What Nyquist Didn't Say

[glen herrmannsfeldt]

I've seen a lot of posts over the last year or so that indicate a lack
of understanding of the implications of the Nyquist theory, and just
where the Nyquist rate fits into the design of sampled systems.

So I decided to write a short little article to make it all clear.

I like it.

As for section 1, for a periodic signal, or one that you only care
about over a finite time, you can (mathematically) sample perfectly in a
finite time. Realistically, quantum mechanics and the uncertainty
principle, in other words noise, will get to you.

The question of < or <= comes up often. There is zero probability
(that is, zero width) so it will never come up in real signals.
(Or consider jitter in the time base.)

Other than that, I think it is pretty good.

-- glen

 

[glen herrmannsfeldt]

glen herrmannsfeldt wrote:

(Actually, Gauss never published it. It was only published
posthumously as part of his notes.)

Then, Gauss wrote the first published paper on FFT?

If you want to put it that way, very few people publish
papers, they just send them to someone else to publish.

But yes, I had forgotten that.

-- glen

 

[Robert Latest]

On Mon, 02 Oct 2006 09:13:52 -0700,

As far as pdf documents go -- I have been doing them in HTML out of fear
that folks will just print out the PDF document and never come back to
my web site.

Don't worry. If you put the link in the pdf, people that like your stuff
and are interested will come to your site for more, even if they only
read it on the print-out. Uninterested people won't go to your site at
all, HTML or not, unless you also put up some free porn.

robert

 

[Joerg]

Hello Glen,

Then, Gauss wrote the first published paper on FFT?

If you want to put it that way, very few people publish
papers, they just send them to someone else to publish.

We have to remember what means there were back in their days. Far fewer
journals with available space. No word processors. Very costly
type-setting process. Etc.

Even nowadays publishing isn't easy. I have done a few and the whole
process is quite laborious. However, we now have an excellent means of
publishing just about anything (legal) we want: The web. Everybody can
set up a web site and go ahead. Also, you can publish your ideas in
newsgroups just like this one. All that provides instant publication.
Gauss, Nyquist and others didn't have all this and I assume Shannon was
too far into retirement by then as well. AFAIR he passed away at old age
around five years ago.

 

[[email protected]]

Then, Gauss wrote the first published paper on FFT?

No. Gauss' work was not published until 1866, as a part of his
collected works. Prior to that, there were various authors who
published related algorithms (e.g. a paper by Everett in 1860, one
published by Archibald Smith in 1846, and one published by F. Carlini
in 1828, although these works only described restricted cases).

What does seem to be true is that Gauss was the first *recorded*
discoverer of an FFT. He was also (apparently) the only author until
Cooley & Tukey in 1965 to describe a general mixed-radix algorithm for
any composite size.

(See the excellent paper, "Gauss and the History of the Fast Fourier
Transform," by Heideman et al., IEEE ASSP Magazine, p. 14, October
1984.)

If you want to put it that way, very few people publish
papers, they just send them to someone else to publish.

You're being a bit too pedantic for my taste; by "publish" in science,
we usually mean "initiate the publication process".

Regards,
Steven G. Johnson

 

[[email protected]]

Then, Gauss wrote the first published paper on FFT?

No. Gauss' work was not published until 1866, as a part of his
collected works. Prior to that, there were various authors who
published related algorithms (e.g. a paper by Everett in 1860, one
published by Archibald Smith in 1846, and one published by F. Carlini
in 1828, although these works only described restricted cases).

What does seem to be true is that Gauss was the first *recorded*
discoverer of an FFT. He was also (apparently) the only author until
Cooley & Tukey in 1965 to describe a general mixed-radix algorithm for
any composite size.

(See the excellent paper, "Gauss and the History of the Fast Fourier
Transform," by Heideman et al., IEEE ASSP Magazine, p. 14, October
1984.)

If you want to put it that way, very few people publish
papers, they just send them to someone else to publish.

You're being a bit too pedantic for my taste; by "publish" in science,
we usually mean "initiate the publication process".

Regards,
Steven G. Johnson