Convolution Calculator
Enter Sequence A and Sequence B to compute their discrete convolution.
Hi, welcome to Hive Calculator! Convolution is a fundamental operation that combines two sequences to produce a third, showing up everywhere from signal processing and image filtering to probability and polynomial multiplication. Our Convolution Calculator lets you find the discrete convolution of any two numeric sequences instantly, without working through the sums by hand.
What Is Convolution?
Discrete convolution takes two sequences, Sequence A
and Sequence B, and combines them term by term to
produce a new sequence. Each output value is a sum of
products of overlapping terms from A and a
time-reversed, shifted copy of B.
If Sequence A has m terms and Sequence B has n terms,
the convolution result has (m + n − 1) terms — for
example, convolving a 3-term sequence with a 3-term
sequence produces a 5-term result.
The Convolution Formula
For two sequences a = [a₀, a₁, …, am−1] and
b = [b₀, b₁, …, bn−1], the discrete
convolution c = a * b is defined as:
c[k] = Σ a[i] × b[k − i], for i = 0 to k
where the sum only includes terms where both a[i] and
b[k − i] exist (i.e., 0 ≤ i < m and
0 ≤ k − i < n). The index k ranges from 0 to
(m + n − 2), giving (m + n − 1) output terms in total.
How to Use the Convolution Calculator
Using our Convolution Calculator is simple.
Enter Sequence A:
Type the numbers of your first sequence, separated by spaces
or commas (for example, 1 2 3).
Enter Sequence B:
Type the numbers of your second sequence the same way (for
example, 0 1 0.5).
Click ‘Calculate’:
The calculator instantly computes every term of the
convolution.
View the Results:
The full convolution sequence appears below, rounded to 6
decimal places, along with its length.
Clear or Share:
Use Clear to reset both inputs, or the Share icon to send your
result to others.
Example Calculation
Sequence A = [1, 2, 3], Sequence B = [0, 1, 0.5]
c[0] = a[0]·b[0] = 1×0 = 0
c[1] = a[0]·b[1] + a[1]·b[0] = 1×1 + 2×0 = 1
c[2] = a[0]·b[2] + a[1]·b[1] + a[2]·b[0] = 1×0.5 + 2×1 + 3×0 = 2.5
c[3] = a[1]·b[2] + a[2]·b[1] = 2×0.5 + 3×1 = 4
c[4] = a[2]·b[2] = 3×0.5 = 1.5
Result: [0, 1, 2.5, 4, 1.5] — a 5-term sequence, as expected from two 3-term inputs.
Applications of Convolution
Convolution is used to apply filters to audio and sensor signals, such as smoothing or echo effects.
2D convolution (an extension of this same idea) underlies blur, sharpen, and edge-detection filters.
Multiplying two polynomials is exactly equivalent to convolving their coefficient sequences.
The distribution of the sum of two independent discrete random variables is the convolution of their individual probability distributions.
Common Mistakes to Avoid
When computing convolution manually, people often:
- Forget to reverse (flip) the second sequence before sliding it across the first
- Miscount the number of output terms — it should always be (length A + length B − 1)
- Mix up convolution with simple term-by-term multiplication, which is a different operation
Using the Hive Calculator eliminates all these issues and computes every term precisely.
Why Use Hive Calculator's Convolution Calculator?
- Instant, accurate convolution for sequences of any length
- Accepts both integer and decimal values
- Results rounded to 6 decimal places
- Clean, user-friendly, horizontally aligned inputs
- Free to use — no signup or download required
If Sequence A has m terms and Sequence B has n terms, the convolution result always has exactly (m + n − 1) terms.
No. Convolution is commutative, so a * b gives the same result as b * a — you can enter your two sequences in either field.
Yes. Both sequences accept positive numbers, negative numbers, and decimals, separated by spaces or commas. Results are rounded to 6 decimal places.