Convolution Calculator

Convolving {1, 2, 3} with {1, 1, 1} gives {1, 3, 6, 5, 3}: a running sum over three samples.

Convolution:

LinearCircular (N-point)

Input x[n]:

Samples from n = 0, separated by commas.

Impulse response h[n]:

Linear convolution y[n] =

{1, 3, 6, 5, 3}

Length: 5
Sum of y = (Σx)(Σh): 18

Output samples
n	y[n]
0	1
1	3
2	6
3	5
4	3

Step-by-step solution

y[n] = Σₖ x[k] h[n − k]
Length = 3 + 3 − 1 = 5
y[0] = x[0]h[0] = 1

Convolution Formula

y[n] = Σₖ x[k] h[n − k]

length of linear convolution = L_x + L_h − 1; circular wraps modulo N

Worked Example

x[n] = {1, 2, 3} through h[n] = {1, 1, 1}:

y[n] = Σₖ x[k] h[n − k]
Length = 3 + 3 − 1 = 5
y[0] = x[0]h[0] = 1

Sum of y = (Σx)(Σh): 18

Output samples
n	y[n]
0	1
1	3
2	6
3	5
4	3

GATE tip

Circular convolution in time equals multiplication of DFTs. To get linear convolution with DFTs, zero-pad both sequences to at least L_x + L_h − 1 points first.

More Signals and Systems Calculators

Sampling Theorem Calculator

fₛ ≥ 2f_max

alias frequency f_a = |f − k fₛ| with k the nearest integer to f/fₛ

Example: A 4 kHz signal sampled at 6 kHz, and a 5 kHz tone in the same system: Nyquist rate 2f_max = 8 kHz.

Open the sampling theorem calculator →
DFT Calculator

X[k] = Σ x[n] e^(−j2πkn/N)

k = 0 … N − 1; X[0] is the sum of the samples

Example: The 4-point sequence x[n] = {1, 2, 3, 4}: X[0] (sum of the samples) = 10.

Open the dft calculator →

Frequently asked questions

How long is the output?

For linear convolution, L_x + L_h − 1 samples. Circular convolution of length N equals linear convolution only if N ≥ L_x + L_h − 1.

What is a quick check?

The sum of y equals the sum of x times the sum of h: 6 × 3 = 18 here.

Sources

The International System of Units (SI), 9th edition (SI Brochure), Bureau International des Poids et Mesures .

Last reviewed: 2026-10-15. Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.

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APA

Scientific Converters. (2026, October 15). Convolution Calculator. Retrieved October 25, 2026, from https://scientificconverters.com/calculators/convolution-calculator/

IEEE

Scientific Converters, "Convolution Calculator," scientificconverters.com. Accessed: Oct. 25, 2026. [Online]. Available: https://scientificconverters.com/calculators/convolution-calculator/