Terminology for instantaneous phase of magnitude component of complex signal In my field, we deal with data that are originally complex-valued. Typically, researchers convert their data from real + imaginary to magnitude + phase, and then discard the phase data (i.e., we generally only deal with the magnitude portion of a complex signal).
There are a number of papers on phase synchrony, in which the Hilbert transform is applied to this magnitude-only data to estimate the analytic signal, from which instantaneous phase is extracted. I am very much a novice in signal processing, so I found this a little confusing. The papers refer to the instantaneous phase signal estimated from magnitude-only data just as &quot;phase signal&quot;. These folks might not be aware of the complex-valued nature of the raw signal, which is reasonable given how these data are generally handled, but I think there should be a more appropriate term for &quot;phase estimated from magnitude signal using the Hilbert transform&quot;, given that there's very little similarity between the real phase signal and this estimated phase signal.
Is this an established concept and, if so, is there a term that would differentiate the two kinds of phase signal?
and then discard the phase data

What's gone is gone. You can't reconstruct the phase of the original data unless you have some additional information.

in which the Hilbert transform is applied to this magnitude-only data to estimate the analytic signal, from which instantaneous phase is extracted.

That's a just mathematical process. You shove data in, and data comes out. Whether the result is useful or not depends on what you want to do with it and why you chose this mathematical process in the first place. If you want to associate any physical meaning with the result you need to justify this with some physical property or law.

To clarify, the Hilbert transform applied to the real component of a complex signal is not a good estimate of the original complex signal?

Correct. In general most signals are NOT analytic unless there is a strong reason why they should. If there is a reason to interpret the results that way, it should be stated in the paper. In order to &quot;estimate&quot; a discarded phase, you need have some additional information or (justified) assumptions about your original signal otherwise you might as well just toss the dice. If there is a additional information, it should be stated AND the phase estimation method should be properly derived from it.