# Convolution Calculator

URL: https://speedycalculator.com/calculators/math/convolution/
Last updated: 2026-10-30

> The convolution calculator computes the discrete convolution of two finite sequences, c = a ∗ b, showing how each term is formed.

## Inputs

- Sequence a (comma-separated)
- Sequence b (comma-separated)

## Results

- a ∗ b
- Length
- Terms

## Formula

```
cₙ = Σₖ aₖ·bₙ₋ₖ,  length = len(a) + len(b) − 1
```

## Explanation

Convolution blends two sequences: it multiplies polynomials, smooths signals with moving averages, blurs and sharpens images, gives the distribution of the sum of two dice and is the core operation of convolutional neural networks.

### How to use the Convolution Calculator

- **Sequence a (comma-separated)**
- **Sequence b (comma-separated)**

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

Enter both sequences as comma-separated numbers. Each output term is the sum of the products of terms whose indices add up to n.

### Tips

- Write out the first few terms to check the pattern before using a formula.
- Decide whether the sequence starts at term 0 or term 1 – formulas differ by one place.
- Arithmetic sequences add a constant; geometric sequences multiply by one.
- An infinite geometric series only has a sum when |r| < 1.

### Common mistakes to avoid

- Off-by-one errors in the term number.
- Using the arithmetic sum formula for a geometric sequence.
- Ignoring the sign of a negative ratio or difference.

## Worked example

Inputs — Sequence a (comma-separated): 1, 2, 3; Sequence b (comma-separated): 0, 1, 0.5.

Results — **a ∗ b = 0, 1, 2.5, 4, 1.5**; Length = 5; Terms = c₀ = 1·0 = 0
c₁ = 1·1 + 2·0 = 1
c₂ = 1·0.5 + 2·1 + 3·0 = 2.5
c₃ = 2·0.5 + 3·1 = 4
c₄ = 3·0.5 = 1.5.

## Key terms

- **Convolution**: A sum of products with reversed indices.

## FAQ

**How is convolution related to polynomial multiplication?**
Multiplying polynomials convolves their coefficient lists.

**Is convolution commutative?**
Yes, a ∗ b = b ∗ a.

**What is the convolution of two dice distributions?**
The distribution of their sum.

**What is the length of the result?**
len(a) + len(b) − 1.

**What formula does the Convolution Calculator use?**
cₙ = Σₖ aₖ·bₙ₋ₖ,  length = len(a) + len(b) − 1.


## Sources

- [Wolfram MathWorld – Convolution](https://mathworld.wolfram.com/Convolution.html)