Convolution Calculator

What convolution computes

Convolution takes two sequences of numbers and blends them into a third. To get each output, flip one sequence, slide it across the other, multiply the numbers that line up, and add the products. Written out, output k equals the sum of a[i] x b[k - i] over every position i where both sequences have an entry.

That flip-and-slide recipe sounds strange until you see what it buys. It is the standard way to answer questions shaped like this: the effect now is the sum of the past effects, each weighted by how long ago it happened. MIT OpenCourseWare's Signals and Systems course builds filters, echoes and smoothing on exactly this operation, and Wolfram MathWorld describes it as the amount of overlap of one function as it shifts over another.

A small example shows the whole idea. Convolving 1, 2, 3 with 4, 5 gives 4, 13, 22, 15. Four outputs come out of six multiply-and-add steps, and the calculator above runs the same steps on any two sequences you type, including decimals and negative numbers.

A worked example, term by term

Take A = (1, 2, 3) and B = (4, 5). The first output lines up 1 with 4 and nothing else, so it is 1 x 4 = 4. The second lines up two pairs, 1 with 5 and 2 with 4, so 1 x 5 + 2 x 4 = 5 + 8 = 13. The third lines up 2 with 5 and 3 with 4, so 2 x 5 + 3 x 4 = 10 + 12 = 22. The last pairs 3 with 5, so 3 x 5 = 15. The result reads 4, 13, 22, 15.

Notice that the entries of B get used in reverse order as the slide progresses. The reversal is not a printing quirk in the textbook: it is what makes the operation fit time systems, where an effect that started k steps ago should be weighted by the response k steps back.

Negative numbers behave like any other entries. The third row mixes one negative value into the second sequence and gets negative outputs in the middle, and nothing about the method changes.

Length and arithmetic rules

Three counting rules cover every discrete convolution. The result has n + m - 1 outputs for inputs of length n and m, so 3 against 2 gives 4 outputs. The total number of multiply-and-add steps is n x m, so the example above uses 3 x 2 = 6 of them. And the outputs add up to the product of the input sums: (1 + 2 + 3) x (4 + 5) = 6 x 9 = 54, which matches 4 + 13 + 22 + 15 = 54. That sum check catches a flipped sign or a dropped term faster than recomputing everything.

Order does not matter either. Convolving 1, 2, 3 with 4, 5 produces the same four numbers as convolving 4, 5 with 1, 2, 3, because each of the n x m products appears exactly once in either direction. The calculator reports the same result if you swap the two boxes, which is a handy way to confirm a hand calculation.

Zero padding explains where the two end outputs come from. Pad the first sequence with zeros at both ends, set the flipped second sequence underneath, and each column of aligned pairs contributes one output. The padding is why the result runs longer than either input, and it is the mental picture to reach for when the counting stops adding up.

Polynomials hide a convolution

Multiply (1 + 2x + 3x^2) by (4 + 5x) and collect the terms. The constant term is 1 x 4 = 4. The x term collects 1 x 5 and 2 x 4, which adds to 13. The x^2 term collects 2 x 5 and 3 x 4, which adds to 22. The x^3 term is 3 x 5 = 15. The product reads 4 + 13x + 22x^2 + 15x^3, the same four numbers as the convolution example, in the same order.

This is why multiplying polynomials by hand and computing a convolution are the same exercise in different notation. It also hands you a free self-check for homework: any coefficient computed both ways has to agree. The calculator serves as that check, since the boxes accept a polynomial's coefficients and nothing else about the powers matters.

Dice, probability and moving averages

Roll two dice and add the faces. The distribution of that total is the convolution of the face distribution with itself. Each face turns up once in six, so convolving six values of 1/6 with six values of 1/6 produces the familiar pyramid: 7 is the most likely total with 6 ways out of 36, while 2 and 12 land once in 36 each.

The same operation smooths data. Convolve the readings 10, 20, 30, 40 with the two-term average 0.5, 0.5 and the outputs are 5, 15, 25, 35, 20: the second output is 10 x 0.5 + 20 x 0.5 = 15, each output averaging a pair of neighbors. Sound engineers run the idea at scale, convolving a dry recording with the impulse response of a concert hall to place the listener inside it.

Sequences that shift, scale and smooth

A few short sequences act like tools, and convolving with them is a quick way to check your work or to build a signal.

The shift row is the one worth memorizing. A sequence with a zero in front delays a signal without changing it, which is how systems textbooks write the sentence "the effect arrives k steps later." Combining a handful of these rows covers a surprising share of homework answers, and each row is fast to confirm in the calculator.

The rows chain, too. Convolving (5, 6, 7) with (0, 1) and then with (1, 1) gives 0, 5, 11, 13, 7, which is the shifted sequence with neighboring pairs added. Each step is easy to check on its own, so a long chain of small convolutions stays as trustworthy as the single four-output example.

Mistakes that produce the wrong sequence

Four habits cause nearly all wrong convolutions, and each one leaves a fingerprint in the output.

The first row is the common one, and skipping the flip even carries its own name, cross-correlation, with its own uses. The outputs differ in length as well as value: correlation returns 2 numbers where convolution returns 3, so a length mismatch is an early warning that the flip went missing.

Where convolution shows up beyond homework

The same sliding multiply-and-add runs a lot of the modern world. Audio plug-ins convolve a dry track with a concert hall's impulse response to add believable echo. Image editors blur or sharpen a photo by convolving it with a small grid of weights. Neural networks named for the operation, convolutional networks, slide learned filters across images to pick out edges and shapes. Statisticians reach for convolution whenever two independent random quantities add, which is why the dice table above looks the way it does.

The encyclopedia entry on convolution catalogs dozens more uses, from probability to differential equations to optics. The common thread is a system whose response spreads over time or space, so effects from different moments pile up. Once you can compute the four-output example by hand, the size of the application stops mattering, because the arithmetic never changes.

Long multiplication is the same arithmetic with digits. The partial rows of 123 x 45 = 5,535 shift and add the way convolution outputs do, which is why the schoolroom method and the convolution formula feel like cousins. The calculator keeps the sequence view instead of the digit view, but the products involved are identical.