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 ATF eLog Not logged in Message ID: 1260     Entry time: Mon Jan 24 17:17:59 2011     In reply to: 1079
 Author: Dmass Type: Misc Category: Doubling Subject: Some notes on previous bounds

IN PROGRESS

[ Notes about my paper and the current questions I am pondering - comments welcome ]

The current semi open (or at least not-quite-closed) question regarding my paper is how exactly to compare it to the other results in a way which is not (unintentionally) scientifically dishonest. More below:

Most results from other groups which contain either direct or indirect bounds on doubling noise are in Allan Deviation (sqrt(Allan Variance)). This is related to the PSD we are familiar with as follows:

1. Take the PSD of a time series of frequency deviations (or make a phase noise PSD and convert it to frequency noise PSD)
2. Apply this filter: 2*pi*sin^4(pi*tau*f)/(pi*tau*f)^2 - this is a very ripply bandpass filter which dies as f^-2 (Wikipedia has this formula under Time and Frequency Filter Properties - this is the correct formula)
3. Take the total rms (integral from zero to measurement BW)
4. This is sigma (Allan Variance) of tau.

Here are some obvious problems that come up when you want to compare Allan Variances

1. If my measurement bandwidth is different than yours, then we can measure different Allan Variances for the same physical process. This becomes a problem because most people don't seem to report their measurement bandwidth when they publish their Allan Variances.
2. Depending on the shape of your PSD, then the above difference might not be very large at all: If your noise is flat or goes down at high frequency then the above difference is negligible
3. If your noise is NOT flat in frequency at high frequency (specifically, if you have a white phase noise floor at high frequency, as with the doubling noise Mach Zehnder), then the rippling bandpass described above flat ripples - and thus the total rms scales ~ linearly with measurement bandwidth - so the Allan Variance scales with measurement bandwidth. (For non white noise such as high frequency features a similar analysis shows how they sample into each Allan Variance point)

The above "problem" can be summarize as: "If you have a frequency noise measurement which has features in it, or increases at high frequencies, the Allan Variance at a particular tau becomes increasingly less dependant on what is going on at 1/tau in the power spectrum." To me this just means it's a bad idea to use Allan Variance for my purpose, as it no longer represents what we intuitively want it to represent.

On comparing with other experiments: The bounds on doubling noise from the clock people are all in Allan Variance. They have some unreported bandwidth which they use for their measurement, and don't actually take frequency PSDs (they use a <machine name here> which generates Allan Variance on the fly). I think they might have measurement bandwidths of up to ~1 GHz (because of the particular heterodyne signals they are capturing). If I scale MY measurement up that high, and extend my white phase noise out to 1 GHz, my Allan Variance would go up by up to (1 GHz/128 Hz) = 8*10^6.

My actual frequency noise at low frequency is LOW, it is likely better than what many (and maybe all) have done before when trying to measure this. My Allan variance is better than them as measured, but I have a smaller measurement bandwidth, which means it might not be "honest" to compare my Allan Variance with theirs.

All frequency noise PSDs I could find were significantly above mine.

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