Convolution Calculator — Discrete Linear Convolution
Enter two finite sequences f and g to compute their linear convolution (f * g), the foundation of digital filtering, polynomial multiplication and signal processing.
len(f) + len(g) − 1
- 1
f length (M)
3 - 2
g length (N)
3 - 3
Output length (M + N − 1)
3 + 3 − 1 = 5
How does this calculator work?
Linear convolution slides a reversed copy of g over f, computing the dot product at each position. The output length is M + N − 1. Enter both sequences as comma-separated numbers; the calculator applies the formula (f * g)[n] = Σ f[k]·g[n−k] and plots the result.
Formula
How this is calculated
The discrete linear convolution of two finite sequences f (length M) and g (length N) produces an output of length M + N − 1. At each output index n, the value is the sum of all products f[k] · g[n − k] for every k where both indices are in range. Think of sliding one sequence (reversed) over the other, multiplying element-wise and summing — this is the standard "flip and drag" convolution.
Convolution is mathematically equivalent to polynomial multiplication: multiply the polynomial with coefficients f by the one with coefficients g, and the product coefficients are the convolution output. It is also the operation that filtering performs in the time domain when you apply a finite-impulse-response (FIR) filter to a signal.
This calculator performs full (linear) convolution with zero-padding at the boundaries, not circular convolution. Inputs are limited to a combined length of 32 to keep results readable. For large or complex-valued sequences, specialized numerical software is more appropriate.
Frequently asked questions
At each output index n, convolution sums the products f[k] · g[n−k] over all valid k — effectively sliding the reversed g over f and computing the dot product at each position. It describes how one sequence "blends" another.
For sequences of lengths M and N, the linear convolution output has exactly M + N − 1 elements — one more than the sum of both lengths minus two.
Multiplying two polynomials whose coefficients are f and g produces a polynomial whose coefficients are exactly the convolution (f * g). So computing convolution is identical to multiplying polynomials coefficient by coefficient.
Also known as
TG we-Calculate Editorial Team. (2026). Convolution Calculator — Discrete Linear Convolution [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/convolution-calculator
TG we-Calculate Editorial Team. "Convolution Calculator — Discrete Linear Convolution." TG we-Calculate. 2026. https://we-calculate.com/calculator/convolution-calculator.
TG we-Calculate Editorial Team, "Convolution Calculator — Discrete Linear Convolution," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/convolution-calculator
@misc{wecalculate_convolution_calculator, title = {Convolution Calculator — Discrete Linear Convolution}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/convolution-calculator}}, year = {2026}, note = {TG we-Calculate} }
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