Convolution Calculator
Convolve two discrete sequences and see the full output, with the first few terms expanded so you can follow how each one is assembled.
How to use it
- Enter both sequences as comma-separated numbers.
- The output is always as long as the two inputs combined, minus one.
- The working shows how the first few output terms are built from overlapping products.
Examples
| [1, 2, 3] ∗ [4, 5, 6] | [4, 13, 28, 27, 18] |
| [1, 1] ∗ [1, 1] | [1, 2, 1] |
Discrete convolution combines two sequences by sliding one across the other, multiplying overlapping terms and summing at each position: y[n] = Σ x[k] · h[n − k]. It's the core operation behind digital filtering, image blurring and sharpening kernels, and — perhaps surprisingly — ordinary polynomial multiplication.
Worked example
Convolving [1, 2, 3] with [4, 5, 6]: the output has 3 + 3 − 1 = 5 terms. y[0] = 1×4 = 4. y[1] = 1×5 + 2×4 = 5 + 8 = 13. y[2] = 1×6 + 2×5 + 3×4 = 6 + 10 + 12 = 28. The remaining terms follow the same sliding pattern, giving [4, 13, 28, 27, 18].
Convolution as polynomial multiplication
Multiplying (x² + 2x + 3) by (4x² + 5x + 6) gives exactly the same coefficient sequence as convolving [1,2,3] with [4,5,6] — convolving coefficient lists is polynomial multiplication in disguise, which is why the same operation shows up in both signal processing and algebra.
Questions people ask
Why is the output longer than either input?
Because the sequences overlap in every possible alignment. Two sequences of length m and n produce m + n − 1 output terms.
What is convolution used for?
Filtering signals, blurring and sharpening images, and polynomial multiplication — multiplying two polynomials is exactly convolving their coefficients.
Is anything sent to a server?
No. Every calculation runs in your browser, so nothing you type is uploaded and the page keeps working offline once it has loaded.
Last updated 22 August 2026