The convolution of ƒ and g is written ƒ∗g, using an asterisk or star. It is defined as the integral of the product of the two functions after one is reversed and shifted. As such, it is a particular kind of integral transform:

	(commutativity)

While the symbol t is used above, it need not represent the time domain. But in that context, the convolution formula can be described as a weighted average of the function ƒ(τ) at the moment t where the weighting is given by g(−τ) simply shifted by amount t. As t changes, the weighting function emphasizes different parts of the input function.

More generally, if f and g are complex-valued functions on R^d, then their convolution may be defined as the integral:

In Digital Signal Processing where convolution is done between two discrete signals the procedure followed is :

There are different methods used in finding convolution (refer Signals And Systems by  Openheim for more details on each method). Here i will demonstrate a C program that uses the matrix method to find convolution.

The CODE

/* Vineeth Kartha
Program To Find Convolution
Released Under GPL */
#include<stdio.h>
int main()
{
printf(“\n\tThis program finds the convolution of a signal and its Impulse response\n”);
printf(“—————————————————————————\n\n”);
int i,j,k,x[10],h[10],y[10],a[10][10],n,m;
printf(“enter the number of elements in x[n]:\t”);
scanf(“%d”,&n);
printf(“enter the elements of x[n]:\n”);
for(i=0;i<n;i++)
{
scanf(“%d”,&x[i]);
}
printf(“enter the number of elements in h[n]:\t”);
scanf(“%d”,&m);
printf(“enter the elements of h[n]:\n”);
for(i=0;i<m;i++)
{
scanf(“%d”,&h[i]);
}
for(k=0;k<m;k++)
{
for(i=0;i<n;i++)
{
a[k][i]=h[k]*x[i];
}
}
for(k=0;k<(n+m-1);k++)
{
y[k]=0;
for(i=0;i<m;i++)
{
for(j=0;j<n;j++)
{
if((i+j)==k)
{
y[k]+=a[i][j];
}
}
}
}
printf(“\n\tx[n]={“);
for(i=0;i<n;i++)
{
printf(“%d “,x[i]);
}
printf(“}\n\th[n]={“);
for(i=0;i<m;i++)
{
printf(“%d “,h[i]);
}printf(“}\n\n\n\ty[n]=x[n]*h[n]\n——————————————\n\n\ty[n]={“);for(i=0;i<(n+m-1);i++)
{
printf(“%d “,y[i]);
}
printf(“}\n\n”);
return 0;
}