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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: <3DB41488.email@example.com> <3DB43497.A32B8380@webaccess.net> <3DB45343.E61C1CCB@webaccess.net> <3DB466BC.593D4D35@webaccess.net> <3DB73CD9.D99E28FF@webaccess.net>
Date: Thu, 24 Oct 2002 11:48:46 GMT
NNTP-Posting-Date: Thu, 24 Oct 2002 04:48:46 PDT
Kevin Aylward wrote:
> "Chuck Simmons" wrote in message
> > Kevin Aylward wrote:
> > >
> > > "John Woodgate" wrote in message
> > > news:TeLxB9BOKst9EwnI@jmwa.demon.co.uk...
> > > > I read in sci.electronics.design that James Meyer
> > > > wrote (in
> > > > > > firstname.lastname@example.org>) about 'How does a mixer work?', on Wed, 23 Oct 2002:
> > > >
> > > > > Correct! Since a Dirac pulse contains all the information
> > > known, or
> > > > >which will be known; all the transmissions that have ever been
> > > produced, or will
> > > > >be produced, are redundant.
> > > >
> > > > Well, not quite correct, because no-one has ever produced a Dirac
> > > pulse.
> > > > We are waiting for some transistors with infinite Vcbo. (;-)
> > >
> > > Well, not even that. A dirac pulse contains no useful information.
> > > certainly true that for a signal to have information content it must
> > > have some randomness associated with it, i.e. log2(/pi), and that as
> > > consequence, a random signal has a frequency spectrum. It does not
> > > necessarily follow that anything with a complete spectrum contains
> > > useful information. White noise, is just that, white noise.
> > It's worse. It is trivial to prove that there does not exist a
> > integrable function having the properties of the "Dirac delta." The so
> > called approximating sequences are all divergent and therefore do not
> > converge to any function. In short, there does not exist a function
> The Dirac function, as you say is not a function. It is a distribution.
> It can be treated entirely rigorously as a limit of a sequence of
> integrals. Agreed, the limit of the sequences themselves diverge, but
> this is not relevant. Its the limit of the integral that matters (e.g.
> Mathematical Methods for Physicists - Arfken, p.84-88) The Dirac
> sequence (function) has no technical meaning outside of an integral.
The approach you are talking about does not define the Dirac delta as a
functional. It is properly referred to as the definition of a unit
approximation (this approach is usual in harmonic analysis where an
exact representation is not needed). It avoids the problem of the
integral not converging properly because the limit is clearly the
integral of a function that does not exist. This avoids defining
anything. Lars Hoermander in the first edition of "Linear Partial
Differential Operators" has a concise treatment of distribution theory
which he expanded to one whole volume in the second edition (four
volumes). Hoermander uses the rigorous method of Laurent Schwartz which
does not use integration. It is true that many functionals are integrals
but not all. In probability theory, the whole idea is simplified by
introducing atomic measures (see various graduate texts in probability).
In this case, one simply takes it that there exist measures that do not
have a corresponding density function. Essentially, you can't really
rigorously define the Dirac delta distribution through the
... The times have been,
That, when the brains were out,
the man would die. ... Macbeth
Chuck Simmons email@example.com
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