GATE 2017 EC – Question 42

Networks, Signals and Systems · Discrete-time Signals · 2 marks · Numerical answer

Two discrete-time signals $x[n]$ and $h[n]$ are both non-zero only for $n = 0, 1, 2$, and are zero otherwise. It is given that $x[0] = 1$, $x[1] = 2$, $x[2] = 1$, $h[0] = 1$. Let $y[n]$ be the linear convolution of $x[n]$ and $h[n]$. Given that $y[1] = 3$ and $y[2] = 4$, the value of the expression $(10y[3] + y[4])$ is ________.

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Correct answer: 31

Explanation

The convolution is $y[n] = \sum_k x[k]h[n - k]$. From $y[1] = x[0]h[1] + x[1]h[0] = h[1] + 2 = 3$ we get $h[1] = 1$. From $y[2] = x[0]h[2] + x[1]h[1] + x[2]h[0] = h[2] + 2 + 1 = 4$ we get $h[2] = 1$. Then $y[3] = x[1]h[2] + x[2]h[1] = 2 + 1 = 3$ and $y[4] = x[2]h[2] = 1$. So $10 \times 3 + 1 = 31$.