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A discrete-time linear shift-invariant system has an impulse response \(h[n]\) with \(h[0]=1\), \(h[1]=-1\), \(h[2]=2\), and zero otherwise. The system is given an input sequence \(x[n]\) with \(x[0]=x[2]=1\), and zero otherwise. The number of nonzero samples in the output sequence \(y[n]\), and the value of \(y[2]\), respectively, are

1. 5, 2
2. 6, 2
3. 6, 1
4. 5, 3

Correct answer: 5, 3

Solution

Since \(x[n]=\delta[n]+\delta[n-2]\), the output is \(y[n]=h[n]+h[n-2]\). This produces nonzero samples at \(n=0,1,2,3,4\), i.e. 5 samples. At \(n=2\), \(y[2]=h[2]+h[0]=2+1=3\).

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