To start with, we should focus our discussion on
**broken link removed** signals. We can extend the discussion to
random (or "stochastic") signals later if required.
So, let's state a simple point first:
If two zero-mean signals are uncorrelated, then their dot products are zero. In other words, zero-mean uncorrelated signals are geometrically "perpendicular" (or "orthogonal").
To visualise this, we can start with a really simple three-sample example. So, let's assume we have 2 signals: x
1(t) and x
2(t), where "t" denotes time. We make a very short digital recording of the signals, lasting for just 3 samples. For our simple example, let's assume these are the values we record:
Code:
x1 = [0;1;-1];
x2 = [-2;1;1];
We can confirm that these signals are zero-mean, using:
and seeing that they are both zero. Then we can check their correlation in the normal way:
Code:
covariance = (x1 - mean(x1)).'*(x2 - mean(x2));
variance1 = sum(abs(x1 - mean(x1)).^2);
variance2 = sum(abs(x2 - mean(x2)).^2);
correlation = covariance/sqrt(variance1*variance2)
which we find is zero.
Now, if we plot the 3D vectors x1 and x2 in Matlab, we can see that they look perpendicular:
and we can confirm this by checking that their inner product (dot product) is zero, using:
[Sorry... I will have to finish this post tomorrow. I have been in surgery today and need to rest now.]