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Convolution Calculator

Convolution Calculator

Enter first data sequence, separated with comma (,):

Enter second data sequence, separated with comma (,):

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An online convolution calculator will combine two different data sequences into a single convolution data sequence quickly. This discrete convolution calculator does point-to-point multiplication of given functions. Read on to know more about how to find convolution function with its formula and example.

What is Convolution?

In mathematics, convolution is the mathematical operation of two functions (a and b), which creates a third function (a * b) that represents how the shape of one function is modified by another function. It is also defined as the integral of the product of any two functions after one function is inverted and shifted. So, calculate the integral of all offset values ​​to obtain the convolution function. This is a special type of integral transformation:

$$  (a * b) (t)  := ∫_-∞^∞b(r) a(t – r) dr $$


However, an Online Integral Calculator helps you to calculate the integrals of the given functions with respect to the variable involved.

How to Calculate Convolution?

You can find the result data sequence with our convolution calculator by substituting the same values.

Example 1:

Write first data sequence (A) = 1, 0, 1, 0, 0

Then, write second data sequence (B) = 0.2, 0.5, 0.6

By using convolution formula

$$ (a * b) (t)  := ∫_-∞^∞b(r) a(t – r) dr $$

Thus, result data sequence (C) = 0.2, 0.5, 0.8, 0.5, 0.6, 0.0, 0.0

Example 2:

Put the first data sequence (A) = 1, 1, 1, 0, 0, 0

Now, put second data sequence (B) = 1, 1, 1, 0, 0, 0

With the help of convolution integral formula

$$ (a * b) (t)  := ∫_-∞^∞b(r) a(t – r) dr $$

Hence, the result data sequence (C) = 1, 2, 3, 2, 1, 0, 0, 0, 0, 0, 0

Circular Convolution:

If a function m is periodic with time period T, then the period for functions n, is the n ∗ mT, so the convolution is an periodic and identical to:

$$ (f * mt) (t)  := ∫_t_0^{t_0 + t} [ ∑_{k=-∞}^∞ f(r + k T)] gt (t – r) dr $$

Where t_0 is an arbitrary so, the summation is known as a periodic summation of a function n.

When mT is a periodic summation of the other function m, then n ∗ mT is known as a cyclic convolution of m and n. And if the periodic summation is replaced by nT, then this operation is called the periodic convolution of nT and mT.

However, an Online Arithmetic Sequence Calculator allows you to evaluate the Arithmetic sequence, nth value, and sum of the arithmetic sequence.

Discrete Convolution:

For the complex-valued functions n, m that defined on the set A of integers, the discrete convolution of n and m is:

$$ (f * g) [n] = ∑^∞_{m= -∞} f [m] g [n – m] $$

The convolution method of two finite sequences is determined by expanding the sequence into a function with limited support in the set of integers. If these sequences are the coefficients of two polynomials, then the coefficients of the usual product of the two polynomials are the convolution of the two original sequences. It is called the Cauchy product of these sequence coefficients.

Algebraic properties:

Convolution defines the product in the linear space of the integrable function. The product satisfies the following algebraic properties. Formally speaking, the space of the product integrable function given by the convolution is a commutative correlation algebra without units.


The discrete convolution calculator use this formula for commutativity of given functions.

$$ a * b = b * a $$

$$ (a * b) (t) := ∫_-∞^∞a(r) b(t – r) dr $$


$$ a * (b * c) = (a * b) * c $$


$$ a * (b + c) = (a * b) + (a * c) $$

Associativity with scalar multiplication

$$ a (b * c) = (ab) * c $$

a for any real number.

Multiplicative Identity:

$$ m * δ = m $$

where δ is delta distribution.

Inverse Element:

Some distributions S have an inverse element \( S_{−1} \) for the convolution which then must satisfy

$$ S^{-1} * S = δ $$

From which an explicit formula for S^{−1} may be obtained.

How Convolution Calculator Works?

An online discrete convolution calculator helps you to compute the convolution function of given functions by following these steps:


  • First, enter given data sequences in both boxes. Remember, sets are separated with a comma.
  • Hit the calculate button to see the convolution function.


  • The convolution calculator online provides given data sequences and using the convolution formula for the result sequence.
  • Click the recalculate button if you want to find more convolution functions of given datasets.


What is deformable convolution?

Convolution is used to build a 2N number of feature maps corresponding to N 2D offsets ∆pn (the x and y direction of each offset).

Define the convolution sum?

The unit step function can be expressed as the sum of offset unit pulses. The overall response of the system is called the CONVOLUTION SUM or superposition sum of the sequence a[n] and b[n].

What is convolution physically?

The physical meaning is that the signal passes through the LTI system. Convolution is defined as flipping (one of the signals), shifting, multiplying, and adding.

What is the difference between circular and linear convolution?

Linear convolution is the basic operation of calculating the output of a linear time-invariant system based on its input and impulse response. Circular convolution is the same but considering that the signal support is periodic.

What are the benefits of circular convolution?

This occurs in a continuous-time when the sum of convolutions is equal to 1, or in discrete time using vectors, where the sum is a sum. This also applies to functions defined from -Inf to Inf, or functions with a specific duration.


Use this Convolution Calculator online because it computes the Convolution matrices of two given matrices with the help of its formula. Apart from this, convolution is a mathematical operation that applies to the two function values such as A and B and produces a third function as a result like C that describes how the shape of one function changed by the other function. So, in convolution, our calculator does point-to-point multiplication of given datasets.


From the source of Wikipedia: Notation, Derivations, Historical developments, Circular convolution, Discrete convolution, Circular discrete convolution.

From the source of Better Explained: Hospital Analogy, Intuition For Convolution, Interactive Demo, Application: COVID Ventilator Usage, Convolution reverses.

From the source of Science Direct: Convolution Theorem, correlation coefficients, convolutions and multiplication of Fourier coefficients, correlation coefficients.