Combinations of characteristic functions: $\alpha\phi_1+(1-\alpha)\phi_2$
If $\phi(t)$ is the characteristic function of a random variable $X$, then $\Re(\phi(t))$ is also a characteristic function of some random variable $Y$ (This can be easily seen by Polya's criterion). How can we find $Y$? ($\Re(\cdot)$ detones the real part of a complex number)
I have already read this related question but I cannot figure out the solution. How can we find the characteristic function of $AX+(1-A)(-X)$?