#
Pulse Height Resolution of TOT

Gus Sinnis
1/22/98
##
Introduction:

In an earlier memo Kelin W. discussed his method of converting from TOT
to PEs. In that memo there is a plot of the "resolution" of TOT.
He indicates that at least for that channel the resolution is roughly 20%.
I wanted to look at this in more detail, on a channel by channel basis
and for all channels. David W. felt that this resolution was
much poorer than expected. Contrary to popular opinion this is an
important conversion. We depend on it to make our sampling corrections.
Systematic channel to channel variations and poor resolution (especially
at low PE values) can dramatically affect our timing widths (TCHI distributions).
The difference in the sampling correction between .1 PE and 2 PE is about
7 ns. With poor resolution and large systematic errors it is quite
easy to mistake a 0.1 PE hit for a 2 PE hit. Since we make a hard
cut on the PE level in our fitter the effect on our angular resolution
can be large.
I have repeated Kelin's work, for Patch 1. I made my own determination
of the pedestals and 1 PE values for all the channels and then did my own
fits of PE vs. TOT (LOW and HI). Below I compare my results with
the currently used online algorithms.

##
Method:

For the adc distributiion I simply plotted the distribution for each channel.
Figure 1 shows a typical channel. I determined the Pedestal and 1
PE level by taking the appropriate peaks.

Using these values for the pedestal and 1 PE levels I plotted a "profile"
histogram of PEs vs. TOT; for each channel and for both LOW and HI TOT
seperately. A typical channel is shown in Figure 2 below. Since
the profile histogram plots the average at each point there will be a systematic
error that is pulse height dependent (getting larger for larger pulse heights).
This is due to the fact that the TOT value for a given PE level is never
too short, but often too long. This systemtaic can be corrected by
fitting to the median or peak of the TOT distribution instead of the average.
This should be done.

I fit the Low TOT distributions to 3rd order polynomials and the High
TOT distributions to fourth order polynomials.

##
Pulse Height Resolution of TOT:

To quantify the resolution I decided to plot:
[PEs(ADC) - PEs(TOT)] / [PEs(ADC) + PEs(TOT)]/2

First I will show the results using the current online algorithms (this
includes the edge finder). I only include an event in the plot if
the PE level as determined by the adc is between 1 and 20. The first
few plots are for individual channels. Note the pathological behavoir
on some channels, the large widths, and different channel-tochannel systematics.

Channel 1:
Channel 3:

Channel 7:
Channel 13:

Channels 1-16:
Systematic vs. PE Level:

Using the full width at half max (to ignore the large tails) I infer
a resolution of about 30% (FWHM = 0.7) from the plot of the sum of channels
1-16. Much of this comes from the channel-to-channel systematic errors.
For one of the better channels in this group (9) the FWHM = 0.4 for a resolution
of 17% - consistent with Kelin's findings.

Below I show the same set of plots from this analysis. In this
analysis I did not use the edge finder. If the channel had 2 edges
I used the LOW TOT, if it had 4 edges I used the HIGH TOT.

Channel 1:
Channel 3:

Channel 7:
Channel 13:

Channels 1-16:
Systematic vs. PE Level:

The resolution from the plot of all channels is nowabout 7% and for
a single channel about the same (a little bit narrower).

Notice that as advertised there is a systematic with increasing PEs.
But for the range of PEs that I plotted this is relatively small (~5%).