What's a Normalized function? I'm studing Representations of Random Processes and the book talks about Orthonormal functions, but doesn't make it clear what is it.
I was able to realize that a set of functions are orthonormal if they are normalized and orthogonal.
I'm familiar with the concept of orthogonal functions, but I couldn't find out what exactly is a normalized function in that case, since is a widely used term in math.
I'm studing using the following book: Detection, Estimation, and Linear Modulation Theory Part 1 by Harry L. Van Trees
Similar to a a vector that is normal when it's magnitude is 1, which is the inner product with itself. In the same fashion, a function is normal over a defined range $[t_1,t_2]$ when the root of the integrated squared magnitude is one (if it's normal the root is trivial but this shows the same form as derived from an inner product):
$$||x(t)|| = \sqrt{\int_{t_1}^{t_2}|x(t)|^2dt} = 1 $$
Normalization of functions is done with a simple scaling: If the above formula results in a number of than one, then the entire function is scaled by that number.