x[n]+x[n+1]$
  3. $x[n]+x[n+1]+x[n+2]$
  4. $x[n+1]+x[n+2]+x[n+3]$

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Show answer and explanation

Correct answer: (A) $x[n]+x[n-1]+x[n-2]$

Explanation

The impulse response is $h[n]=\delta[n]+\delta[n-1]+\delta[n-2]$. For an LTI system $y[n]=x[n]*h[n]=x[n]+x[n-1]+x[n-2]$.