Why does the Hilbert-Huang transform not suffer from the uncertainty principle, and what are the implications of this?

Question

I have been studying the Hilbert-Huang transform, which I understand is a frequency analysis technique that uses the adaptive IMFs basis components to decompose a signal using EMD.

I understand that frequency analysis techniques such as the STFT (short-time Fourier transform) and Wavelet transform involve the convolution of a signal with a predefined basis in order to extract frequency information.

Several articles from Huang and others state "convolution processes involve integration, which makes the results suffering the limitation imposed by the uncertainty principle".

http://web-static-aws.seas.harvard.edu/climate/pdf/Zhaohua.pdf https://pyhht.readthedocs.io/en/latest/tutorials/hilbert_view_nonlinearity.html

I also understand that the frequency of each IMF is extracted by finding the time rate of change of phase of the analytic signal for each IMF (which would involve differentiation of the phase with respect to time).

Why does this differentiation mean that the uncertainty principle is avoided?

I have a vague idea that it's due to operators associated with time and frequency that don't commute, but I have struggled to find any sources that approach this from a signal processing/mathematical perceptive rather than in a quantum mechanical sense.

Furthermore, if this kind of signal processing circumvents the uncertainty principle, does that mean time and frequency events can be resolved with arbitrary precision?

(In my head, this, in turn, would have implications in quantum mechanics, but I imagine this would be outside the topic of this site.)

Any insights and/or articles are gratefully received.

Answer 1

I'm not that familiar with the Hilbert-Huang transform, so I won't comment on that.

You seem to be under the impression that the DFT "suffers from the uncertainty principle". This is not true. Don't feel bad about thinking that as it is a widespread misunderstanding. This is a quote from an email to me from a well known expert in the field after I tried to get them to look at my exact frequency formula for a single (non-integer) pure real tone in a DFT:

An "exact" computation of frequency based on FTs is not possible for some very simple reasons: The Fourier Transform is itself an estimator, and the time-frequency uncertainty principle means frequency can't be determined exactly without an infinitely long observation window. So essentially all frequency computations of time series or signals are estimates, it's just that some are more accurate or more efficient than others.

So claiming that you have an exact solution just isn't going to get the attention of everyone interested in the field.

The first paragraph is entirely hogwash. The last sentence was prophetic, hence I started my blog so no one could dispute the math. You will find my original exact solution here:

• Exact Frequency Formula for a Pure Real Tone in a DFT

I have discovered, and written about, several more. You will find them among my articles. There are also a few in the pipeline, including a correction to Macleod's three bin formula which makes it exact also.

The FT does apply to the uncertainty principle in physics, but in a different manner. There they are talking about the transform of a Gaussian curve which happens to be an eigenfunction of the FT (but not the DFT, close though) and how it is the "narrowest function" in the transform.

I'm not an expert in Physics either, but I dabble. I consider this to be my greatest mathematical discovery ever (exclipses the simple math of the frequency formulas by a mile):

• Off Topic: Refraction in a Varying Medium

This is the vector form of Snell's law in a continuously varying index of refraction medium. It is a theoretical particle model so its applicability to light is still a subject of study for me. I believe it has the potential to illuminate subatomic behavior (i.e. the quantum world) but I haven't gotten there yet.