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Noise Occupancy Fits


Introduction

The current analysis shows "fits" in the Noise Occupancy that do not look very good at all.  I have (briefly) investigated the reasons for this and the affect on the value for the noise generated.  I have also looked at how to improve the fits.  The summary is that the current procedure produces very similar results to SCTDAQ in terms of the noise it calculates.

Explanation

The noise occupancy is calculated for a whole chip.  When SCTDAQ calculates the error on each data point, it divides the total occupancy by the number of channels rather than increasing the number of triggers by the number of channels.  This results in an error which is sqrt(nChannels) too large.  This can easily be seen from the SCTDAQ plots - it is certainly not true that the fit does not go through 1/3 of the error bars.

Algorithm

This is a brief overview of the algorithm.  The intent is to measure the noise of the occupancy plot at 1.0fC.  The occupancy is very low here when there is no injected charge, so a high limit approximation to the erfc functional form of the occupancy is used.  For very high x, erfc(x) ~ exp(-x^2) so ln (occ) against threshold^2 is a straight line.  This graph is formed and a straight line is fitted.

It is worth noting here that the problem is essentially that this approximation is very bad.  Because the errors are smallest for low threshold, the fit is pulled here and deviates significantly away from the data towards higher threshold.  This gives a measurement of the noise that is a lower bound.  This can be mitigated somewhat by raising the lower bound on the fit function, but this does not help much.  Ultimately of course it is best to fit to the correct function (ie erfc) rather than some approximation.

Test

To investigate this, I used SCTDAQ raw data for module 20220330200020.  A summary of the SCTDAQ output is here and the SCTDAQ plots are here.  I compared the noise to the output from our analysis modifying the algorithm.  The different alogorithms tested were:

  1. Linear fit with correct errors
  2. Linear fit with errors scaled by a factor of 11.3 (this should be similar/identical to SCTDAQ)
  3. Fit function a + bx + 0.5*log x together with lower limit increased to 0.5 fC^2.
  4. Fit function a+log (erfc (sqrt(b*x))).  This is the "correct" function with no approximation

Results

This table shows the results (follow the links for the plots).  I have included approximate fitting errors.

Chip
 SCTDAQ
 1. Correct Errors
 (±1-2 ENC)
 2. Large Errors
 (±10-20 ENC) 		3. Better approx
 (±3-4 ENC) 		4. Correct Function
 (±3-4 ENC)
0
 1469 1470 1471 1601 1591
1
 1440 1441 1442 1533 1525
2
 1482 1483 1483 1552 1543
3
 1453 1454 1454 1576 1567
4
 1426 1427 1428 1538 1529
5
 1453 1454 1454 1616 1609
6
 1459 1460 1459 1598 1589
7
 1462 1463 1464 1570 1561
8
 1451 1453 1453 1574 1565
9
 1441 1442 1443 1584 1574
10
 1412 1412 1413 1560 1552
11
 1461 1463 1464 1625 1616

Comment


Reducing the errors makes no difference to the fit.  This is because they are all scaled by a constant factor.  It does of course make the plots look worse, however, I think this is good - we have consistency with SCTDAQ but the poor quality of the plots serves as a reminder that the numbers are not too meaningful.  Fitting using the correct function gives significantly higher noise which may cause some complaints!  Of course, it is possible to provide both fits.