[Thank you, @leftaroundabout for your helpful comments to my related question. I'm trying to ask a less confusing question here.]
$0\leq\phi\leq 2\pi$ is a parameter that theoretically describes a system. $\alpha=\sin(\phi)$ and $\alpha' = \cos(\phi)$ are observables.
In a measurement, $\phi(t)$ is a continuous periodic function of the time $t$. $\alpha$ can be directly measured and $A_i$ is a noisy over-sampled digital representation of such a measurement. $\beta$ is a different observable, the relation between $\beta$ and $\phi$ shall be studied. $B_i$ is a sampled synchronously with $A_i$ and noisy.
Three questions are interesting and the answer shall be found from $A_i$ and $B_i$:
What is $\delta = \beta(\phi=-\pi/2) - \beta(\phi=+\pi/2)$? This the difference of the $\beta$ values corresponding to the extrema of $\alpha$.
Is $\beta = a * \alpha + c$ a valid model and what is the coefficient $a$?
Can we give a confidence interval for $a$ or $\delta$?
Inspection of the data shows that a better model for $B_i$ is $\beta' = a * \alpha + a' * \alpha' + c$. Does this affect the recipe to find $\delta$ or $a$?
A third alternative model is $\beta'' = f(\phi) + c$. How to reliably find $\delta$?
The nature of the noise is unknown. The noise spectrum has both frequencies less then the modulation frequency of $\phi$ and higher frequencies. A Fourier filter (e.g. lowpass) renders $A_i$ and $B_i$ reasonably smooth.