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If I have two periodic functions so that $X = X(t)$ and $Y = X(t+\alpha)$, with $\alpha$ unknown is there a way to extract the value of $\alpha$ just from the difference of the two functions $X-Y$?

$X$ and $Y$ are not necessarily sine or cosine, but whatever periodic function.

311411
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Fabio Fuiano
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No, let $X(t)$ be any periodic function with period $p$. Then any $\alpha=np$ for integer $n$ results in the zero function:

$$X(t)-Y(t)=X(t)-X(t+\alpha)=0.$$

Golden_Ratio
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  • This is true when $\alpha=np$. I'm asking if it is possible to find $\alpha$ value with a mathematical expression starting from the difference of two delayed functions when $\alpha$ is not necessarily an integer multiple of p. – Fabio Fuiano Feb 20 '22 at 06:48
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    @FabioFuiano I suggest you specify your question more clearly in that case. $\alpha$ being a multiple of $p$ is not essential to this example. Let's say $\alpha=k$ was used to construct the difference (where $k$ is not a period multiple); then at best, you could infer $\alpha=k+np$ from observing the difference. – Golden_Ratio Feb 20 '22 at 10:27
  • Thank you very much for your precious help. Sorry if I wasn't clear in my question. – Fabio Fuiano Feb 21 '22 at 11:34