Separate Simple Signal into Exponential Components
Your problem seems to be finding the analytic signal to a given real signal.

What you apparently need is the Hilbert transform. This will allow you to "recreate" the imaginary part of $y(t)$ from the real part alone, which is essentially your measurement $x(t)$.

Note that the transform can only be approximated as it is a acausal filter, and only works best when the signal $x(t)$ is available in the computer for all $t$.

A simple implementation is to Fourier transform $x(t)$ and set the negative frequency components to zero. After transforming back you obtain (the complex signal) $y(t)$.