PMT/Linear Fan-in noise study

D. Lawrence
Nov. 20, 2001
Here are some results from a quick Monte Carlo calculation I did to estimate the rate at which we might expect a discriminator to fire due to simply to summing the noise of a large number of channels. For the case of the PrimEx trigger logic, the original plan called for summing up to 176 channels. If we increase the number of PbWO4 crystals by 600, this may increase to as much as 256 channels (though no serious work has been done yet on scaling up the trigger design).
PS Here is a back-of-the-envelope calculation for what we might expect for the minimum signal size which the discriminator might see for real signals. The values shown assume the summing was already done and the timing is reasonably tight. The gap between this value and what is obtained from the calculation below indicate how much room we'll have to play with when setting thresholds.
PS The noise was randomly sampled from a signed exponential distribution as shown in the plot to the left. A new, independant sample was generated for each channel (256 total) at every point. Each point was taken to represent 1/2 nanosecond. (The 1/2 nanosecond was chosen arbitrarily based on my experience with looking at PMT signals on oscilloscopes.)

A total of 500,000 points were generated representing a total time of 250 microseconds. The exponential used a decay constant of 1mV.

PS Here is a plot showing the simulated noise for a single channel over a 50ns range.
PS Here is a plot showing the simulated noise for the same channel as above, but over a 50ms range.
PS This plot shows the linear sum of 256 independant channels over a 50ms time range. Note that the Y-axis has increased an order of magnitude over the single channel noise plot above. I assume here that the linear fan-in is adding the noise over the input channels and introduces none of its own.
PS Projecting the previous plot onto the amplitude (Y) axis gives the distribution of the sum of 256 channels. This shape is fit extremely well by a Gaussian function.
PS Assuming the sum of 256 channels is truly Gaussian in nature (the important part being the tails), the sigma and complementary error function (erfc) can be used to calculate the rate as a function of threshold.

This plot shows the rate at 100mV is already at less than 1Hz.

Note that this simple calculation goes like (0.5ns)-1 , my arbitrary choice for time scale. One might imagine the most realistic timescale value could differ from this as much as a factor of ten. In this case, the rate (Y-axis) would scale like one over the value of the timescale.

PS This plot is the same as the one above except I've changed to a linear scale and zoomed in to the part above 100mV. This shows you really get no rate from adding random noise for a threshold above 120mV.
PS This plot shows the rate as a function of threshold for the sum of only 100 channels. This is for comparison to the 256 channel case in the previous 2 plots.
PS Same as above but on a zoomed-in linear scale.
PS PS These two plots are from the presentation of my PrimEx trigger simulation study from 2 years ago.

The first plot here is the more relvant. It shows expected discriminator rates from different areas of the calorimeter as a function of particle energy. Taking the values given at the top of this page, the 800MeV point would correspond to about the 200mV point on the above plots. This indicates that discriminator rates due just to real backgrounds will far outweigh anything from additive noise in the region we expect to set the thresholds.

Summary and Conclusions

This study is based on several arbitrary, but I believe, reasonable estimates. Namely, the timescale and amplitude of the noise input to the linear fan-in modules. I believe the parameterizations used were quite conservative and can be taken as an upper limit on the noise rates.

One thing notably missing from this calculation is a contribution due to common mode noise. This can be a significant effect, but it will also affect the anode signals and thus, our ultimate resolution. It is assumed great care will need to be taken to minimize any contributions of this type simply in order to do the experiment.

David Lawrence
davidl@jlab.org
Tue Nov 20 15:31:26 EST 2001