Mechanics of correlation

To demonstrate the mechanics of correlation, we are going to calculate the convolution of two small signals by hand

first signal                         1   2   3
second signal                        2   1   3

First of all we subtract the mean of both signals.

first signal (minus mean)           -1   0   1
second signal (minus mean)           0  -1   1

Then we have to displace progressively the second signal and to add the product of both signals

first signal (minus mean)           -1   0   1
second signal (minus mean)           0  -1   1 

product                              0   0   1
sum                         1 <---------------


first signal (minus mean)           -1   0   1
second signal (minus mean)       0  -1   1        

product                              1   0  
sum                         1 <---------------

first signal (minus mean)           -1   0   1
second signal (minus mean)   0  -1   1                

product                             -1 
sum                        -1 <---------------

And finally we divide the result by the length of the array

previous result                   1     1    -1
divided by 3 (end result)        1/3   1/3  -1/3

We can check the previous result with the next code

-->a = [1 2 3]                
 a  =
 
!   1.    2.    3. !
 
-->b = [2 1 3 ]               
 b  =
 
!   2.    1.    3. !
 
-->corr(a,b,3)
 ans  =
 
!   0.3333333    0.3333333  - 0.3333333 !

We have to notice that correlation is not commutative

-->corr(a,b,3)
 ans  =
 
!   0.3333333    0.3333333  - 0.3333333 !
 
-->corr(b,a,3)
 ans  =
 
!   0.3333333  - 0.3333333    0. !