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From: Chuck Simmons
Organization: You jest.
X-Mailer: Mozilla 4.61 [en] (X11; U; Linux 2.0.33 i586)
Subject: Re: How does a mixer work?
References: <email@example.com> <firstname.lastname@example.org> <3DB88D9C.50F75FA0@webaccess.net> <0f5u9.161$OM6.email@example.com>
Date: Fri, 25 Oct 2002 12:16:00 GMT
NNTP-Posting-Date: Fri, 25 Oct 2002 05:16:00 PDT
Kevin Aylward wrote:
> "Chuck Simmons" wrote in message
> > Kevin Aylward wrote:
> > >
> > > "john jardine" wrote in message
> > > news:firstname.lastname@example.org...
> > > > "Kevin Aylward" wrote in message
> > > news:...
> > > > > "Kevin Aylward" wrote in message
> > > > > news:o%Mt9.435$Zk2.email@example.com...
> > > > > > Well, I did explain that the frequency and time domain are
> > > > > fundamentally
> > > > > > incompatible by its uncertainty relation Sigma.Sigma >=1/2.
> And in
> > > > > view
> > > > >
> > > > > Sigma_F.Sigma_T>=1/2
> > >
> > > > That equation looks a bit portentious to me.
> > >
> > > That's only became you are not familiar with what is quite a basic
> > > result. Its pretty fundamental really.
> > >
> > > >Where did it come from?.
> > >
> > > A book. Any math text on fourier transforms for example.
> > >
> > > > Why the ">"?. Who figured it out?.
> > >
> > > Apparently Gabor.
> > >
> > > >Any references on the web?
> > >
> > > search on the combined "words time frequency uncertainty relation"
> > >
> > > http://www.mathpages.com/home/kmath152.htm
> > > http://sepwww.stanford.edu/sep/prof/fgdp/c4/paper_html/node2.html
> > Math text books on Fourier transforms, harmonic analysis, analysis,
> > functional analysis, differential equations (ordinary and partial),
> > integral equations and related areas are very unlikely to have the
> > relationship above because noise does not exist in these areas.
> Noise is not relevant. You need to try and understand what the relation
> really means.
> > Your
> > first reference was to sampling which is irrelevant to harmonic
> > and both mentioned noise also completely irrelevant to harmonic
> > and related mathematical subjects where the Fourier transform is
> > mentioned. The proper realm for noise is in statistics which is a
> > subject optional for math graduate students (I read probability
> > instead). No wonder I never saw it.
> These were simple two references are found to illustrate the concept.
> The relation itself has nothing to do with noise. It stands on its own.
> I don't see why anyone would have an issue with this, its obvious.
The issue is not with what the references said but with the specific
interpretation. There is no fundamental uncertainty in the Fourier
transformation itself. Uncertainty exists only if the time over which
integration takes place is not infinite. Fundamentally, the function of
time can be know a priori (no uncertainty) but obtaining the spectrum
still requires integration over infinite limits. Your second reference
clearly brings in a physical context in the first paragraph. The second
paragraph considers truncation of the intervals to meet conditions of
the physical context. The fact is that if you have a priori knowledge of
either the spectrum or the function of time, there is no uncertainty in
going from one to the other. In other words, the uncertainty
relationship is an expression of the fact that given uncertainty in the
function of time there is uncertainty in the spectrum. The uncertainty
relation is not fundamental to Fourier analysis itself but rather to
particular applications of it.
In the case of modulation, there does not exist uncertainty at the
transmitting end because the carrier and the message are known. The
uncertainty only exists at the receiving end. However, if part of the
message expresses the carrier function and the duration of the message,
the uncertainty vanishes at the receiving end.
None of this applies in any way to elementary understanding of amplitude
modulation in which the carrier is known and the message is taken to be
a constant amplitude constant frequency sinusoidal signal. The
complications of random information blah blah blah are gone and the
trigonometric identities are the full story.
... The times have been,
That, when the brains were out,
the man would die. ... Macbeth
Chuck Simmons firstname.lastname@example.org
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