# Johnson Nyquist Noise

Started by December 30, 2006
```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

Thanks,

Clay

```
```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.
```
```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,

Clay

```
```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
>
>
>
>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
```
```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

```
```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
>

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 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
> >
>
> 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

```
```"Clay" <physics@bellsouth.net> wrote in message
>
> 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

```
```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 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

```