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

Chosen origin: first sample of each sequence at $n = 0$: $x[n] = \{\underset{\uparrow}{1}, 2, 2, 1\}$ and $h[n] = \{\underset{\uparrow}{1}, 2, 2, -1\}$, both for $0 \le n \le 3$.

$$y[n] = \sum_k x[k]\,h[n-k]$$

Output starts at $0 + 0 = 0$, length $4 + 4 - 1 = 7$, so $0 \le n \le 6$.

Tabular method

Direct working

• $y[0] = 1(1) = 1$
• $y[1] = 1(2) + 2(1) = 4$
• $y[2] = 1(2) + 2(2) + 2(1) = 8$
• $y[3] = 1(-1) + 2(2) + 2(2) + 1(1) = 8$
• $y[4] = 2(-1) + 2(2) + 1(2) = 4$
• $y[5] = 2(-1) + 1(2) = 0$
• $y[6] = 1(-1) = -1$

Sum check: $(\sum x)(\sum h) = 6 \times 4 = 24$, and $1 + 4 + 8 + 8 + 4 + 0 - 1 = 24$.

Answer:

$$y[n] = \{\underset{\uparrow}{1},\ 4,\ 8,\ 8,\ 4,\ 0,\ -1\}$$

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

 n    y[n]
 0       1    |##
 1       4    |########
 2       8    |################
 3       8    |################
 4       4    |########
 5       0    |
 6      -1  ##|