# 1/f noise generation

Started by July 4, 2003
```Hi All,

Is there anyone know how to generate 1/f noise for simulating the phase
noise. Someone told me though passing a white noise to a FIR the noise can
be generated. Any clue about the FIR?

Thanks alot

R. Y.

```
```"Yu Xiaodong" <yuxdntu@yahoo.com> wrote in message news:<be481k\$tug\$1@nobel2.pacific.net.sg>...
> Hi All,
>
> Is there anyone know how to generate 1/f noise for simulating the phase
> noise. Someone told me though passing a white noise to a FIR the noise can
> be generated. Any clue about the FIR?

why do it with an FIR when an IIR can do it very cheaply and just as
well (because phase response of the "pinking filter" has zero
importance).

anyway, if you type "pink noise" (which is the audiohead term for
"1/f" or "flicker noise") into Google or Google Groups you will get
lotsa references.  the most common reference is

http://www.firstpr.com.au/dsp/pink-noise/

> Thanks alot

do it with an 3rd order (3 poles, 3 zeros) IIR and you'll be withing
+/- 0.3 dB of true 1/f over a 10 octave range.  if you require higher
precision, there are other higher order IIR designs on that page.

r b-j
```
```Yu Xiaodong wrote:
> Hi All,
>
> Is there anyone know how to generate 1/f noise for simulating the phase
> noise.

How do you define 1/f noise? How is does DC behave in this noise?

Regards,
Andor

```
```Andor wrote:
>
> Yu Xiaodong wrote:
> > Hi All,
> >
> > Is there anyone know how to generate 1/f noise for simulating the phase
> > noise.
>
> How do you define 1/f noise? How is does DC behave in this noise?
>
> Regards,
> Andor

Ideal 1/f ("pink") noise increases 3 dB/octave as the frequency goes
down. Just as ideal white noise doesn't exist because infinite power
would come from extending it at the high end, ideal pink noise can't
exist because of trouble as DC is approached. Nevertheless, both are
useful abstractions.

Jerry
--
Engineering is the art of making what you want from things you can get.
&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;
```
```"Jerry Avins" <jya@ieee.org> wrote in message
news:3F089FB3.47729442@ieee.org...
> Andor wrote:
> >
> > Yu Xiaodong wrote:
> > > Hi All,
> > >
> > > Is there anyone know how to generate 1/f noise for simulating
the phase
> > > noise.
> >
> > How do you define 1/f noise? How is does DC behave in this noise?
> >
> > Regards,
> > Andor
>
> Ideal 1/f ("pink") noise increases 3 dB/octave as the frequency goes
> down. Just as ideal white noise doesn't exist because infinite power
> would come from extending it at the high end, ideal pink noise can't
> exist because of trouble as DC is approached. Nevertheless, both are
> useful abstractions.
>
> Jerry
> --
> Engineering is the art of making what you want from things you can
get.
>
&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;
&#4294967295;
As Jerry says, ideal pink noise can't exist, as the total power goes
infinite (equal power per octave, infinite octaves as you reduce
frequency).
However, nobody has found a lower frequency limit yet. I recall some
measurements done on galactic background radiation which went down
to some ridiculously low frequencies without deviating from 1/f.

If you are simulating phase noise you may wish to include a 1/f^2 term
as well.

Regards
Ian

```
```Jerry Avins wrote:
> Andor wrote:
> >
> > Yu Xiaodong wrote:
> > > Hi All,
> > >
> > > Is there anyone know how to generate 1/f noise for simulating the phase
> > > noise.
> >
> > How do you define 1/f noise? How is does DC behave in this noise?
> >
> > Regards,
> > Andor
>
> Ideal 1/f ("pink") noise increases 3 dB/octave as the frequency goes
> down. Just as ideal white noise doesn't exist because infinite power
> would come from extending it at the high end, ideal pink noise can't
> exist because of trouble as DC is approached.

Is anybody familiar with fractional gaussian noise? It also has a pole
at DC in the PSD and an exponential drop-off (for 1/2 < H < 1, H the
Hurst parameter), but it is integrable.

Perhaps some people mean fractional gaussian noise when they say pink
noise (or 1/f noise)?

Regards,
Andor
```
```On 7 Jul 2003 04:12:31 -0700, an2or@mailcircuit.com (Andor) wrote:

>Jerry Avins wrote:
>> Andor wrote:
>> >
>> > Yu Xiaodong wrote:
>> > > Hi All,
>> > >
>> > > Is there anyone know how to generate 1/f noise for simulating the phase
>> > > noise.
>> >
>> > How do you define 1/f noise? How is does DC behave in this noise?
>> >
>> > Regards,
>> > Andor
>>
>> Ideal 1/f ("pink") noise increases 3 dB/octave as the frequency goes
>> down. Just as ideal white noise doesn't exist because infinite power
>> would come from extending it at the high end, ideal pink noise can't
>> exist because of trouble as DC is approached.
>
>Is anybody familiar with fractional gaussian noise? It also has a pole
>at DC in the PSD and an exponential drop-off (for 1/2 < H < 1, H the
>Hurst parameter), but it is integrable.
>
>Perhaps some people mean fractional gaussian noise when they say pink
>noise (or 1/f noise)?

I don't think that applies in this case.  There is usually an easily
identifiable 1/f section in the phase noise of many oscillators.
This comes from 1/f noise in transistors affecting the phase directly.

There is also 1/f^3 noise, which comes from 1/f noise on the control
voltage of a VCO.  For a VCO though, the 1/f^2 noise (which comes from
white noise on the control voltage) is usually dominant over most of
the spectrum.  It's only for low offset frequencies that the 1/f^3
noise dominates.

For a reference crystal oscillator, white phase noise (1/f^0) is
usually dominant at offset frequencies > a few kHz.

From http://www.wenzel.com/documents/ulntests.html (which describes
tests on a reference oscillator):

"The following model agrees with all the phase noise measurements for
oscillator 8339045 to within 1.5 dB for Fourier frequencies from 1 to
100,000 Hz.

&#4294967295;(f) (8339045) = 10^-l2.86 / f^3 + 10^-15.0 / f + 10^-178.7 .
"

(units are dB)

Regards,
Allan.
```
```Ian Buckner wrote:
>
> "Jerry Avins" <jya@ieee.org> wrote in message
> news:3F089FB3.47729442@ieee.org...
> > Andor wrote:
> > >
> > > Yu Xiaodong wrote:
> > > > Hi All,
> > > >
> > > > Is there anyone know how to generate 1/f noise for simulating
> the phase
> > > > noise.
> > >
> > > How do you define 1/f noise? How is does DC behave in this noise?
> > >
> > > Regards,
> > > Andor
> >
> > Ideal 1/f ("pink") noise increases 3 dB/octave as the frequency goes
> > down. Just as ideal white noise doesn't exist because infinite power
> > would come from extending it at the high end, ideal pink noise can't
> > exist because of trouble as DC is approached. Nevertheless, both are
> > useful abstractions.
> >
> > Jerry
> > --
> > Engineering is the art of making what you want from things you can
> get.
> >
> &#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;
> &#4294967295;
> As Jerry says, ideal pink noise can't exist, as the total power goes
> infinite (equal power per octave, infinite octaves as you reduce
> frequency).
> However, nobody has found a lower frequency limit yet. I recall some
> measurements done on galactic background radiation which went down
> to some ridiculously low frequencies without deviating from 1/f.
>
> If you are simulating phase noise you may wish to include a 1/f^2 term
> as well.
>
> Regards
>     Ian

Some of those galactic sources are pretty powerful! (But even f = 1/A,
where A is the age of the universe, isn't thought to be zero.)

Jerry
--
Engineering is the art of making what you want from things you can get.
&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;
```
```Allan Herriman wrote:
>
...
>
> I don't think that applies in this case.  There is usually an easily
> identifiable 1/f section in the phase noise of many oscillators.
> This comes from 1/f noise in transistors affecting the phase directly.
>
> There is also 1/f^3 noise, which comes from 1/f noise on the control
> voltage of a VCO.  For a VCO though, the 1/f^2 noise (which comes from
> white noise on the control voltage) is usually dominant over most of
> the spectrum.  It's only for low offset frequencies that the 1/f^3
> noise dominates.
>
> For a reference crystal oscillator, white phase noise (1/f^0) is
> usually dominant at offset frequencies > a few kHz.
>
...

Allan,

What is an "offset frequency"? For that matter, what is meant by
"Fourier frequencies" in the part I snipped?

Jerry
--
Engineering is the art of making what you want from things you can get.
&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;
```
```On Mon, 07 Jul 2003 09:32:00 -0400, Jerry Avins <jya@ieee.org> wrote:

>Allan Herriman wrote:
>>
>  ...
>>
>> I don't think that applies in this case.  There is usually an easily
>> identifiable 1/f section in the phase noise of many oscillators.
>> This comes from 1/f noise in transistors affecting the phase directly.
>>
>> There is also 1/f^3 noise, which comes from 1/f noise on the control
>> voltage of a VCO.  For a VCO though, the 1/f^2 noise (which comes from
>> white noise on the control voltage) is usually dominant over most of
>> the spectrum.  It's only for low offset frequencies that the 1/f^3
>> noise dominates.
>>
>> For a reference crystal oscillator, white phase noise (1/f^0) is
>> usually dominant at offset frequencies > a few kHz.
>>
>  ...
>
>Allan,
>
>What is an "offset frequency"? For that matter, what is meant by
>"Fourier frequencies" in the part I snipped?

The same thing.
Phase noise is a measure of the modulation of a carrier.
We can measure phase noise with a spectrum analyser.  There will be a
peak on the spectrum analyser display at the carrier frequency (e.g.
10MHz).  Move an offset frequency (e.g. 1kHz) away from the carrier
(to 10.001 MHz), and ideally we would have no power.  In practice, we
have a certain noise density, which can be expressed in
dB(whatever)/Hz, e.g. dBm/Hz.  We can also express the power relative
to the carrier power to get dBc/Hz, which is the (one sided) phase
noise.

Note that the phase noise is independent of the carrier frequency.  We
can heterodyne it up and down all day, and the phase noise will be the
same (well... the phase noise of the local oscillators will add to the
phase noise of the signal, and the mixers will add a little noise
too.)
Note also that an amplifier will add some noise, and some of this
(half the power) will be phase noise.

The phase noise is, of course, a fuction of the offset frequency, so
it's usually specified as dBc/Hz @ so many Hz,
e.g.  -170dBc/Hz @ 1kHz (which incidentally is only a few dB above the
thermal noise floor of a 0dBm carrier).

-170dBc/Hz is actually a lot better than the phase noise floor of any
spectrum analyser that I've ever used.  Measurements of this sort
require a phase noise test set (e.g. HP3048), which is really just an
extra low noise spectrum analyser with a bad user interface.  It will
have an analog quadrature mixer that mixes to DC, and the result is
analysed with the help of an FFT.
Note: the mixer requires a local oscillator, of course.  This either
has to be a lot better than the device under test, or perhaps it could
be a copy of the dut, and the software will divide the result by 2
(i.e. take off 3db).

BTW, When the OP asked for 1/f noise, the 'f' was the offset
frequency.

Here's another example:
http://www.miteq.com/micro/fresourc/frequencysynthesizers/mfs_4.htm

Regards,
Allan.
```