About the Convolution Calculator
Convolution combines two sequences into a third by sliding one across the other and adding up the products where they overlap. It is the basic operation behind filters in signal processing, blurring in image editing, probability of sums and the multiplication of polynomials. This convolution calculator takes two lists of numbers and gives the full result, or the same-length or valid-only parts used in numerical libraries. It also shows the sequences as polynomials so you can see that convolving coefficient lists is exactly the same as multiplying the polynomials.
Updated October 2026
How to use the Convolution Calculator
- 1Type sequence A as numbers separated by commas or spaces.
- 2Type sequence B the same way.
- 3Choose the full, same-length or valid output.
- 4Read the result, which has one number per output position.
- 5Check the polynomial line to see the same calculation as algebra.
A worked example
Convolve a = [1, 2, 3] with b = [0, 1, 0.5]. The first output is 1 times 0, which is 0. The second is 1 times 1 plus 2 times 0, which is 1. The third is 1 times 0.5 plus 2 times 1 plus 3 times 0, which is 2.5. Continuing gives [0, 1, 2.5, 4, 1.5]. You can check the middle values by writing the sequences as polynomials and multiplying them out.
Common patterns to try
- Convolve with [1, 1, 1] and divide by 3 for a three-point moving average.
- Convolve [1, 1, 1, 1, 1, 1] with itself to see how two six-sided dice add up.
- Convolve with [1] and the sequence is unchanged, so [1] is the identity.
- Convolve with [0, 1] to shift a sequence one step later.
Related tools
The Matrix Calculator handles linear algebra, the Probability Calculator covers dice and sums, the Standard Deviation Calculator and Average Calculator summarise the data you filter, and the Complex Number Calculator is useful for frequency-domain work. Large sequences of thousands of values are better handled in dedicated software.
Features
- Discrete convolution of any two sequences
- Full, same-length and valid outputs
- Polynomial multiplication view
- Handles decimals and negative values
- Sequences up to 500 numbers
- Copy the result as a list
- Runs in your browser
Frequently asked questions
What is convolution?
Discrete convolution of sequences a and b gives a new sequence where each element is the sum of a[k] times b[n minus k] over all k. Picture flipping one sequence, sliding it along the other and adding the products at each position. The result is longer than either input.
How long is the result?
The full convolution of a sequence of length m with one of length n has m plus n minus 1 values. Convolving 3 numbers with 3 numbers gives 5. The same-length and valid options trim the full result to match the first sequence, or to only the part where the two fully overlap.
What is the difference between full, same and valid?
Full returns every position where the sequences overlap at all. Same returns the central part with the same length as the first sequence, as used in filtering. Valid returns only the positions where the shorter sequence lies completely inside the longer one, which avoids edge effects.
How does convolution relate to multiplying polynomials?
If the sequences are the coefficients of two polynomials, their convolution gives the coefficients of the product. For example, the polynomial 1 + x times 1 + x equals 1 + 2x + x squared, and the convolution of 1, 1 with 1, 1 is 1, 2, 1. The tool prints this view for you.
Where is convolution used?
In audio and image filters, neural networks, smoothing data with a moving average, finding the distribution of the sum of two dice, and in the theory of linear systems, where the output equals the input convolved with the system's impulse response.
Is convolution the same as correlation?
They are close but not identical. Correlation slides one sequence over the other without flipping it, while convolution flips one first. For symmetric sequences they give the same result, and for others you can reverse one input to turn one into the other.