GATE 2021 EC – Question 50
For a unit step input $u[n]$, a discrete-time LTI system produces an output signal $(2\delta[n+1]+\delta[n]+\delta[n-1])$. Let $y[n]$ be the output of the system for an input $\left(\frac12\right)^nu[n]$. The value of $y[0]$ is ________.
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Correct answer: 0
Explanation
The step response is $s[n]=2\delta[n+1]+\delta[n]+\delta[n-1]$, so the impulse response is $h[n]=s[n]-s[n-1]=2\delta[n+1]-\delta[n]-\delta[n-2]$. For $x[n]=\left(\frac12\right)^nu[n]$, $y[0]=\sum_kh[k]x[-k]=h[-1]x[1]+h[0]x[0]+h[2]x[-2]=2\cdot\frac12-1\cdot1-1\cdot0=0$.