Re: Hilbert envelope bandwidth

```Yadong Wang <ydwang@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).

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
```