Compute the discrete convolution of two number sequences of any length.

How to Use the Convolution Calculator

Build Sequence A and Sequence B independently, adding or removing terms with the button under each list — the convolved output recalculates every time either sequence changes.

Convolution is more specialized than most of the calculators in this batch, so it's worth being concrete about what it does before the formula: flip one sequence, slide it across the other one position at a time, and at each position multiply the overlapping terms together and add them up. That single number becomes one entry in the output; sliding one more step produces the next.

output[k] = Σ A[i] × B[k−i],  for every i where both A[i] and B[k−i] exist

With the calculator's defaults, A = [1, 2, 3] and B = [0, 1, 0.5], the output comes out to [0, 1, 2.5, 4, 1.5] — five values, since two sequences of length 3 convolve into one of length 3 + 3 − 1.

One clean, checkable way to see convolution at work: it's exactly how you'd multiply two polynomials by their coefficients. Treat A as 1 + 2x + 3x² and B as 0 + 1x + 0.5x², multiply them out the long way, and the coefficients of the product land on that same [0, 1, 2.5, 4, 1.5] — convolution and polynomial multiplication are the same operation wearing different notation.

Outside pure math, this is the mechanism behind a reverb effect layered onto audio (the dry signal convolved with a recorded "impulse response" of a physical room), and behind blurring or sharpening an image in graphics software, where a small grid of numbers — a kernel — gets convolved across every pixel in turn.