How this instrument works
Convolution combines two sequences into a new one by summing the products of every pair of terms whose positions add up to the same output index. For two 2-term sequences (a₁,a₂) and (b₁,b₂), the result has three terms: c₀=a₁b₁, c₁=a₁b₂+a₂b₁, and c₂=a₂b₂ — each output collecting every input pair that 'lines up' at that position as one sequence conceptually slides past the other.
This same sliding-and-summing structure underlies several seemingly unrelated calculations: multiplying two polynomials produces coefficients that are exactly a convolution of the two original coefficient lists; combining two independent probability distributions (like the result of rolling two dice) to find the distribution of their sum uses the identical operation; and signal processing filters a signal by convolving it with a filter's own response.
This page is scoped to the smallest non-trivial case, two 2-term sequences, specifically to make the pattern visible without the sheer bulk of a longer real-world convolution — the identical multiply-and-sum idea extends to sequences of any length, just with more terms landing at each output position as the sequences grow.
- Enter the first sequence's two terms into the Sequence a fields.
- Enter the second sequence's two terms into the Sequence b fields.
- Read Result: c₀, c₁, and c₂: the sheet computes the three-term convolution directly.
Worked example — (1,2) convolved with (3,4)
Convolving (1,2) with (3,4) gives (3, 10, 8): c₀=1×3=3, c₁=1×4+2×3=10, and c₂=2×4=8 — each output term summing every input pair whose positions add up to that same index.
Convolving (2,3) with itself gives (4, 12, 9) — matching exactly the coefficients of (2+3x)² expanded, 4+12x+9x², confirming that convolution and polynomial multiplication are the identical operation viewed from two different angles.
Questions
What is convolution?
An operation that combines two sequences into a new one by summing the products of every pair of terms whose positions add up to the same output index — a sliding, multiply-and-sum process.
How is convolution related to polynomial multiplication?
They're the identical operation — multiplying two polynomials and collecting like terms produces coefficients that are exactly the convolution of the two original coefficient sequences, term for term.
Where does convolution show up in probability?
Combining two independent random quantities (like the results of rolling two dice) to find the probability distribution of their SUM uses convolution directly — the probability of each possible total is a sum of products across every combination that reaches it.
Why is this page limited to 2-term sequences?
To make the underlying pattern visible at the smallest useful scale, without the bulk of a longer real-world signal or polynomial. The identical multiply-and-sum structure extends to sequences of any length, with more terms contributing to each output position.
Does the order of the two sequences matter?
No — convolution is commutative, meaning (a) convolved with (b) always gives the identical result as (b) convolved with (a), the same way ordinary multiplication doesn't care which number comes first.