Averaging of phase noise?

Hi all,

My most basic question can be summarized as: "Can phase noise be
averaged out?" And subsequentially, is there a difference if the input
signal is a sinusoid or wideband? What happens after averaging
infinitely many times?

My first reaction is: No, it can't (at least not in a similar manner as
white noise where SNR is increased by 10dB for every additional 10x of
averaging) because:

- Phase noise is multiplicative
- http://www.bitsofbits.com/2015/07/07/signal-jitter-and-averaging/
  argues that after infinite many averages, the result would be the PSD
  of the phase noise skirt convolved with the sinusoid

My assumption changed once I went to lab and tested it.
How do I define averaging and how do I measure it?

- An RF signal generator generates a 2001.01 MHz signal which is
  downconverted by a 2 GHz LO (resulting in 1.01MHz) and sampled by a 4
  MHz ADC
- I take "NumAvgs" traces, each with N=2^15 samples and average them to
  get a single N-length, averaged trace
- However, the exact start of the sampling point in each trace is
  unknown. Hence I time-align subsequent traces with the first trace in
  digital domain
- I use two types of time alignment
  a) polyphase interpolation: A coarse alignment with crosscorrelation
     is followed by obtaining the fractional delay using 2nd order
     polynomial interpolation. Then an 8-tap FIR filter is used to
     interpolate for this fractional delay
  b) Markus Nentwigs MATLAB function which is based on frequency domain
     phase estimation:
     https://www.dsprelated.com/blogimages/MarkusNentwig/bl_fitSignal2
     /fitSignal_120825.m


With the first method I obtain this result:

https://snag.gy/sNdvH1.jpg

For 100 averages, the entire noise floor shifts down by 20dB, including
the phase noise skirt! With more averaging I can shift this down
arbitrarily low and hence make the SNR arbitrarily good. This implies
that I can average phase noise out!

Now if I use Markus Nentwigs fitSignal_120825 I get this result:

https://snag.gy/npPwZU.jpg

It can be seen that the noise floor shifts again by 20dB. While the
phase noise skirt is somewhat reduced, it does not shift by 20 dB.
Increasing the number of averages, the SNR saturates.



1. Can Phase noise be averaged out? I.e., can the error introduced by
   phase noise be reduced by averaging over multiple signal traces?

2. What is the expected improvement? Is it 10dB/10averages, as for white
   noise?

3. If not, how could my observations be explained?

4. What could be a reason why there is a difference between the time
   alignment methods (which qualitatively both look OK)?

5. Is there a difference in the behavior for a sinusoid vs. a wideband
   signal?
   - If the SNR due to *phase noise only* is 45 dB for a sinusoid, would
     the SNR for a wideband signal be also 45 dB?
   - If averaging as described above works for a sinusoid, would it work
     for a wideband signal as well?


Thanks.
Peter

My first impression is that while phase noise is multiplicative,
for low levels of phase noise it can be approximated as
additive noise and, therefore, can be averaged.

If the phase noise is -10 dBc I am betting you cannot average
it out.


Steve

On Wed, 8 Aug 2018 15:40:42 -0700, Peter Mairhofer <63832452@gmx.net>
wrote:

>Hi all,
>
>My most basic question can be summarized as: "Can phase noise be
>averaged out?" And subsequentially, is there a difference if the input
>signal is a sinusoid or wideband? What happens after averaging
>infinitely many times?
>
>My first reaction is: No, it can't (at least not in a similar manner as
>white noise where SNR is increased by 10dB for every additional 10x of
>averaging) because:
>
>- Phase noise is multiplicative
>- http://www.bitsofbits.com/2015/07/07/signal-jitter-and-averaging/
>  argues that after infinite many averages, the result would be the PSD
>  of the phase noise skirt convolved with the sinusoid
>
>My assumption changed once I went to lab and tested it.
>How do I define averaging and how do I measure it?
>
>- An RF signal generator generates a 2001.01 MHz signal which is
>  downconverted by a 2 GHz LO (resulting in 1.01MHz) and sampled by a 4
>  MHz ADC
>- I take "NumAvgs" traces, each with N=2^15 samples and average them to
>  get a single N-length, averaged trace
>- However, the exact start of the sampling point in each trace is
>  unknown. Hence I time-align subsequent traces with the first trace in
>  digital domain
>- I use two types of time alignment
>  a) polyphase interpolation: A coarse alignment with crosscorrelation
>     is followed by obtaining the fractional delay using 2nd order
>     polynomial interpolation. Then an 8-tap FIR filter is used to
>     interpolate for this fractional delay
>  b) Markus Nentwigs MATLAB function which is based on frequency domain
>     phase estimation:
>     https://www.dsprelated.com/blogimages/MarkusNentwig/bl_fitSignal2
>     /fitSignal_120825.m
>
>
>With the first method I obtain this result:
>
>https://snag.gy/sNdvH1.jpg
>
>For 100 averages, the entire noise floor shifts down by 20dB, including
>the phase noise skirt! With more averaging I can shift this down
>arbitrarily low and hence make the SNR arbitrarily good. This implies
>that I can average phase noise out!
>
>Now if I use Markus Nentwigs fitSignal_120825 I get this result:
>
>https://snag.gy/npPwZU.jpg
>
>It can be seen that the noise floor shifts again by 20dB. While the
>phase noise skirt is somewhat reduced, it does not shift by 20 dB.
>Increasing the number of averages, the SNR saturates.
>
>
>
>1. Can Phase noise be averaged out? I.e., can the error introduced by
>   phase noise be reduced by averaging over multiple signal traces?
>
>2. What is the expected improvement? Is it 10dB/10averages, as for white
>   noise?
>
>3. If not, how could my observations be explained?
>
>4. What could be a reason why there is a difference between the time
>   alignment methods (which qualitatively both look OK)?
>
>5. Is there a difference in the behavior for a sinusoid vs. a wideband
>   signal?
>   - If the SNR due to *phase noise only* is 45 dB for a sinusoid, would
>     the SNR for a wideband signal be also 45 dB?
>   - If averaging as described above works for a sinusoid, would it work
>     for a wideband signal as well?
>
>
>Thanks.
>Peter
>


You might want to look into Phase Locked Loop design and analysis, as
the loop filter pretty much filters out or shapes phase noise in a
predictable way.   There are a number of analytical methods that
provide means to quantify the effect.  If you find some texts or
papers on PLL design and analysis it may be helpful to understand
what's possible from that perspective.

Am 09.08.2018 um 00:40 schrieb Peter Mairhofer:
> 1. Can Phase noise be averaged out? I.e., can the error introduced by
>     phase noise be reduced by averaging over multiple signal traces?

Yes, but there are restrictions. The most important one is aliasing. BTDT

You should not assume that averaging samples is an appropriate method. 
This will decrease the amplitude.
To reduce phase noise you need to do the averaging in polar space. This 
will work unless you have so much noise that the phase is indeterminate.

> 2. What is the expected improvement? Is it 10dB/10averages, as for white
>     noise?

I did not calculate that. I will likely be non-linear depending on the SNR.

> 4. What could be a reason why there is a difference between the time
>     alignment methods (which qualitatively both look OK)?

Side effects.

> 5. Is there a difference in the behavior for a sinusoid vs. a wideband
>     signal?

I don't think so.
But in practice a wideband signal might be much more sensitive to 
aliasing, i.e. indeterminate phase because it has less energy per 
frequency bin.

However, if the spectrum of the phase noise and your signal spectrumare 
sufficiently different, e.g due to short time stability of your 
oscillators, you may still get reasonable results.


Marcel

Am 09.08.2018 um 00:40 schrieb Peter Mairhofer:
> Hi all,

Go to www.rubiola.org.
Feels like drinking from a hydrant.

regards,
Gergard