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Everyone here should be familiar with the trignonometric identity

$$2 \cos(x)=e^{jx}+e^{-jx}$$

Let's say I have a real world analog signal given by the following function

$$x(t)=2\cos (\omega_1 t)$$

and I only want a single exponential component, specifically

$$y(t) = e^{j\omega_1 t}$$

where $\omega_1$ is a real constant. How to do this? Is it possible?

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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) $y(t)$.

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