In the tabular method, what does each row represent?

One x[k] multiplied by h and shifted by k

Why is sum of y equal to 0 here?

sum y = (sum x)(sum h) and sum h = 1+2-3 = 0

Because x has a negative sample

Because convolution always sums to zero

Short answer

y[n] = {1, 4, 4, 4, -1, -12} for n=0..5. Tabular method with both origins at index 0. Sum check: ∑y=0.

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

Choose the origin at the first sample of each sequence: $x[n]=\{1,2,3,4\}$ , $n=0..3$ and $h[n]=\{1,2,-3\}$ , $n=0..2$ .

$y[n]=\sum_k x[k]h[n-k]$ has $4+3-1=6$ samples, $n=0..5$ .

	$n=0$	1	2	3	4	5
$1\cdot h$	1	2	−3
$2\cdot h$		2	4	−6
$3\cdot h$			3	6	−9
$4\cdot h$				4	8	−12
$y[n]$	1	4	4	4	−1	−12

\begin{aligned} y[2]&=x[0]h[2]+x[1]h[1]+x[2]h[0]=-3+4+3=4\\ y[4]&=x[2]h[2]+x[3]h[1]=-9+8=-1 \end{aligned}

Sum check: $\sum y=0=(\sum x)(\sum h)=10\times0$ .

Answer: $y[n]=\{1,\ 4,\ 4,\ 4,\ -1,\ -12\}$ for $n=0,\dots,5$ (↑ at the first 1). With another origin the values are the same, only shifted.

✓ Answer checked by Claude Opus 5.5 (high effort)

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