I have a question regarding autocorrelation function and detecting periods of a signal. I would like to know/prove if the periods of the autocorrelation function can be found in the signal and vice versa. Any periodicity of the signal is a period of the autocorrelation function. The proof can be found here (second case) and is just to input the definition of a periodic signal into the definition of the autocorrelation function: https://de.wikipedia.org/w/index.php?title=Autokorrelation&diff=199179620&oldid=188846938#AFK_und_Periodizit%C3%A4ten
In the same proof they say, that the other way round also holds, meaning that any periodicity of the autocorrelation function is a period of the signal. I am not convinced by this part of the proof since they seem to use that if the integral is zero, one of the two product integrand functions has to be the zero function. Is that part right? If yes, what is the argument that you can conclude this in this case because in the general case of two convoluted functions it does not hold that if the convolution is zero that one of the convoluted functions has to be the zero function. If this part of the proof is not correct, does the statement still hold? If yes, what is the basic idea of the proof? Or can it be that the period of the signal is only 10s and the period of the corresponding autocorrelation function is 5s and 10s?
Thanks a lot Tim