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