DSC waveform fitting

by S. Ritt

Introduction

The Domino Sampling Chip was used sucessfully for the first time in our 1999 beam time on sixteen beam counters. It ran at ~500MHz thus producing waveforms of 256ns width in steps of 2ns. The waveforms were pre-analyzed already in the DAQ frontend and zero supression was applied in software to reduce the data rate by a factor of ten, while still keeping the interesting sectios of the waveforms for later off-line analysis.

This document now describes which methods were applied in the off-line analysis to evaluate the waveform data.

Speed determination

The domino speed and thus the sampling frequency varies from channel to channel. While they have all been set to ~500MHz by means of the voltage controlling the domino speed, the need a fine-adjustment in software due to inaccuracies of the voltage and drifts over time.

To determine the exact sampling speed, a 100 ns wide test pulse was used which was capacitively coupled to the DSC input at a rate of ~1 Hz. This pulse induced two spikes 100 ns apart, which can be used to determine the DSC speed. Following picture shows a typical event and the derived domino speed:

The exact position of the peaks was determined by fitting a second order parabola through the three points at the peaks, which gives a sub-bin resolution. To determine the current speed, all events from a run are analyzed and averaged. Since the calibration events come at ~1Hz interlaced without our normal data, the calibration runs permanently in the analyzer code and updates the calibration constants for every run. Here is a list of typical domino speeds for all sixteen channels in a given run:

                run 22370	  run 22406
channel         ns/bin     	  ns/bin

[0]             1.94106		  1.93549
[1]             2.00366		  1.99911
[2]             1.96029		  1.9561 
[3]             1.99133		  1.98513
[4]             2.00226		  1.99841
[5]             2.00538		  2.00102
[6]             1.99917		  1.99499
[7]             1.96288		  1.95852
[8]             1.95804		  1.95312
[9]             2.03156		  2.02833
[10]            1.97487		  1.96807
[11]            1.96509		  1.96134
[12]            1.94905		  1.9447 
[13]            1.98898		  1.98396
[14]            2.13812		  2.13358
[15]            2.07931		  2.07277

The two runs were taken one week apart so one sees that the domino speed is pretty stable over time. The maximum deviation is 0.3% which corresponds to 0.4ns for the average peak, which lies about in the center of the waveform at 128ns.

Waveform fitting

Assuming that both pions and positrons produce identical waveforms, everything which is conainted in a given waveform are a number of ADC and TDC values corresponding to the number of hits in that waveform. The best waveform analysis one therefore could think of is to determine the exact shape of an average waveform ("system function") for each channel, and then fit all events with this waveform. If there is more than one hit in a given event, one makes a sum of N system functions, all with different multiplicators (corresponding to the ADC value) and differently shifted (corresponding to the TDC value):

where g(x) is the system function for a given channel and f(x) is the function which is fitted to the waveform by varying ADC_i and TDC_i in order to minimize the chi^2 of the fit.

Following picture shows a typical system function derived by averaging over > 100,000 events which contained only one hit in that particular channel:

Note the ripple around t=175ns which corresponds to a signal reflection due to imperfect 50 Ohm termination in the trigger branch of our electronics. This ripple is present in every event and correctly taken care of by including it in the system function, while other techniques like multi-hit TDC's would identify this peak falsly as a real hit. The system functions for all channels are stored in the file waveform.dat in the analyzer directory and must be present to sucessfully run the analyzer.

Following displays show some fitted waveforms using this system function. Note how well the multiple hits can be fitted. The blue line is the originial waveform. The "gap" between separated hits come from the front-end zero supression, which only stores the waveform around real hits. This region with y=0 is of course not taken into account when doing the fit, which is displayed in green. Red circles indicate the peak value. The x-axis is not yet calibrated, so one unit corresponds to ~2ns.

The used channel in these figures is the active degrader, which shows a monoenergetic line of ~20MeV for degraded pions. The pile-up shown in these figures comes from the 20ns beam structure of the PSI accelerator. We used a stopping rate of ~1MHz, which leads to a probability of ~1/50 for having two adjacent pions.

Comparison of ADC/TDC and DSC data

Only very basic comparisons between the DSC derived data and traditional ADC/TDC data have been performed. Following four panes show some correlation plots between various data. Pane 1 shows DSC TDC value vs. Fastbus LRS1877 TDC values. The gap around zero comes from the trigger, which supresses prompt events. To compare ADC data, further cuts are necessary. Pane 2 (top right) shows raw DSC ADC data (Y-scale) vs. CAMAC LRS2249 ADC data (X-axis). Since our ADC gate is only 25ns wide, singnals which come at t<>0 are not covered completely by the ADC gate and therefore show smaller ADC values. This artefact is of course not present in the DSC data, which covers the full range of 256ns. On pane 2 one can therefore see a distribution which, when projected onto the Y-axis, shows a peak at ~270 which corresponds to valid pion signals. The projection onto the X-axis only produces a broad distribution going all the way to zero, which corresponds to events where the inoming pion came >25ns before the trigger and therefore is not included in the CAMAC ADC data. Only the small band at the main diagonal shows a clean correlation between DSC and ADC data. If one cuts on prompt events (Pane 3), only this band is shown. The correlation comes from the fact that for prompt events the signal always comes at t=0 and is therefore contained fully in the ADC gate.

Pane 4 shows DSC ADC vs. DSC TDC data. There is no visible dependency, as it should be, unlike at the CAMAC ADC data.

For a closer look at the timing resolution, pane 1 has been enlarged:

One can see the 0.5ns resolution of the LRS1877 TDC's as vertical bands and some time jitter between the two variables. The value DSC TDC - FASTBUS TDC has a distribution with a sigma of 0.4ns which comes mainly from the limited LRS1877 TDC resolution. To make a more careful study of the DSC timing resolution one would need TDCs with a better timing resolution. A rule of thumb is that with a sampling speed of 2ns one can obtain a time accuracy of 0.2ns, taken into account the full waveform during the fitting process.

S. Ritt, 31 Jan 2000