NOTES ON BQI   8/2/2024
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These notes give a brief description of Burst Quality Indicators
that, as the name implies, are intended to describe the quality
of bursts. Below the BQI are identified in capitals.
This set of plots, Fig 1, show a
set of figures that illustrate the general  burst characteristics. 



1) GTK measurements of Bursts
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The distribution of the number of hits per trigger  in GTK 1 (Fig 3a,b)
is, to a good approximation, Gaussian in shape (Fig 3a),
but a tail is present in many cases as shown by the 
log plot (fig 3b).
 
The Gaussian distribution  is described by the MEAN and STANDARD DEVIATION(SD or RMS).
These are ~ 50 hits/trigger and ~15 hits/trigger, respectively.
Notice that the RMS is about a factor 2 larger than expected from the 
value of the mean.  This is due to the lack uf uniformity of 
the spill (see Fig 0, Fig 4) and is further  discussed below in relation
to the spill duty factor.

The deviation of the hit distribution from a gaussian distribution
is measured by the SKEWNESS and KURTOSIS parameters.

SKEWNESS  = (3rd moment of distribution)/SD**3        0. for Gaussian
KURTOSIS  = (4th moment)/SD*4                         3. for Gaussian

Skewness measures the asymmetry of the distribution, whereas
kurtosis measures the the tails of the distribution:
leptokurtic = long tails , platykurtic = more peaked than Gaussian.
SKEWNESS and KURTOSIS are highly correlated for these data.


Another parameter that is used is the %HITS GREATER THAN 100.
This gives another measure of the tail of the distribution and is 
highly correlated with the skewness and RMS. 



The parameter CRENellation measures an aspect of the performance of
the GTK readout.

CREN = 100*(E-O)/E    expected value 0.

where E = sum hits on even 1/4 chips
      O = sum hits on odd  1/4 chips

For details see Alan's Naples talk.



2) The spill distribution.
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Fig 0 shows the spill in 2D.  The X axis is the trigger time  modulo the SPS period
(the folded spill time) with a cubic correction to partially linearize the distribution
in folded  spill time. The Y axis  is the trigger time.
This plot has been  used to determine the SPS period. 
Evidently the spill is not debunched and there is substantial low frequency noise
modulating the spill.

Fig 4 shows the number of hits per trigger in 5ms bins of trigger time.
(An alternative would be number of triggers in 5ms of trigger time 
but this shows a reduced variablity due to trigger dead-time, see Fig. 5).
The distribution is characterised by the MEAN , RMS(SD), MAX/MIN 
of the spill. The MEAN, RMS and MIN are determined from the 2-5 sec region
of the spill.

The  'noise' in the spill is  measured by a Fourier analysis (Fig 7).
This Figure shows that there is a substantial level of low frequency noise,
together with resonances at 50, 100, and 150 Hz.
The  BQIs have been chosen to 
be the amplitudes of the  LOW , 50 and 100 Hz signals since these are the
main contributors to the high RMS of the spill ( and GTK hit distribution).   

  

3)Spill Duty Factor 
-------------------

Machine physicists measure the overall characteristics of a beam
by the spill duty factor(SDF). This is defined  as 
 (mean intensity)**2/(mean of  intensity**2).
Thus a uniform beam  would have a SDF of unity and any variations in 
in intensty would reduce the SDF.

See  Page 8 of  Fig 2, Fig 3
for plots of the SDF  for low (12288)  and higher (12567) beam intensity
runs, respectively.

For run 12288 with statistical noise only, the SDF would be expected to
be  ~0.98 .  A value of ~0.92 is found due to the RMS of the beam being 
a factor ~2 larger than the statistical expectation.  In addition, some bursts have lower
values of the DF due to the 100 Hz noise ( see  plots on Page 4, Fig 2).