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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 "phase signal". 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 "phase estimated from magnitude signal using the Hilbert transform", 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?

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  • $\begingroup$ hm, I don't see why you need a term. One is the original phase signal, the other is the estimated one. They aren't the same. That happens to any estimator. $\endgroup$ – Marcus Müller Aug 7 '20 at 16:57
  • $\begingroup$ Also, what's the motivation to ask us for a name? Is there something you want to research? Maybe we can help, if you told us what kind of signal you have. $\endgroup$ – Marcus Müller Aug 7 '20 at 16:58
  • $\begingroup$ The reason I'd like to have a term is because the issue is muddied in the field. If someone says "phase" with fMRI data, no one knows if they're referring to actual phase signal or this Hilbert-based phase signal. I'm also not 100% sure that analytic signal estimated from magnitude data is conceptually an estimate of the original complex data, but I don't know enough about it to say. Wouldn't applying the transform to the real signal be more like estimating the original complex data? $\endgroup$ – Taylor Aug 7 '20 at 17:23
  • $\begingroup$ no not really. The Hilbert transform really just gives you the analytical signal to a real-valued signal. That's not a sensible estimate of the phase of your signal, unless you know the original signal was the analytical signal before. So, people will say "analytical signal" when they mean that. $\endgroup$ – Marcus Müller Aug 7 '20 at 17:40
  • $\begingroup$ To clarify, the Hilbert transform applied to the real component of a complex signal is not a good estimate of the original complex signal? If so, then is there some way to make this clear, as well as to clarify that it's phase estimated by treating magnitude signal like real signal? Like calling it "pseudo-phase", "Hilbert-derived phase", "estimated phase", etc.? $\endgroup$ – Taylor Aug 7 '20 at 22:59
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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 "estimate" 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.

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    $\begingroup$ Maybe this is just me but I can’t even think of a model by which taking the Hilbert transform of a magnitude signal would give you a reasonable estimate of phase. It feels like someone read enough about it to say, “oh, the Hilbert transform gives phase”, without checking to see if it actually makes sense. $\endgroup$ – Dan Szabo Aug 8 '20 at 15:05

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