Say I have a function, $C=C\left(x\right)$, whose fourier transform is denoted by $c=c\left(k\right)$, i.e. $C\left(x\right)=\sum_{k=-\infty}^{\infty}c\left(k\right)\chi\left(x\right)$, where $\chi\left(x\right)$ is some complex valued basis function.
I know that the autocorrelation is given by $M\left(\Delta x\right)=\sum_{k=-\infty}^{\infty}m\left(k\right)\chi\left(\Delta x\right)$, where $\Delta x = x_2-x_1$ and by the convolution theorem I know that $m\left(k\right)=\left(c\left(k\right)\right)^*c\left(k\right)$. This means that knowing the coefficients of the autocorrelation immediately gives me the squared magnitude of the original coefficients. This implies that any two functions $C'\left(x\right)=C\left(x-x'\right)$, that differ only by a translation, will possess the same autocorrelation, $M\left(\Delta x\right)$, and so if I am trying to obtain the $c\left(k\right)$ from the $m\left(k\right)$, then I can choose an arbitrary phase for the $c\left(k\right)$. However, I don't think I can choose the phase of $c\left(k\right)$ independently for each $k$, it seems like there needs to be some way to consistently choose the phase for all $k$ (which corresponds to specifying a translation for $C\left(x\right)$).
My question is: How do I choose a consistent phase for each of the coefficients for the original function, i.e. for all $k$?