Discussion
The convolution of
\[ x(m) = h(n) = (\underset{\uparrow}{1}, 2, -1) \]
will be :
Answer: A
The convolution is {1, 4, 2, -4, 1}.
Both sequences are {1, 2, -1}, starting at n = 0 (the arrow). The result has 3 + 3 - 1 = 5 samples:
\[ \begin{aligned} y(0) &= 1 \times 1 = 1 \\ y(1) &= 1 \times 2 + 2 \times 1 = 4 \\ y(2) &= 1(-1) + 2 \times 2 + (-1)1 = 2 \\ y(3) &= 2(-1) + (-1)2 = -4 \\ y(4) &= (-1)(-1) = 1 \end{aligned} \]
{1, 4, 6, 4, 1} is the result for {1, 2, 1} convolved with itself, which ignores the minus sign on the last sample. {1, 4, 4, 2, 1} gets the last three values wrong, and {1, 4, 1} has too few samples: the convolution of two 3-sample sequences has 5.
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