Understanding the Concept:
The linear convolution of two discrete-time sequences \(x(n)\) and \(h(n)\) is mathematically defined by the summation equation:
\[
y(n) = x(n) * h(n) = \sum_{k=-\infty}^{\infty} x(k)h(n-k)
\]
The arrow symbol (\(\uparrow\)) denotes the index location where time index \(n = 0\). If no arrow is explicitly shown, the first element of the sequence is assumed by default to be at \(n = 0\). The boundaries of the resulting output sequence are determined by summing indices:
\[
n_{\text{start}} = n_{x,\text{start}} + n_{h,\text{start}}
\]
\[
n_{\text{end}} = n_{x,\text{end}} + n_{h,\text{end}}
\]
Step 1: Identify index boundaries and element values.
For sequence \(x(n) = \{1 \uparrow, 2, 0, 1\}\):
• \(x(0) = 1\) (marked by arrow)
• \(x(1) = 2\)
• \(x(2) = 0\)
• \(x(3) = 1\)
Thus, \(n\) ranges from \(0\) to \(3\).
For sequence \(h(n) = \{2, 2, 3\}\):
Since there is no arrow, the first element defaults to index \(0\):
• \(h(0) = 2\)
• \(h(1) = 2\)
• \(h(2) = 3\)
Thus, \(n\) ranges from \(0\) to \(2\).
The output sequence \(y(n)\) index boundaries will be:
• Start index = \(0 + 0 = 0\)
• End index = \(3 + 2 = 5\)
The total length of the convolution sequence is \(L_x + L_h - 1 = 4 + 3 - 1 = 6\) elements, covering indices from \(n = 0\) to \(n = 5\).
Step 2: Use the tabular (matrix) multiplication method for accuracy.
We construct a multiplication grid to systematically collect products of elements:
{c|cccc}
& x(0)=1 & x(1)=2 & x(2)=0 & x(3)=1
h(0)=2 & 2 & 4 & 0 & 2
h(1)=2 & 2 & 4 & 0 & 2
h(2)=3 & 3 & 6 & 0 & 3
Now, sum along the anti-diagonals to compute each individual index value for \(y(n)\):
• For \(n = 0\):
\[ y(0) = 2 \]
• For \(n = 1\):
\[ y(1) = 4 + 2 = 6 \]
• For \(n = 2\):
\[ y(2) = 0 + 4 + 3 = 7 \]
• For \(n = 3\):
\[ y(3) = 2 + 0 + 6 = 8 \]
• For \(n = 4\):
\[ y(4) = 2 + 0 = 2 \]
• For \(n = 5\):
\[ y(5) = 3 \]
Step 3: Assemble the final sequence with its origin arrow.
Putting the computed values together starting from index \(n = 0\):
\[
y(n) = \{2 \uparrow, 6, 7, 8, 2, 3\}
\]
This precisely matches option (B).