Andor wrote:
>
> You must admit that something like
>
> integral_0^b P(f)
>
> looks awful. I never did like statistical physics (except quantum
> information theory of course :-).
>
> Regards,
> Andor

Hello Andor,

Yes I admit, it looks odd to say the least. I'll see if I can work out
a "cleaner" way of doing this. Stat. Phys. is a slippery subject to say
the least. I have a book[1] that takes a nonstandard approach, and the
author wants to refute much of the standard paradigm about how Stat.
Phys. is approached. When I get time I'm going to revisit his ideas and
see if I can sort them out.

Thanks,
Clay

[1] Lavenda, Bernard, "Statistical Physics, a Probabilistic Approach"
John Wiley, 1991

Clay wrote:
> Andor wrote:
> > Hello Clay,
>
> > thanks for your public domain work (including the other papers on your
> > site - I especially liked the oscillator paper).You are welcome and thanks.
> I hope people find them useful.

I certainly did.

> Thanks for pointing out my sloppiness. The thing about the df term is
> wanting to show during the density of states transformation that we are
> going from the summation to an integration, so I really don't want to
> drop the df. So I fixed the last set of equations so as to not have a
> df squared term. I think this is okay since we modified something
> inside of a summation to become something inside of an integration.

Except that the notation of summation does not require a "df". I would
prefer that P(f) be a proper power density function, which one can
integrate over f and use the standard notation

P_bar = integral_0^b P(f) df.

Also, when you plot P(f) the "df" operator magically disappears, and
P(f) becomes a standard function. Strictly speaking, you are plotting

P(f)  / df = h f / ( exp(h f / k T) - 1)

and not

P(f) = h f / ( exp(h f / k T) - 1)T df,

except that P(f) / df has a different meaning altogether (never mind
the factor k T).

> This pretty much follows the standard development in statistical
> physics texts for handling the summation.

You must admit that something like

integral_0^b P(f)

looks awful. I never did like statistical physics (except quantum
information theory of course :-).

Regards,
Andor

Andor wrote:
> Hello Clay,
>
> thanks for your public domain work (including the other papers on your
> site - I especially liked the oscillator paper).

You are welcome and thanks. I hope people find them useful.


>
> I have a query with the Nyquist noise paper. You develop your formular
> in differential form, for example
>
> P(f) = h f / ( exp(h f / k T) - 1) df,
>
> keeping the integral operator df. Further down you integrate and get
>
> P_bar = integral_0^b P(f) df.
>
> which, when substituting the term above becomes
>
> P_bar = integral_0^b h f / ( exp(h f / k T) - 1) df^2,
>
> which is probably not what you want (note the df^2 term). Can't you
> just delete the differential operator in your derivation and set for
> example
>
> P(f) = h f / ( exp(h f / k T) - 1)?
>
> Regards,
> Andor


Hello Andor,

Thanks for pointing out my sloppiness. The thing about the df term is
wanting to show during the density of states transformation that we are
going from the summation to an integration, so I really don't want to
drop the df. So I fixed the last set of equations so as to not have a
df squared term. I think this is okay since we modified something
inside of a summation to become something inside of an integration.
This pretty much follows the standard development in statistical
physics texts for handling the summation.

Thanks,
Clay

Clay schrieb:

> Hello All,
>
> I had a recent situation where I needed to write a paper explaining the
> why's and wherefores of Johnson noise. So if you are interested, the
> following link will take you to my paper.
>
> http://www.claysturner.com/dsp/Johnson-Nyquist%20Noise.pdf

Hello Clay,

thanks for your public domain work (including the other papers on your
site - I especially liked the oscillator paper).

I have a query with the Nyquist noise paper. You develop your formular
in differential form, for example

P(f) = h f / ( exp(h f / k T) - 1) df,

keeping the integral operator df. Further down you integrate and get

P_bar = integral_0^b P(f) df.

which, when substituting the term above becomes

P_bar = integral_0^b h f / ( exp(h f / k T) - 1) df^2,

which is probably not what you want (note the df^2 term). Can't you
just delete the differential operator in your derivation and set for
example

P(f) = h f / ( exp(h f / k T) - 1)?

Regards,
Andor

Mike Yarwood wrote:

> Hi Clay - thanks - it's nice!  I know it's just a matter of personal taste
> but I usee -228.6 dBW/K/Hz for Boltzmann's and the  0.6 makes a difference
> in some of my stuff....
>
> Best of Luck - Mike

Thanks Mike,

For exactness I went ahead and reworked the numbers with the
fundamental constants utilized out to their currently known limits! The
paper has been updated.

Clay

"Clay" <physics@bellsouth.net> wrote in message 
news:1167584506.683040.232840@h40g2000cwb.googlegroups.com...
>
> Martin Eisenberg wrote:
>> Clay wrote:
>>
>> > http://www.claysturner.com/dsp/Johnson-Nyquist%20Noise.pdf
>>
>> I'll keep that, thanks! Noticed a typo though -- the left equation in
>> the penultimate formula on page 3 has a factor 1/2 missing.
>>
>>
>> Martin
>>
>> --
>> The difference between genius and
>> stupidity is that genius has limits.
>
> Hello Martin,
>
> Thanks for reading my paper. And yes I see the typo and I'm fixing it.
> Again thanks,
Hi Clay - thanks - it's nice!  I know it's just a matter of personal taste 
but I usee -228.6 dBW/K/Hz for Boltzmann's and the  0.6 makes a difference 
in some of my stuff....

Best of Luck - Mike

Eric Jacobsen wrote:
> On 31 Dec 2006 11:24:18 -0800, "Clay" <physics@bellsouth.net> wrote:
>
> >
> >Eric Jacobsen wrote:
> >> On 30 Dec 2006 17:10:02 -0800, "Clay" <physics@bellsouth.net> wrote:
> >>
> >> >
> >> >Hello All,
> >> >
> >> >I had a recent situation where I needed to write a paper explaining the
> >> >why's and wherefores of Johnson noise. So if you are interested, the
> >> >following link will take you to my paper.
> >> >
> >> >http://www.claysturner.com/dsp/Johnson-Nyquist%20Noise.pdf
> >> >
> >> >
> >> >Any and all comments welcome.
> >> >
> >> >Thanks,
> >> >
> >> >Clay
> >>
> >>
> >> Definitely a keeper!
> >>
> >> According to the analysis and last figure the noise is reduced at
> >> 60GHz?    Do I understand correctly that that's really just a failing
> >> of the approximation and the thermal noise is really independent of
> >> frequency?
> >>
> >
> >Hello Eric,
> >
> >Actually, the Johnson formula is the approximation - it assumes the
> >noise power is constant across all frequencies. Basically a conductor
> >at room temperature would have little noise power up above 60GHz. At
> >light frequencies (~10^14 Hz), there is almost none. Of course, just
> >heat up the conductor to several thousand Kelvins, and the situation
> >changes.
> >
> >But at room temperature and constraining one's self to frequencies <
> >10GHz, the Johnson formula is quite good. The exact power is given by
> >an integral of  hf/(exp(hf/kT)-1) df over the band of interest. It is
> >just that this integral does not have a nice closed form solution, so
> >we look to approximations or numerical solutions.
> >
> >Clay
> >
> >p.s. Thanks for reading.
>
> That's pretty cool and something that I didn't realize: the thermal
> noise would be less at 60GHz than at typical lower frequencies.   That
> might be another motivator for people looking at 60GHz technology, but
> not one that I ever heard articulated before.
>

Hello Eric,

My earlier statement was a bit misleading - the power density at 60GHz
is reduced compared to lower freqs at room temperature, but the
difference isn't worth writing home about. At T=300K, the -3 dB point
for the power densitity is 5.45*10^12 Hz. So the desire to use 60GHz
will be motivated by other reasons.

Clay

On 31 Dec 2006 11:24:18 -0800, "Clay" <physics@bellsouth.net> wrote:

>
>Eric Jacobsen wrote:
>> On 30 Dec 2006 17:10:02 -0800, "Clay" <physics@bellsouth.net> wrote:
>>
>> >
>> >Hello All,
>> >
>> >I had a recent situation where I needed to write a paper explaining the
>> >why's and wherefores of Johnson noise. So if you are interested, the
>> >following link will take you to my paper.
>> >
>> >http://www.claysturner.com/dsp/Johnson-Nyquist%20Noise.pdf
>> >
>> >
>> >Any and all comments welcome.
>> >
>> >Thanks,
>> >
>> >Clay
>>
>>
>> Definitely a keeper!
>>
>> According to the analysis and last figure the noise is reduced at
>> 60GHz?    Do I understand correctly that that's really just a failing
>> of the approximation and the thermal noise is really independent of
>> frequency?
>>
>
>Hello Eric,
>
>Actually, the Johnson formula is the approximation - it assumes the
>noise power is constant across all frequencies. Basically a conductor
>at room temperature would have little noise power up above 60GHz. At
>light frequencies (~10^14 Hz), there is almost none. Of course, just
>heat up the conductor to several thousand Kelvins, and the situation
>changes.
>
>But at room temperature and constraining one's self to frequencies <
>10GHz, the Johnson formula is quite good. The exact power is given by
>an integral of  hf/(exp(hf/kT)-1) df over the band of interest. It is
>just that this integral does not have a nice closed form solution, so
>we look to approximations or numerical solutions.
>
>Clay
>
>p.s. Thanks for reading.

That's pretty cool and something that I didn't realize: the thermal
noise would be less at 60GHz than at typical lower frequencies.   That
might be another motivator for people looking at 60GHz technology, but
not one that I ever heard articulated before.

Eric Jacobsen
Minister of Algorithms, Intel Corp.
My opinions may not be Intel's opinions.
http://www.ericjacobsen.org

Eric Jacobsen wrote:
> On 30 Dec 2006 17:10:02 -0800, "Clay" <physics@bellsouth.net> wrote:
>
> >
> >Hello All,
> >
> >I had a recent situation where I needed to write a paper explaining the
> >why's and wherefores of Johnson noise. So if you are interested, the
> >following link will take you to my paper.
> >
> >http://www.claysturner.com/dsp/Johnson-Nyquist%20Noise.pdf
> >
> >
> >Any and all comments welcome.
> >
> >Thanks,
> >
> >Clay
>
>
> Definitely a keeper!
>
> According to the analysis and last figure the noise is reduced at
> 60GHz?    Do I understand correctly that that's really just a failing
> of the approximation and the thermal noise is really independent of
> frequency?
>

Hello Eric,

Actually, the Johnson formula is the approximation - it assumes the
noise power is constant across all frequencies. Basically a conductor
at room temperature would have little noise power up above 60GHz. At
light frequencies (~10^14 Hz), there is almost none. Of course, just
heat up the conductor to several thousand Kelvins, and the situation
changes.

But at room temperature and constraining one's self to frequencies <
10GHz, the Johnson formula is quite good. The exact power is given by
an integral of  hf/(exp(hf/kT)-1) df over the band of interest. It is
just that this integral does not have a nice closed form solution, so
we look to approximations or numerical solutions.

Clay

p.s. Thanks for reading.

On 30 Dec 2006 17:10:02 -0800, "Clay" <physics@bellsouth.net> wrote:

>
>Hello All,
>
>I had a recent situation where I needed to write a paper explaining the
>why's and wherefores of Johnson noise. So if you are interested, the
>following link will take you to my paper.
>
>http://www.claysturner.com/dsp/Johnson-Nyquist%20Noise.pdf
>
>
>Any and all comments welcome.
>
>Thanks,
>
>Clay


Definitely a keeper!

According to the analysis and last figure the noise is reduced at
60GHz?    Do I understand correctly that that's really just a failing
of the approximation and the thermal noise is really independent of
frequency?

Eric Jacobsen
Minister of Algorithms, Intel Corp.
My opinions may not be Intel's opinions.
http://www.ericjacobsen.org