I would like to better understand why the instantaneous frequency estimation by Hilbert transformation works (and especially why it doesn't work / lead to precise results in many cases).
The motivation is to estimate signal $x(t)$ by decomposing it into an amplitude envelope $m(t)$ and phase of cosine $\omega_c (t)$ (or carrier waveform):
$$x(t) = m(t) \cos\left(\omega_c (t)\right)$$
Now, assume that $x(t)$ indeed is a result from such a process.
1) There are two "parameters" to be estimated for any $t$, as such some constraints are needed. What are the constraints regarding $m(t)$ and $\omega_c (t)$ that are selected when applying the Hilbert transform decomposition?
2) Is there a proof available somewhere that given the constraints, the estimation indeed finds the correct amplitude envelope and carrier (for continuous and also discrete case)?