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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: fft in hc11 again
References: <email@example.com> <3DA87610.DD963CF3@webaccess.net> <3DA883C2.222BF1A@SpamMeSenseless.pergamos.net> <3DA8A583.17B233A3@webaccess.net> <3DA8BFA7.B4D44823@SpamMeSenseless.pergamos.net>
Date: Sun, 13 Oct 2002 13:18:59 GMT
NNTP-Posting-Date: Sun, 13 Oct 2002 06:18:59 PDT
Phil Hobbs wrote:
> Chuck Simmons wrote:
> > False for several reasons. There does not exist a meaningful FFT for a
> > buffer length of 3998, for example. As far as I can remember, roundoff
> > error is theoretically identical except that the FFT is more sensitive
> > to systematic roundoff error. Roundoff error is determined by
> > computational depth, number of multiply-adds per point in the final
> > result. The FFT reduces computational width by reusing results at all
> > depths. The depth is the same.
> That isn't true. Chirp-z transforms can be used to compute DFTs
> efficiently even for prime lengths, and there are fast algorithms
> available for any highly-composite N.
My poimt was about roundoff error being due to computational depth. It
was astute of you to notice that 3998 happens to be twice a prime. What
about my point? You skipped that.
> >Data interpretation is further complicated by misconceptions. Too many
> >engineers think of the DFT and the FFT as giving results that relate to
> >the Fourier transform on the real line. Actually, the FFT and the DFT
> >give results identical to a truncated Fourier transform on the unit
> >circle. I've seen engineers chase wild geese for weeks because they
> >didn't understand that fundamental fact.
> (You're confusing DFTs and Z-transforms. It's the Z-transform that
> happens on the unit circle--the DFT happens along the real frequency
> axis. One is a conformal mapping of the other, so this is a fine
You are confused. The Fourier transform on the unit circle is usually
called the Fourier series. I direct your attention to any elementary
graduate textbook on harmonic analysis. I think that you are thinking
that the unit circle in what I said refers to the complex plane. I can't
imagine how you got that idea. I in no way implied that I meant the unit
circle in the complex plane. As to the z-transform, there are a lot of
ways to look at it. It naturally, of course, comes from probability
theory. It, like the Laplace transform which also cames from probability
theory, is useful in other problems.
> The DFT gives exact samples of the true, continuous-time Fourier
> transform of a function that is band-limited and periodic with period N
> samples. The error comes from failing to meet the conditions of the
> sampling theorem and failing to window the data to limit the wraparound
> error. It's true that people often aren't careful enough, but that
> doesn't make the DFT wrong.
Windowing is a method of smearing the buffer period. It is only useful
if the data has unknown period or is truly aperiodic. If a natural
period is known, it is a horrendous blunder not to use it.
The errors I see people make are quite different. In one case, I saw an
engineer insist that a glue joint was bad. He failed to realize that the
lack of 1/n amplitudes implied a magnetic problem in a motor. In the
long run, after weeks of messing with glue, the motor manufacturer was
contacted and the problem was soon corrected by a minor change in the
magnetics. The original glue was perfectly fine.
> With care, our ordinary continuous-time notions of bandwidth and 1-Hz
> SNR survive the transition into the sampled domain.
Not entirely. For example, one of my requirements for a dynamic analyzer
is that it use swept sine. This is because some systems cannot be
accuratly neasured with white noise and DFT (FFT) methods. The
manufacturers of such analyzers are perfectly aware of this fact and
supply instruments that use swept sine. Essentially, swept sine uses a
series of fixed frequency sinusoidal stimuli and then uses a single
point DFT to obtain the amplitude. The realization that performing the
complete transformation is futile (because it will be horribly in error)
Welcome to the real world of applied harmonic analysis.
... 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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