These are concepts I learned during a signal theory course. I don't remember them perfectly, but I will try to clarify your ideas and, for simplicity, I will use comparisons from the world of music so that simple examples can be made that everyone can understand, and which will surely interest other users who are not particularly fond of acoustics and signal theory; otherwise, it becomes a discussion between me, you, and a few others, whereas we should try to find a way to interest the rest of the users.
Correlation functions are functions that give us a statistical idea (regarding random signals) of how much two signals resemble each other. It is more important in random signal phenomena because finding the autocorrelation of a random signal means finding that signal, dictated by statistical laws, which is "simpler" and more linear than the initial signal. In short, to give you an example (since most topics are understood through examples), it is as if we took a recording in which there is a lot of background noise. Finding the autocorrelated signal would mean cleaning the recording of the background noise.
Obviously, this approach can be applied to both continuous and discrete signals, so in both the acoustic and digital fields.
The important approach for evaluating a signal is, first of all, knowing the signal we are going to work on. If the signal is deterministic, then one proceeds with mathematical analysis approaches; if the signal is random, then with statistical ones.
How do we proceed, however?
Suppose we have, for example, a continuous random signal. Autocorrelation is defined as the mean value of x(t) and x(t + tau). Therefore, if the wave is very slow, the result of the autocorrelation will probably be a signal very similar to the given one because x(t + tau) is nothing more than x(t) shifted to the right if tau is negative and shifted to the left if tau is positive. Since tau is very small, the mean value of two very close values falls statistically 99 times out of 100 in a very small neighborhood of the original value. E.g., (The mean value between 4 and 4.1 is 4.05, which is practically very close to 4, the original value).
If the signal is very fast instead, there is a very rapid change in sign (I remind you that the signal is always defined by an alternation of positive and negative values; therefore, in this case, since x(t) is probably very different from x(t + tau), the autocorrelation will have a sign statistically very close to zero. E.g., X(t) = 4.3; x(t + tau) = -3.7. The mean value will be 0.3, which is very close to zero. There are also other cases where both are positive, but in a random approach, what matters is the statistics that tend toward the differentiation of signs.
What does all this have to do with the point I made at the beginning about background noise? Now that we have clarified the result of autocorrelation in fast and slow continuous signals (anyway random ones, which are the interesting ones to study), we can assume that within a certain value of tau, the voice recording (which has frequencies between 500 and 2000 Hz) is a slow signal and the background noise (in the order of a few KHz) is instead a fast signal. Applying autocorrelation, we said that the resulting autocorrelated signal is a signal that statistically most of the time approximates to zero, and what remains is the voice which, being slower, produces a result that statistically is always similar to the original signal. We have attenuated the background noise. However, I want to point out another interesting application. By varying the value of tau, we could notice any periodicity in a signal; this is why in anti-noise filters, even in deEssers, it is possible to adjust temporal parameters to act on a certain type of noise present in some frequencies and not in others.
Finally, I remind you that in mathematical calculation, one must know the nature of the signal. In particular, in the case of continuous signals, one operates with the integral of the mean value, while in the case of discrete signals, obviously (to put it simply), with the summation operation.
I hope I have been helpful to you. Speaking about it brought many concepts back to me; it was also a way for me to dig this topic out of my memory.
