The related information of correlation is mostly from David Dorran video at youtube, it is very clear and easy understanding.

What Is Correlation

Correlation is a measure of how similar signals are. They are present in things like discrete Fourier transform.

Which is always present as equation:

Eq. (1)

$$Cor{r}_{x,y}=\sum _{n=0}^{N-1}x[n]\cdot y[n]$$

Where $x[n]$ and $y[n]$ are two discrte function.

Here gives an example,

$x=$ $$\text{Missing end{bmatrix}}$$

$y=$ $$\text{Missing end{bmatrix}}$$

$z=$ $$\text{Missing end{bmatrix}}$$

plot them in matlab, we get this

From this figure we can find x is similar with y but not similar with z. How we can describe this in mathematics?

We substitute x, y and y, z into Eq. (1),

$$\begin{array}{rl}{\mathrm{Corr}}_{x,y}& =x[0]y[0]+x[1]y[1]+x[2]y[2]+x[3]y[3]\\ & =(1)(2)+(3)(3)+(-2)(-1)+(4)(3)\\ & =2+9+2+12=25\end{array}$$

and

$$\begin{array}{rl}{\mathrm{Corr}}_{y,z}& =y[0]z[0]+y[1]z[1]+y[2]z[2]+y[3]z[3]\\ & =(2)(2)+(3)(-1)+(-1)(4)+(3)(-2)\\ & =-9\end{array}$$

However if we substitute the first 2 in Z into 100, which will lead to $Cor{r}_{y,z}=187$, and we will become confused, because we can’t get clearly which two are more similar with the other. as seen in the figure:

So here we need Normalised Correlation.

$$ \operatorname{Corr}{norm{x,y}}=\frac{\sum_{n=0}^{N-1} x[n]y[n]}{\sqrt{\sum_{n=0}^{N-1} x^{2}[n] \sum_{n=0}^{N-1} y^{2}[n]}} $$

The two terms at denominator are the measure of energy in signal X and Y respectively, which make the denominator the overall scaling factor.

Basically, this normalised correlation function is this result divided by a scaling factor which is related to the energy that’s contained in the signals that you are measuring the similarity.

For $Cor{r}_{x,y}=0.95$ and $Cor{r}_{y,z}=0.38$, $Corr$ ranges in $[-1,1]$. Which can well score the similarity.

It’s worth noting that, sometimes, as seen in the code demo:

% This code highlights where non normalised correlation is
% be more beneficial than normalised correlation
t = [0:100-1]/100;
s1 = cos(2*pi*1*t);
s2 = cos(2*pi*4*t);
s3 = cos(2*pi*10*t);

% The following signals contain the three sinusoids above
a = 2*s1+4*s2+s3;
b = s1 + s2;

% Comparing the results it can be seen that non-normalised
% correlation is useful for identifying how strongly present
% one signal is in another
corr_res1 = sum(a.*s1)
norm_corr_res1 = sum(a.*s1)/sqrt((sum(a.^2).*sum(s1.^2)))

corr_res2 = sum(b.*s1)
nor_corr_res2 = sum(b.*s1)/sqrt((sum(b.^2).*sum(s1.^2)))

The output is:

>> NormalisedCorrelation

corr_res1 =

   100


   norm_corr_res1 =

       0.4364


       corr_res2 =

          50.0000


          nor_corr_res2 =

              0.7071