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Entry  Tue Jun 28 16:59:35 2022, Paco, Summary, ALS, ALS beat allan deviation (XARM) xbeat_1340469968.pdffrac_avar_xbeat.pdf
    Reply  Tue Jun 28 18:53:16 2022, rana, Summary, ALS, ALS beat allan deviation (XARM) 
       Reply  Wed Jun 29 14:28:06 2022, Paco, Summary, ALS, ALS beat allan deviation (XARM) frac_adev_xbeat.pdf
Message ID: 16956     Entry time: Tue Jun 28 16:59:35 2022     Reply to this: 16959
Author: Paco 
Type: Summary 
Category: ALS 
Subject: ALS beat allan deviation (XARM) 


I took ~ 7 minutes of XALS beatnote data with the XAUX laser locked to the XARM cavity, and the XARM locked to PSL to develop an allan deviation estimator. The resulting timeseries for the channel C1:ALS-BEATX_FINE_PHASE_OUT_HZ_DQ (decimated timeseries in Attachment #1) was turned into an allan variance using the "overlapped variable tau estimator":

\sigma_y^2(n\tau_0, N) = \frac{1}{2n^2\tau_0^2(N - 2n)} \sum_{i=0}^{N-2n-1} (x_{i+2n} - 2x_{i+n} + x_i)^2

Where x_k represents the k-th data point in the raw timeseries, and n\tau_0 are the variable integration intervals under which two point variances are computed (the allan variance is a special case of M-point variance, where M=2). Then, the allan deviation is just the square root of that. Attachment #2 shows the fractional deviation (normalized by the mean beat frequency ~ 3 MHz for this measurement) for 100 integration times spanning the full duration (~ 7 min = 420 s).

The code used for this lives in Git/40m/labutils/measuremens/ALS/

If this estimate is any good, wherever the fractional beatnote deviation reaches a minimum value can be used as a proxy for the longest averaging time that give a statistical increase in SNR. After this timescale, the frequency comparison is usually taken over by "environmental instabilities" which I don't think I can comment further on. In our particular estimate, the 100 second integration gives a fractional deviation of ~ 0.44 %, or absolute deviation of 12.925 kHz.

Attachment 1: xbeat_1340469968.pdf  63 kB  Uploaded Tue Jun 28 18:22:03 2022  | Hide | Hide all
Attachment 2: frac_avar_xbeat.pdf  20 kB  Uploaded Tue Jun 28 18:54:34 2022  | Hide | Hide all
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