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Convolve {2, 0, −1, 2} and {1, 2, −1, 3}

Short answer

y[n] = {2, 4, -3, 6, 5, -5, 6} starting at n = 0 (origin at first sample).

Find the convolution between the following sequences. h[n] = {2, 0, −1, 2}, x[n] = {1, 2, −1, 3}. Choose origin as per your choice.

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

The output has 4+4−1=74+4-1=7 samples, from n=0n=0 to n=6n=6, with y[n]=∑kx[k]h[n−k]y[n]=\sum_k x[k]h[n-k].

Tabular method

n=0n=0123456
x[0]h=1⋅hx[0]h = 1\cdot h20−12
x[1]h=2⋅hx[1]h = 2\cdot h40−24
x[2]h=−1⋅hx[2]h = -1\cdot h−201−2
x[3]h=3⋅hx[3]h = 3\cdot h60−36
y[n]y[n]24−365−56

Step-by-step values

y[0]=1(2)=2y[1]=1(0)+2(2)=4y[2]=1(−1)+2(0)+(−1)(2)=−3y[3]=1(2)+2(−1)+(−1)(0)+3(2)=6y[4]=2(2)+(−1)(−1)+3(0)=5y[5]=(−1)(2)+3(−1)=−5y[6]=3(2)=6\begin{aligned} y[0]&=1(2)=2\\ y[1]&=1(0)+2(2)=4\\ y[2]&=1(-1)+2(0)+(-1)(2)=-3\\ y[3]&=1(2)+2(-1)+(-1)(0)+3(2)=6\\ y[4]&=2(2)+(-1)(-1)+3(0)=5\\ y[5]&=(-1)(2)+3(-1)=-5\\ y[6]&=3(2)=6 \end{aligned}

Check: ∑y=15=(∑x)(∑h)=5×3\sum y=15=(\sum x)(\sum h)=5\times3.

Answer: y[n]={2, 4, −3, 6, 5, −5, 6}y[n]=\{2,\ 4,\ -3,\ 6,\ 5,\ -5,\ 6\}, with y[0]=2y[0]=2 (↑ at the first value).

If a different origin is chosen, the values stay the same; only the starting index changes (start index of yy = start of xx + start of hh).

✓ Answer checked by Claude Opus 5.5 (high effort)

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