Question

There are 2 signals A(t), B(t) representing output power. A is a time delayed output. How to implement this formula

 Cross_cor(delay)=[A(t + delay) − <A(t)>]*[B(t) − <B(t)>] / {[A(t)-<A(t)]^2 * [B(t)-<B(t)]^2} ^2

1. I do not understand what the numerator implies?
2. How to plot a color map or something like the surf plot whereby the different regions can be seen with different colors.
3. How to obtain a plot of the cross correlation as a function of the delay shift time (delay X axis, Cross_corr on Y axis) so that the peak can be visualized?

Answer 1

I’m gonna say [X] is the mean of X then:

R(dt) = [(A(t + dt) – [A(t)]) * (B(t) – [B(t)])]/(std(A) * std(B))

The numerator is taking two functions A(t) and B(t) and centering them, this is what subtracting the mean does. I can create two new functions that make this all look a little bit easier:

C(t) = A(t) – [A(t)]
D(t) = B(t) – [B(t)]

then:

R(dt) = [C(t + dt) * D(t)]/(std(C) * std(D))

Now I’m muliplying these two functions and taking the mean. If these two functions are moving exactly with one another this is going to be 1 which mean positive correlation.

You can do this in Matlab by doing the following:

a = randn(1,50);
b = randn(1,50);

x1 = mean((a - mean(a)) .* (b - mean(b)))/(std(a) * std(b))

This give me the crosscor at dt = 0, for another dt do:

x2 = mean((a(2:50) - mean(a(2:50))) .* (b(1:49) - mean(b(1:49))))/(std(a) * std(b))