> stevenj@alum.mit.edu wrote:
> > glen herrmannsfeldt wrote:
> >>By the way, Gauss published the first paper on the FFT.
> > (Actually, Gauss never published it. It was only published
> > posthumously as part of his notes.)
>
> 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

Reply by ●October 5, 20062006-10-05

glen herrmannsfeldt wrote:

> stevenj@alum.mit.edu wrote:
> > glen herrmannsfeldt wrote:
> >>By the way, Gauss published the first paper on the FFT.
> > (Actually, Gauss never published it. It was only published
> > posthumously as part of his notes.)
>
> 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

Reply by Joerg●October 5, 20062006-10-05

Hello Glen,

>
>>> By the way, Gauss published the first paper on the FFT.
>
>> (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.
>

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.
--
Regards, Joerg
http://www.analogconsultants.com

Reply by glen herrmannsfeldt●October 4, 20062006-10-04

stevenj@alum.mit.edu wrote:

> glen herrmannsfeldt wrote:

>>By the way, Gauss published the first paper on the FFT.

> (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

Reply by glen herrmannsfeldt●October 4, 20062006-10-04

Tim Wescott wrote:

> 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

Reply by ●October 4, 20062006-10-04

glen herrmannsfeldt wrote:

> By the way, Gauss published the first paper on the FFT.

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

Reply by Jerry Avins●October 4, 20062006-10-04

Ban wrote:
...

> Well done analog filters are of the *minimum phase* type, having just the
> lowest possible delay for that shape of output response. This is possible to
> realize digitally with IIR filters only.

Minimum-phase (or nearly minimum) FIRs are possible, just not symmetric
FIRs. You can make maximum-phase FIRs too. Then *all* the ringing is on
the leading edge.
Jerry
--
"The rights of the best of men are secured only as the
rights of the vilest and most abhorrent are protected."
- Chief Justice Charles Evans Hughes, 1927
���������������������������������������������������������������������

Reply by glen herrmannsfeldt●October 4, 20062006-10-04

Joerg wrote:
(someone wrote)

>> It was actually Shannon (among others) that did the sampling theorem;
>> Nyquist made an observation. Your bibliography doesn't cite either of
>> them. It's probably correct to use "Nyquist rate" but not "Nyquist
>> theorem."

> Nyquist published his paper about the minimum required sample rate in
> 1928. Shannon was a kid of 12 years back then. The paper wasn't about
> ADCs or sampling in today's sense but about how many pulses per second
> could be passed through a telegraph channel of a given bandwidth.

(and be distinguished on the other end).
The important point being that the math is the same even though the
goal is different. I suppose, then, the sample rate should be
a lemma to Nyquist's telegraph channel theorem.
By the way, Gauss published the first paper on the FFT.
-- glen

Reply by Jeroen Belleman●October 3, 20062006-10-03

Ban wrote:

>>Here is the line verbatim:
>>"Linear phase has a very undesirable side effect, it rings *before and
>>after*
>>the step, supposed to be more audible."
>>
>
> It's called *pre-ringing* and it appears because the chunks are processed
> forward and backward in a row, so a unity pulse will have an identical
> rising and falling edge. If the filter is of the ringing type, thus the
> ringing occurrs twice.

Even steep linear phase analogue filters will exhibit pre-ringing.
If you were to linearise the phase response of, say, a Butterworth
filter, by adding one or more all-pass sections, its impulse
repsonse will ring before and after the main output. Of course,
the overall delay must go up for the filter to remain causal.
Jeroen Belleman

Reply by ●October 3, 20062006-10-03

Ban wrote:

> miso@sushi.com wrote:
> > Jerry Avins wrote:
> >> miso@sushi.com wrote:
> >>
> >>> Ring before the signal arrives? That sound non-causal to me.
> >>
> >> Please read more carefully. The filter rings before the main part of
> >> the output step *emerges* but after the step arrives at the input.
> >> The filter's inherent delay makes that quite possible.
> >>
> >> Jerry
> >> --
> >> "The rights of the best of men are secured only as the
> >> rights of the vilest and most abhorrent are protected."
> >> - Chief Justice Charles Evans Hughes, 1927
> >> =AF=AF=AF=AF=AF
> >
> > =AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=

> > Here is the line verbatim:
> > "Linear phase has a very undesirable side effect, it rings *before and
> > after*
> > the step, supposed to be more audible."
> >
> > Nothing wrong with my reading. Now if you are somehow looking at the
> > output to interpret where the large transition occurred, that is a
> > different story. However, any filter where the impulse response goes
> > negative will have such ringing, be it linear phase or not. You need
> > to visualize the convolution.
>
> It's called *pre-ringing* and it appears because the chunks are processed
> forward and backward in a row, so a unity pulse will have an identical
> rising and falling edge. If the filter is of the ringing type, thus the
> ringing occurrs twice.
> You are right in saying it's impossible, but only in an analog world.
> Digital filters do have a latency which will always be longer than the de=

lay

> of the corresponding analog filter; with linear phase it will be twice the
> FIR size plus twice the conversion time and more than double than the ana=

log

> counterpart.
> Well done analog filters are of the *minimum phase* type, having just the
> lowest possible delay for that shape of output response. This is possible=

to

> realize digitally with IIR filters only.
> And do not think that even a Gauss filter has only positive
> FIR-coefficients. This would be only true for a filter of infinite length,
> which apparently isn't that desirable at all. For practicable sizes the
> location of the poles and zeros has to be modified and one might get even
> negative coefficients, depending on the ratio of sampling- and filter
> frequency and filter length.
>
> --
> ciao Ban
> Apricale, Italy

The Gaussian to which I refer is S domain. If you mapped it to Z
domain, it would have to be IIR, not FIR.