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Convolution Calculator

This calculator computes the outcome of convolving two sets of data sequences. You can input up to 9 data terms for each sequence, and the resulting convoluted sequence will be outputted.

Calculation

Input

First Data Sequence:


a1
:1.00​



a2
:2.00​



a3
:3.00​



a4
:4.00​



a5
:5.00​



a6
:0.00​



a7
:7.00​



a8
:8.00​



a9
:2.00​

Second Data Sequence:


b1
:1.00​



b2
:2.00​



b3
:3.00​



b4
:4.00​



b5
:5.00​



b6
:6.00​



b7
:7.00​



b8
:8.00​



b9
:3.00​

Where:
  1. 
    
    are the variables/data input for the first data sequence
  2. 
    
    are the variables/data inputs for the second data sequence
  3. The output is the resulting sequence from the convolution of the two data sequences

Output

Sequence_1 
1
2
3
4
5
6
7
8
Sequence_2 
1
2
3
4
5
6
7
8
Convolution_results 
1
4
10
20
35
50
72
102
123
134
134
122
97
112
64


Explanation

Convolution is a widely used mathematical concept with applications in pure mathematics, such as probability theory and differential equations, as well as practical applications in acoustics, geophysics, signal processing, and computer vision. In signal processing, the impulse response of a linear time-invariant system (like an amplifier) can be used to predict the system's output for any input using convolution.
Let's take a look at convolution in detail. First, we can consider two sequences represented like:

a=[a]n=0∞a=[a]_{n=0}^{\infty}

b=[b]n=0∞b=[b]_{n=0}^{\infty}
👉Note, the use of

here means that their lengths can be arbitrarily long.
We define the convolution of sequence

and

as the sequence

, represented by:

c=[c]n=0∞c=[c]_{n=0}^{\infty}

where:

cn=∑k=0nakbn−kc_n=\sum_{k=0}^{n}a_kb_{n-k}
The term

can be calculated by computing the products of terms whose indices sum up to

. Compute the products

,

, etc up to

.
👉Note, the indices change simultaneously, i.e. the first increases while the second decreases.
Add the terms above to get

. The first few terms are:
  1. 
    
    
  2. 
    
    
  3. 
    
    
  4. 
    
    

Example

We will compute the convolution of two data sets,

and

, using the formula above to find the cn terms.
  1. 
    
    
    
    
  2. 
    
    
  3. 
    
    
  4. 
    
    
  5. 
    
    
Therefore, the convoluted sequence from the two data sets will be

.

Related Resources

  1. 🔗 Prime Number Checker
  2. 🔗 Poisson Distribution Calculator
  3. 🔗 Polar-Cartesian Coordinate Calculator