Convolve {−0.5, 1, −1, 0.5} and {1, 2, −1, 0, 2, −1}

No origin is marked, so take the first sample of each at $n = 0$: $x[n] = \{\underset{\uparrow}{-0.5}, 1, -1, 0.5\}$ ($0 \le n \le 3$) and $y[n] = \{\underset{\uparrow}{1}, 2, -1, 0, 2, -1\}$ ($0 \le n \le 5$).

Let the result be $z[n] = x[n]*y[n] = \sum_k x[k]\,y[n-k]$.

Output starts at $n = 0$; length $4 + 6 - 1 = 9$, so $0 \le n \le 8$.

Tabular method

Each row is one sample of $x$ times $y$ shifted to that position (last row is the column sum $z[n]$):

Direct working

• $z[0] = -0.5(1) = -0.5$
• $z[1] = -0.5(2) + 1(1) = 0$
• $z[2] = -0.5(-1) + 1(2) + (-1)(1) = 1.5$
• $z[3] = -0.5(0) + 1(-1) + (-1)(2) + 0.5(1) = -2.5$
• $z[4] = -0.5(2) + 1(0) + (-1)(-1) + 0.5(2) = 1$
• $z[5] = -0.5(-1) + 1(2) + (-1)(0) + 0.5(-1) = 2$
• $z[6] = 1(-1) + (-1)(2) + 0.5(0) = -3$
• $z[7] = (-1)(-1) + 0.5(2) = 2$
• $z[8] = 0.5(-1) = -0.5$

Sum check: $(\sum x)(\sum y) = 0 \times 3 = 0$, and $-0.5 + 0 + 1.5 - 2.5 + 1 + 2 - 3 + 2 - 0.5 = 0$.

Answer:

$$z[n] = \{\underset{\uparrow}{-0.5},\ 0,\ 1.5,\ -2.5,\ 1,\ 2,\ -3,\ 2,\ -0.5\}$$

Plot

Stem plot drawn sideways (each # = 0.5; left of | is negative):

 n   z[n]
 0  -0.5       #|
 1   0.0        |
 2   1.5        |###
 3  -2.5   #####|
 4   1.0        |##
 5   2.0        |####
 6  -3.0  ######|
 7   2.0        |####
 8  -0.5       #|