Mechanics of convolution

To demonstrate the mechanics of convolution we are going to convolute two small signals by hand

first signal                       1 2 3
second signal                      2 1 3

The first step is to reflect one of them

first signal (reflected)     3 2 1 
second signal                      2 1 3

The second step is to displace one of them progressively and to add the product of both signals

\ttfamily

first signal                   3 2 1
second signal                      2 1 3

product                        0 0 2 0 0 0 0
sum                       2 <---------------



first signal                     3 2 1
second signal                      2 1 3

product                        0 0 4 1 0 0 0
sum                       5 <--------------- 



first signal                       3 2 1
second signal                      2 1 3

product                        0 0 6 2 3 0 0
sum                       11 <--------------



first signal                         3 2 1 
second signal                      2 1 3

product                        0 0 0 3 6 1 0
sum                        9 <--------------



first signal                           3 2 1
second signal                      2 1 3

product                        0 0 0 0 9 0 0
sum                       9 <---------------

We can check it with Scilab.

-->convol ([1 2 3], [2 1 3])
 ans  =
 
!   2.    5.    11.    9.    9. !