Richard Clark wrote:

On Tue, 08 Feb 2005 23:45:52 GMT, Gene Fuller
wrote:


Richard,

It is clear that you are not discussing a general Fourier Transform.
Everything you state below, while correct, refers to Discrete Fourier
Series analysis and Discrete Fourier Transforms, including FFT.

More generally, integral Fourier Transforms are widely, rigorously, and
correctly used to analyze pulse phenomena that are definitely not periodic.

The original spark for this thread was an aperiodic pulse. You chose to
"mandate" all sorts of conditions that may be useful in a particular
case but are not required for theoretical correctness.

73,
Gene
W4SZ


Hi Gene,

The simple truth to the matter would be resolved in your offering the
"general" Fourier Transform that could accomplish this feat of
rendering the spectrum of an aperiodic pulse without having to tailor
the waveform. I will lead the way instead.

I would note, ironically, that this data would be discrete (not
continuous) and would necessarily drive peripheral processes to
approach this "general" Fourier Transform. In other words, to
accomplish this generality you would be required to describe the
aperiodic function mathematically from discrete data. I've done tons
of multivariate regressions, and there are any number of "solutions"
that each would exhibit quite different Fourier results. Nearly every
regression suffers from the same issues I've already discussed for
Fourier analysis - aperiodicity. Frankly such an approach would be
inferior to rather simple windowing and performing standard FFTs.
This, of course, strips D.C. from the data set.

Windowing has been studied for its qualities since Blackman and
Tukey's seminal work "The Measurement of Power Spectra" written in
1958. This is the mathematical work that predates FFTs. Every
constraint and reservation that I have describe arises from the pages
of this slim volume in terms of what you describe as "general"
Fourier. As I've said, I have worked with both the Integral solutions
to Fourier Analysis and discrete FFTs for some 20 years. The cautions
and constraints are absolutely identical. Nyquist teaches us this,
Shannon further instructs us.

The authors offer three methods to perform Fourier analysis: Spaced,
Mixed, and Continuous. They report:
"The choice among these types will depend on their particular
advantages and disadvantages, and on the availability of
equipment, both for recording and analysis. In almost every case,
however, the detailed problems will be surprisingly similar."
pg. 55

The concepts of windowing are revealed by Blackman and Tukey; and
their necessity described at great length - per my summarized
cautions. The notion of "prewhitening" is discussed so that a finite
data record can be transformed (no practical Fourier analysis consists
of an infinite record). Aliasing is revealed as a problem as a
consequence of sampling (I cannot imagine our correspondent made a
continuous recording with a 2.5GHz baseband recorder). Windowing is
described in "general" Fourier (and Laplace) math (no one here wants
to deal with these abstractions). Impulse data (Dirac functions)
litter the pages. Analysis goes miles beyond power spectra to include
autocorrelation math- all done in integral calculus. Should we go
into covariability? How about Coherence? This last would be useful
to prove that the data is even real (and likely as not, it fails this
transform).

The long and short of this obviates the hugely erroneous report of
measuring DC at the terminals of an antenna. There is absolutely no
Fourier Cavalry coming to the rescue of a poorly stated problem with
equally problematic data.

Right. It's probably just a rusty bolt somewhere.

ac6xg