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




Richard Clark wrote:
On Mon, 07 Feb 2005 14:34:38 GMT, Gene Fuller
wrote:


I can ignore the name-dropping, but I cannot ignore the incorrect
statement about "Pure Fourier". There is no mandate of constancy even
for the purest Fourier transform. The function needs only to be
moderately well-behaved, including single valued and integrable.


Hi Gene,

In this case you are seriously wrong. There are no IFs ANDs or BUTs.
The loop hole of well-behaved is not enough with it being far too
inspecific.

The ONLY case where the Fourier Series resolves a correct
transformation is if you limit your data set (or for an Integration,
you define your limits) over an interval of n · 2 · PI for a periodic
function where n is an integer from 1..m. Further, you are resolution
limited if you fail to observe Nyquist's laws and under sample, or
fail to frequency limit your real data. This also segues into
Shannon's laws where you can observe the S+N/N in the transform
(discussed below). These concerns are EXTERNAL to the simple act of
transforming data, but are necessary correlatives that MUST be taken
into account.

If you fail even in this simple regard for periodicity (say looking at
only 359 degrees of the periodic function), the result is quite
dramatically different in the Fourier output. Even the casual
observer can immediately see the difference between the correct and
incorrect results, there is nothing ambiguous about it at all.


Perhaps you are confusing Fourier series analysis with Fourier transform
analysis?


No, I have done both, and I will drop the name again, at HP with their
work on Fourier Analysis equipment where I tested their FFT algorithms
(call them what you may, the basic underlying requirements do not
change). I was working with 24 Mathematicians AND Engineers - there
was nothing sloppy about the quality of up-front preparation. This
was a project 5 years in the making. They even wrote their own Pascal
compiler for 1000000 lines of code. I have also done IIRs and FIRs,
Wavelets, and a host of other frequency/time series decimation
analysis.

ALL Fourier techniques have requirements that go beyond the Fourier
math. These requirements (if you have any interest in accuracy)
cannot be ignored. If you have no interest in accuracy, you still
have to perform some of them, which is to say there are trade offs as
I mentioned previously. Ignoring them all simply reduces real data
into transformed garbage.

I have written FFT software that has resolved pure sine waves into a
transformation to a single bin with a statistical noise floor and ALL
spurious response down 200dB. To give an example of what 1° of
decimation error will do, it will inject 120dB of noise into the
product and spurs that are barely 10 to 20 dB down from the principle
bin (which also exhibits about 3dB error).

Much of what is available through college texts and on the web are
seriously under powered in their scope. College is not very
interested in scope, simply introduction. That is, unless you find
yourself in a undergrad (more probably grad school with the additional
considerations taken into account) engineering course dedicated to
modern implementations (practical Fourier) now largely focused on DSP
(which had its genesis in the IIR and FIR earlier implementations).

73's
Richard Clark, KB7QHC