The GATE Grind

GATE 2021 EE – Question 17

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

Two discrete-time linear time-invariant systems with impulse responses $h_1[n]=\delta[n-1]+\delta[n+1]$ and $h_2[n]=\delta[n]+\delta[n-1]$ are connected in cascade, where $\delta[n]$ is the Kronecker delta. The impulse response of the cascaded system is

  1. $\delta[n-2]+\delta[n+1]$
  2. $\delta[n-1]\delta[n]+\delta[n+1]\delta[n-1]$
  3. $\delta[n-2]+\delta[n-1]+\delta[n]+\delta[n+1]$
  4. $\delta[n]\delta[n-1]+\delta[n-2]\delta[n+1]$

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Correct answer: (C) $\delta[n-2]+\delta[n-1]+\delta[n]+\delta[n+1]$

Explanation

The cascade has impulse response $h_1*h_2=(\delta[n-1]+\delta[n+1])*(\delta[n]+\delta[n-1])=\delta[n-1]+\delta[n-2]+\delta[n+1]+\delta[n]$.