Convolve {1, 3, 4, 3, 1} and {2, 2, −2}

Short answer

The convolution sum gives y[n] = {2, 8, 12, 8, 0, -4, -2} for n = 0 to 6 (origin at first sample).

Find the convolution sum of x[n] = {1, 3, 4, 3, 1} and h[n] = {2, 2, −2}

Take the origin at the first sample of each sequence (none is marked): $x[n]=\{1,3,4,3,1\}$ , $n=0..4$ and $h[n]=\{2,2,-2\}$ , $n=0..2$ .

Output length $=5+3-1=7$ , for $n=0$ to $6$ ; $y[n]=\sum_k x[k]h[n-k]$ .

	$n=0$	1	2	3	4	5	6
$1\cdot h$	2	2	−2
$3\cdot h$		6	6	−6
$4\cdot h$			8	8	−8
$3\cdot h$				6	6	−6
$1\cdot h$					2	2	−2
$y[n]$	2	8	12	8	0	−4	−2

\begin{aligned} y[2]&=x[0]h[2]+x[1]h[1]+x[2]h[0]=-2+6+8=12\\ y[4]&=x[2]h[2]+x[3]h[1]+x[4]h[0]=-8+6+2=0 \end{aligned}

Sum check: $\sum y=24=(\sum x)(\sum h)=12\times2$ .

Answer: $y[n]=\{2,\ 8,\ 12,\ 8,\ 0,\ -4,\ -2\}$ for $n=0,\dots,6$ (↑ at 2).

✓ Answer checked by Claude Opus 5.5 (high effort)

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