Convolve {3, 5, 6, 9} and {−1, 3, 5}

Chosen origin: first sample of each sequence at $n = 0$: $x[n] = \{\underset{\uparrow}{3}, 5, 6, 9\}$ ($0 \le n \le 3$) and $h[n] = \{\underset{\uparrow}{-1}, 3, 5\}$ ($0 \le n \le 2$).

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

Output starts at $n = 0$, length $= 4 + 3 - 1 = 6$, so $0 \le n \le 5$.

Tabular method

Direct working

• $y[0] = 3(-1) = -3$
• $y[1] = 3(3) + 5(-1) = 4$
• $y[2] = 3(5) + 5(3) + 6(-1) = 24$
• $y[3] = 5(5) + 6(3) + 9(-1) = 34$
• $y[4] = 6(5) + 9(3) = 57$
• $y[5] = 9(5) = 45$

Sum check: $(\sum x)(\sum h) = 23 \times 7 = 161$ and $-3 + 4 + 24 + 34 + 57 + 45 = 161$.

Answer (output signal):

$$y[n] = \{\underset{\uparrow}{-3},\ 4,\ 24,\ 34,\ 57,\ 45\}$$

Stem plot drawn sideways (each # = 3; left of | is negative):

 n    y[n]
 0      -3  #|
 1       4   |#
 2      24   |########
 3      34   |###########
 4      57   |###################
 5      45   |###############