An Interesting Historical Paper on Sampling

Hi guys,
  I recently ran across an interesting article
on the origins of the Sampling Theorem.

If such things interest you, the article is 
at both of the following two web sites:

http://webee.technion.ac.il/courses/044130/00755459.pdf
http://www.hit.bme.hu/~papay/edu/Conv/pdf/origins.pdf

See Ya',
[-Rick-]

On Sep 6, 7:19=A0am, Rick Lyons <R.Lyons@_BOGUS_ieee.org> wrote:
> Hi guys,
> =A0 I recently ran across an interesting article
> on the origins of the Sampling Theorem.
>
> If such things interest you, the article is
> at both of the following two web sites:
>
> http://webee.technion.ac.il/courses/044130/00755459.pdfhttp://www.hit.bme=
.hu/~papay/edu/Conv/pdf/origins.pdf
>
> See Ya',
> [-Rick-]

I like at the end that they say the Sampling theory is attributed to
Shannon-Nyquist etc etc but don't put the guy who was Historically
first first! Whittaker.

On Sep 6, 12:07 pm, HardySpicer <gyansor...@gmail.com> wrote:
>...
>
> I like at the end that they say the Sampling theory is attributed to
> Shannon-Nyquist etc etc but don't put the guy who was Historically
> first first! Whittaker.

This is a characteristic tendency of human groups to award naming
status to those whose publications have made a discovery more
immediately convenient to a user group instead of to earlier
discoverers. Examples include the DSP window known as Kaiser-Bessel
which had earlier been published as 'Taylor one parameter', but has
been credited to Kaiser since his publication of his rediscovery of
Bessel's convenient series formula for the modified Bessel function of
the first kind on the blackboard of one of his associates at Bell
Labs.There is also the case of the continent and country where I live
taking their name, in part, from the name of Amerigo Vespucci instead
of from the name of Christopher Columbus or any of the others who have
been suggested as early visitors to the New World.

Dale B. Dalrymple

>Hi guys,
>  I recently ran across an interesting article
>on the origins of the Sampling Theorem.
>
>If such things interest you, the article is 
>at both of the following two web sites:
>
>http://webee.technion.ac.il/courses/044130/00755459.pdf
>http://www.hit.bme.hu/~papay/edu/Conv/pdf/origins.pdf

That paper says who who wrote about the baseband and bandpass cases. It
doesn't say who wrote about the complex sampling case. That would be
interesting.

I'm not sure this paper goes back far enough. I seem to remember reading
about a statistician way back in the 19th century figuring out the basics
of sampling, though I can't remember who it was. Of course, a part of the
picture must have been obvious to the earliest investigators of
stroboscopic effects.

Steve

On Sep 6, 3:07&#4294967295;pm, HardySpicer <gyansor...@gmail.com> wrote:
> On Sep 6, 7:19&#4294967295;am, Rick Lyons <R.Lyons@_BOGUS_ieee.org> wrote:
>
> > Hi guys,
> > &#4294967295; I recently ran across an interesting article
> > on the origins of the Sampling Theorem.
>
> > If such things interest you, the article is
> > at both of the following two web sites:
>
> >http://webee.technion.ac.il/courses/044130/00755459.pdfhttp://www.hit...
>
> > See Ya',
> > [-Rick-]
>
> I like at the end that they say the Sampling theory is attributed to
> Shannon-Nyquist etc etc but don't put the guy who was Historically
> first first! Whittaker.

Actually, Whittaker is mentioned under the section "The
Mathematicians".

Cheers,
David

On 7 &#4294967295;&#4294967295;&#4294967295;, 07:00, "steveu" <ste...@coppice.org> wrote:
> >Hi guys,
> >  I recently ran across an interesting article
> >on the origins of the Sampling Theorem.
>
> >If such things interest you, the article is
> >at both of the following two web sites:
>
> >http://webee.technion.ac.il/courses/044130/00755459.pdf
> >http://www.hit.bme.hu/~papay/edu/Conv/pdf/origins.pdf
>
> That paper says who who wrote about the baseband and bandpass cases. It
> doesn't say who wrote about the complex sampling case. That would be
> interesting.
>
> Steve

On the one hand Kotelnikov in "On the transmission capacity of 'ether'
and wire in electrical communications", issued in 1933, has proved the
theorem for a minimum sampling frequency of quadratures (Theorem IV):

"If x(t) has frequencies from F1 to F2, then

x(t) = x1(t)*cos(2piF0t) + x2(t)*sin(2piF0t),

where F0=(F1+F2)/2, x1(t) and x2(t) have frequencies from 0 to (F2-F1)/
2.
So x1(t) and x2(t) can be sampled with Fs => F2-F1."

In modern DSP books x1(t) and -x2(t) are known as "inphase" and
"quadrature" components of x(t), while Kotelnikov did not name it so.
Complex envelope ("minimal" complex forms of x(t)) is defined as:

y(t) = x1(t) + j(-x2(t))

So sampling frequency for y(t) is the same as for x1(t) and x2(t).

It seems, Kotelnikov`s Theorem IV is the first mention of an
opportunity of such decomposition.

On the other hand in original work Kotelnikov has proved all theorems
for real signals. He used the math of real Fourier transform, so the
proof seems difficult. But if to use the math of complex Fourier
transform the proofs of theorems become simple and obvious. Also it
becomes clear, that theorems are true for both real and complex
signals. In this case Fs => Fh-Fl, where Fh is highest frequency in
spectrum, Fl is lowest frequency in spectrum (for real signal spectrum
includes negative frequencies, so  Fl = -Fh).

Andrew.