Subject: Re: Hilbert envelope bandwidth From: Eckard Blumschein <Eckard.Blumschein(at)E-Technik.Uni-Magdeburg.DE> Date: Fri, 1 Oct 2004 11:15:08 +0200Yadong Wang <ydwang(at)ELE.URI.EDU> wrote: > If bandwidth of bandpass filtered signal is B > > Then: > > The envelope (as defiend by Hilbert transform), log-envelope, > instantaneous frequency (time derivative of phasee) are not > band-limited. > > But it can be shown that: envelope squre and intensity weighted > instantaneous frequency (IWIF) are bandlimited with bandwidth = B. >> Given a signal x(n) with >> X(f) = 0 for |f| < f1 or |f| > f2 >> (bandpass filtered signal with bandwidth B = f2-f1) >> >> e(n) is the Hilbert envelope of x(n) which can then be written as: >> x(n) = e(n)y(n), >> >> where y(n) is the "temporally flattened" version of x(n). Dear Yadong Wang, Perhaps, I am not the only one here who would like to understand how Hilbert envelope differs from temporal envelope and what "temporally flattened" does mean. I am aware of Dan Ellis and others who calculate squared Hilbert envelope as squared magnitude of the analytic signal in order to depict hearing as determined by envelope and fine structure within a number of frequency bands. Smith, Delgutte, and Oxenham (letter to nature 2002) even spoke of an 'alternative signal decomposition by Hilbert slowly varying envelope and rapidly varying fine time structure'. Who introduced the term Hilbert envelope? I respect those who create new tools. However, I cannot confirm that so many confusing redundancy in theory is really justified. Let's tear down a lot of unnecessary sophistication after restricting to either really elapsed time or time to come after a given point. In other words, let's abandon the wrong belief that complex calculus must be immediately merged with frequency analysis. Complex modulator envelopes, as demanded by Atlas, Li, and Thompson at ICASSP 2004, are only then necessary prerequisites of unambiguous demultiplication if Fourier transform is used instead of cosine transform. Isn't it absurd to declare the modulating signal non-negative but operate with unreal negative frequency? Incidentally, misconception concerning band-limitation is widespread in science. It even led to 'measurement' of signals propagating with superluminal speed. Eckard Blumschein