I've googled and looked around on StackExchange but I can't quite find the answer. I'm trying to calculate the correlation matrix R. Below I've included a screenshot of some text that explains how. This is exactly my scenario. My actual textbook uses the same equation as found in 3.26, but I don't understand how to go from 3.26 to 3.27. It's clear to me that multiplying the column vector times the row vector will produce an L-by-L matrix, but I don't understand how I can take the expected value of this. To me it seems that taking the expected value would give me a L-by-1 matrix of the form:
$$ {1 \over L} \begin{pmatrix} (x_1(t)\times x_1^*(t))+(x_1(t)\times x_2^*(t))+...+(x_1(t)\times x_L^*(t)) \\ (x_2(t)\times x_1^*(t))+(x_2(t)\times x_2^*(t))+...+(x_2(t)\times x_L^*(t)) \\ \vdots \\ (x_L(t)\times x_1^*(t))+(x_L(t)\times x_2^*(t))+...+(x_L(t)\times x_L^*(t)) \\ \end{pmatrix} $$
I know this is wrong, but I don't really see any other way to do it. I don't have a really strong grasp of statistics but I would think this is how you take the expected value of a random variable. Thank you for your help!
