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1\section{Calibration \label{sec:calibration}}
2
3
4In this section, we describe the tests performed using light pulses of different colour,
5pulse shapes and intensities with the MAGIC LED Calibration Pulser Box \cite{hardware-manual}.
6\par
7The LED pulser system is able to provide fast light pulses of 3--4\,ns FWHM
8with intensities ranging from 3--4 to more than 500 photo-electrons in one inner photo-multiplier of the
9camera. These pulses can be produced in three colours {\textit {\bf green, blue}} and
10{\textit{\bf UV}}.
11
12\begin{table}[htp]
13\centering
14\begin{tabular}{|c|c|c|c|c|c|c|}
15\hline
16\hline
17\multicolumn{7}{|c|}{The possible pulsed light colours} \\
18\hline
19\hline
20Colour & Wavelength & Spectral Width & Min. Nr. & Max. Nr. & Secondary & FWHM \\
21 & [nm] & [nm] & Phe's & Phe's & Pulses & Pulse [ns]\\
22\hline
23Green & 520 & 40 & 6 & 120 & yes & 3--4 \\
24\hline
25Blue & 460 & 30 & 6 & 500 & yes & 3--4 \\
26\hline
27UV & 375 & 12 & 3 & 50 & no & 2--3 \\
28\hline
29\hline
30\end{tabular}
31\caption{The pulser colours available from the calibration system}
32\label{tab:pulsercolours}
33\end{table}
34
35Table~\ref{tab:pulsercolours} lists the available colours and intensities and
36figures~\ref{fig:pulseexample1leduv} and~\ref{fig:pulseexample23ledblue} show exemplary pulses
37as registered by the FADCs.
38Whereas the UV-pulse is very stable, the green and blue pulses show sometimes smaller secondary
39pulses after about 10--40\,ns from the main pulse.
40One can see that the very stable UV-pulses are unfortunately only available in such intensities as to
41not saturate the high-gain readout channel. However, the brightest combination of light pulses easily
42saturates all channels in the camera, but does not reach a saturation of the low-gain readout.
43\par
44Our tests can be classified into three subsections:
45
46\begin{enumerate}
47\item Un-calibrated pixels and events: These tests measure the percentage of failures of the extractor
48resulting either in a pixel declared as un-calibrated or in an event which produces a signal outside
49of the expected Gaussian distribution.
50\item Number of photo-electrons: These tests measure the reconstructed numbers of photo-electrons, their
51spread over the camera and the ratio of the obtained mean values for outer and inner pixels, respectively.
52\item Linearity tests: These tests measure the linearity of the extractor with respect to pulses of
53different intensity and colour.
54\item Time resolution: These tests show the time resolution and stability obtained with different
55intensities and colours.
56\end{enumerate}
57
58\begin{figure}[htp]
59\centering
60\includegraphics[width=0.48\linewidth]{1LedUV_Pulse_Inner.eps}
61\includegraphics[width=0.48\linewidth]{1LedUV_Pulse_Outer.eps}
62\caption{Example of a calibration pulse from the lowest available intensity (1\,Led UV).
63The left plot shows the signal obtained in an inner pixel, the right one the signal in an outer pixel.
64Note that the pulse height fluctuates much more than suggested from these pictures. Especially, a
65zero-pulse is also possible.}
66\label{fig:pulseexample1leduv}
67\end{figure}
68
69\begin{figure}[htp]
70\centering
71\includegraphics[width=0.48\linewidth]{23LedsBlue_Pulse_Inner.eps}
72\includegraphics[width=0.48\linewidth]{23LedsBlue_Pulse_Outer.eps}
73\caption{Example of a calibration pulse from the highest available mono-chromatic intensity (23\,Leds Blue).
74The left plot shows the signal obtained in an inner pixel, the right one the signal in an outer pixel.
75One the left side of both plots, the (saturated) high-gain channel is visible,
76on the right side from FADC slice 18 on,
77the delayed low-gain
78pulse appears. Note that in the left plot, there is a secondary pulses visible in the tail of the
79high-gain pulse. }
80\label{fig:pulseexample23ledblue}
81\end{figure}
82
83We used data taken on the 7$^{th}$ of June, 2004 with different pulser LED combinations, each taken with
8416384 events. 19 different calibration configurations have been tested.
85The corresponding MAGIC data run numbers range from nr. 31741 to 31772. These data was taken
86before the latest camera repair access which resulted in a replacement of about 2\% of the pixels known to be
87mal-functioning at that time.
88There is thus a lower limit to the number of un-calibrated pixels of about 1.5--2\% of known
89mal-functioning photo-multipliers.
90\par
91Although we had looked at and tested all colour and extractor combinations resulting from these data,
92we refrain ourselves to show here only exemplary behaviour and results of extractors.
93All plots, including those which are not displayed in this TDAS, can be retrieved from the following
94locations:
95
96\begin{verbatim}
97http://www.magic.ifae.es/~markus/pheplots/
98http://www.magic.ifae.es/~markus/timeplots/
99\end{verbatim}
100
101%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
102
103\subsection{Un-Calibrated Pixels and Events \label{sec:uncalibrated}}
104
105The MAGIC calibration software incorporates a series of checks to sort out mal-functioning pixels.
106Except for the software bug searching criteria, the following exclusion criteria can apply:
107
108\begin{enumerate}
109\item The reconstructed mean signal is less than 2.5 times the extractor resolution $R$ from zero.
110(2.5 Pedestal RMS in the case of the simple fixed window extractors, see section~\ref{sec:pedestals}).
111This criterium essentially cuts out
112dead pixels.
113\item The reconstructed mean signal error is smaller than its value. This criterium cuts out
114signal distributions which fluctuate so much that their RMS is bigger than its mean value. This
115criterium cuts out ``ringing'' pixels or mal-functioning extractors.
116\item The reconstructed mean number of photo-electrons lies 4.5 sigma outside
117the distribution of photo-electrons obtained with the inner or outer pixels in the camera, respectively.
118This criterium cuts out channels with apparently deviating (hardware) behaviour compared to
119the rest of the camera readout\footnote{This criteria is not applied any more in the standard analysis,
120although we kept using it here}.
121\item All pixels with reconstructed negative mean signal or with a
122mean numbers of photo-electrons smaller than one. Pixels with a negative pedestal RMS subtracted
123sigma occur, especially when stars are focused onto that pixel during the pedestal taking (resulting
124in a large pedestal RMS), but have moved to another pixel during the calibration run. In this case, the
125number of photo-electrons would result artificially negative. If these
126channels do not show any other deviating behaviour, their number of photo-electrons gets replaced by the
127mean number of photo-electrons in the camera, and the channel is further calibrated as normal.
128\end{enumerate}
129
130Moreover, the number of events are counted which have been reconstructed outside a 5$\sigma$ region
131from the mean signal. These events are called ``outliers''. Figure~\ref{fig:outlier} shows a typical
132outlier obtained with the digital filter applied on a low-gain signal, and figure~\ref{fig:unsuited:all}
133shows the average number of all excluded pixels and outliers obtained from all 19 calibration configurations.
134One can already see that the largest window sizes yield a high number of un-calibrated pixels, mostly
135due to the missing ability to recognize the low-intensity pulses (see later). One can also see that
136the amplitude extracting spline yields a higher number of outliers than the rest of the extractors.
137\par
138The global champion in lowest number of un-calibrated pixels results to be
139{\textit{\bf MExtractTimeAndChargeSpline}} extracting the integral over two FADC slices (extractor \#25).
140The one with the lowest number of outliers is
141{\textit{\bf MExtractFixedWindowPeakSearch}} with an extraction range of 2 slices (extractor \#11).
142
143\begin{figure}[htp]
144\centering
145\includegraphics[width=0.95\linewidth]{Outlier.eps}
146\caption{Example of an event classified as ``un-calibrated''. The histogram has been obtained
147using the digital filter (extractor \#32) applied to a high-intensity blue pulse (run 31772).
148The event marked as ``outlier'' clearly has been mis-reconstructed. It lies outside the 5 sigma
149region from the fitted mean.}
150\label{fig:outlier}
151\end{figure}
152
153\begin{figure}[htp]
154\centering
155\includegraphics[height=0.75\textheight]{UnsuitVsExtractor-all.eps}
156\caption{Un-calibrated pixels and outlier events averaged over all available
157calibration runs.}
158\label{fig:unsuited:all}
159\end{figure}
160
161The following figures~\ref{fig:unsuited:5ledsuv},~\ref{fig:unsuited:1leduv},~\ref{fig:unsuited:2ledsgreen}
162and~\ref{fig:unsuited:23ledsblue} show the resulting numbers of un-calibrated pixels and events for
163different colours and intensities. Because there is a strong anti-correlation between the number of
164excluded pixels and the number of outliers per event, we have chosen to show these numbers together.
165
166\par
167
168\begin{figure}[htp]
169\centering
170\includegraphics[height=0.95\textheight]{UnsuitVsExtractor-5LedsUV-Colour-12.eps}
171\caption{Un-calibrated pixels and outlier events for a typical calibration
172pulse of UV-light which does not saturate the high-gain readout.}
173\label{fig:unsuited:5ledsuv}
174\end{figure}
175
176\begin{figure}[htp]
177\centering
178\includegraphics[height=0.95\textheight]{UnsuitVsExtractor-1LedUV-Colour-04.eps}
179\caption{Un-calibrated pixels and outlier events for a very low
180intensity pulse.}
181\label{fig:unsuited:1leduv}
182\end{figure}
183
184\begin{figure}[htp]
185\centering
186\includegraphics[height=0.95\textheight]{UnsuitVsExtractor-2LedsGreen-Colour-02.eps}
187\caption{Un-calibrated pixels and outlier events for a typical green pulse.}
188\label{fig:unsuited:2ledsgreen}
189\end{figure}
190
191\begin{figure}[htp]
192\centering
193\includegraphics[height=0.95\textheight]{UnsuitVsExtractor-23LedsBlue-Colour-00.eps}
194\caption{Un-calibrated pixels and outlier events for a high-intensity blue pulse.}
195\label{fig:unsuited:23ledsblue}
196\end{figure}
197
198One can see that in general, big extraction windows raise the
199number of un-calibrated pixels and are thus less stable. Especially for the very low-intensity
200\textit{\bf 1\,Led\,UV}-pulse, the big extraction windows -- summing 8 or more slices -- cannot calibrate more
201than 50\% of the inner pixels (fig.~\ref{fig:unsuited:1leduv}).
202This is an expected behavior since big windows
203sum up more noise which in turn makes the search for the small signal more difficult.
204\par
205In general, one can also find that all ``sliding window''-algorithms (extractors \#17-32) discard
206less pixels than the corresponding ``fixed window''-ones (extractors \#1--16).
207
208The spline (extractors \#23--27) and the digital filter with the correct weights (extractors \#30-33) discard
209the least number of pixels and are also robust against slight modifications of the pulse form
210(of the weights for the digital filter).
211\par
212Concerning the numbers of outliers, one can conclude that in general, the numbers are very low never exceeding
2130.1\% except for the amplitude-extracting spline which seems to mis-reconstruct a certain type of events.
214It seems however that the spline algorithm extracting the amplitude of the signal produces an over-proportional
215\par
216In conclusion, already this first test excludes all extractors with too large window sizes because
217they are not able to extract cleanly small signals produced by about 4 photo-electrons. Moreover,
218the amplitude extracting spline produces a significantly higher number of outlier events.
219
220\clearpage
221
222%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
223
224\subsection{Number of Photo-Electrons \label{sec:photo-electrons}}
225
226Assuming that the readout chain adds only negligible noise to the one
227introduced by the photo-multiplier itself, one can make the assumption that the variance of the
228true signal, $S$, is the amplified Poisson variance of the number of photo-electrons,
229multiplied with the excess noise of the photo-multiplier which itself is
230characterized by the excess-noise factor $F$:
231
232\begin{equation}
233Var(S) = F^2 \cdot Var(N_{phe}) \cdot \frac{<S>^2}{<N_{phe}>^2}
234\label{eq:excessnoise}
235\end{equation}
236
237After introducing the effect of the night-sky background (eq.~\ref{eq:rmssubtraction})
238and assuming that the variance of the number of photo-electrons is equal
239to the mean number of photo-electrons (because of the Poisson distribution),
240one obtains an expression to retrieve the mean number of photo-electrons impinging on the photo-multiplier from the
241mean extracted signal, $\widehat{S}$, and the RMS of the extracted signal obtained from
242pure pedestal runs $R$ (see section~\ref{sec:ffactor}):
243
244\begin{equation}
245<N_{phe}> \approx F^2 \cdot \frac{<\widehat{S}>^2}{Var(\widehat{S}) - R^2}
246\label{eq:pheffactor}
247\end{equation}
248
249In theory, eq.~\ref{eq:pheffactor} must not depend on the extractor! Effectively, we will use it to test the
250quality of our extractors by requiring that a valid extractor yields the same number of photo-electrons
251for all pixels of a same type and does not deviate from the number obtained with other extractors.
252As the camera is flat-fielded, but the number of photo-electrons impinging on an inner and an outer pixel is
253different, we also use the ratio of the mean numbers of photo-electrons from the outer pixels to the one
254obtained from the inner pixels as a test variable. In the ideal case, it should always yield its central
255value of about 2.6$\pm$0.1~\cite{michele-diploma}.
256\par
257In our case, there is an additional complication due to the fact that the green and blue coloured light pulses
258show secondary pulses which destroy the Poisson behaviour of the number of photo-electrons. We will
259have to split our sample of extractors into those being affected by the secondary pulses and those
260being immune to this effect.
261\par
262Figures~\ref{fig:phe:5ledsuv},~\ref{fig:phe:1leduv},~\ref{fig:phe:2ledsgreen}~and~\ref{fig:phe:23ledsblue} show
263some of the obtained results. One can see a rather good stability for the standard
264{\textit{\bf 5\,Leds\,UV}}\ pulse, except for the extractors {\textit{\bf MExtractFixedWindowPeakSearch}}, initialized
265with an extraction window of 2 slices.
266\par
267There is a considerable difference for all shown non-standard pulses. Especially the pulses from green
268and blue LEDs
269show a clear dependency of the number of photo-electrons on the extraction window. Only the largest
270extraction windows seem to catch the entire range of (jittering) secondary pulses and get the ratio
271of outer vs. inner pixels right. However, they (obviously) over-estimate the number of photo-electrons
272in the primary pulse.
273\par
274The strongest discrepancy is observed in the low-gain extraction (fig.~\ref{fig:phe:23ledsblue}) where all
275fixed window extractors with extraction windows smaller than 8 FADC slices fail to reconstruct the correct numbers.
276This has to do with the fact that
277the fixed window extractors fail to catch a significant part of the (larger) pulse because of the
2781~FADC slice event-to-event jitter. Also the sliding windows smaller than 6 FADC slices and the spline smaller than
2792 FADC slices reproduce too small numbers of photo-electrons. Moreover, the digital filter shows a small dependency
280of the number of photo-electrons w.r.t. the extration window.
281\par
282
283
284\begin{figure}[htp]
285\centering
286\includegraphics[height=0.92\textheight]{PheVsExtractor-5LedsUV-Colour-12.eps}
287\caption{Number of photo-electrons from a typical, not saturating calibration pulse of colour UV,
288reconstructed with each of the tested signal extractors.
289The first plots shows the number of photo-electrons obtained for the inner pixels, the second one
290for the outer pixels and the third shows the ratio of the mean number of photo-electrons for the
291outer pixels divided by the mean number of photo-electrons for the inner pixels. Points
292denote the mean of all not-excluded pixels, the error bars their RMS.}
293\label{fig:phe:5ledsuv}
294\end{figure}
295
296\begin{figure}[htp]
297\centering
298\includegraphics[height=0.92\textheight]{PheVsExtractor-1LedUV-Colour-04.eps}
299\caption{Number of photo-electrons from a typical, very low-intensity calibration pulse of colour UV,
300reconstructed with each of the tested signal extractors.
301The first plots shows the number of photo-electrons obtained for the inner pixels, the second one
302for the outer pixels and the third shows the ratio of the mean number of photo-electrons for the
303outer pixels divided by the mean number of photo-electrons for the inner pixels. Points
304denote the mean of all not-excluded pixels, the error bars their RMS.}
305\label{fig:phe:1leduv}
306\end{figure}
307
308\begin{figure}[htp]
309\centering
310\includegraphics[height=0.92\textheight]{PheVsExtractor-2LedsGreen-Colour-02.eps}
311\caption{Number of photo-electrons from a typical, not saturating calibration pulse of colour green,
312reconstructed with each of the tested signal extractors.
313The first plots shows the number of photo-electrons obtained for the inner pixels, the second one
314for the outer pixels and the third shows the ratio of the mean number of photo-electrons for the
315outer pixels divided by the mean number of photo-electrons for the inner pixels. Points
316denote the mean of all not-excluded pixels, the error bars their RMS.}
317\label{fig:phe:2ledsgreen}
318\end{figure}
319
320
321\begin{figure}[htp]
322\centering
323\includegraphics[height=0.92\textheight]{PheVsExtractor-23LedsBlue-Colour-00.eps}
324\caption{Number of photo-electrons from a typical, high-gain saturating calibration pulse of colour blue,
325reconstructed with each of the tested signal extractors.
326The first plots shows the number of photo-electrons obtained for the inner pixels, the second one
327for the outer pixels and the third shows the ratio of the mean number of photo-electrons for the
328outer pixels divided by the mean number of photo-electrons for the inner pixels. Points
329denote the mean of all not-excluded pixels, the error bars their RMS.}
330\label{fig:phe:23ledsblue}
331\end{figure}
332
333One can see that all extractors using a large window belong to the class of extractors being affected
334by the secondary pulses, except for the digital filter.
335\par
336The extractor {\textit{\bf MExtractTimeAndChargeDigitalFilter}} seems to be stable against modifications in the
337exact form of the weights in the high-gain readout channel since all applied weights yield about
338the same number of photo-electrons and the same ratio of outer vs. inner pixels, except if one applies the cosmics
339weights to the very low-intensity pulse $1\,LED\,UV$ where a slight increase in photo-electrons is observed.
340\par
341All sliding window and spline algorithms yield a stable ratio of outer vs. inner pixels in the high and the low-gain.
342\par
343Concluding, there is no fixed window extractor yielding always the correct number of photo-electrons,
344except for the extraction window of 8 FADC slices.
345Either the number of photo-electrons itself is wrong or the ratio of outer vs. inner pixels is
346not correct. All sliding window algorithms seem to reproduce the correct numbers if one takes into
347account the after-pulse behaviour of the light pulser itself. The digital filter seems to be
348stable against changing the pulse width from 1~to~4\,ns.
349
350\clearpage
351
352%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
353
354\subsection{Linearity \label{sec:calibration:linearity}}
355
356\begin{figure}[htp]
357\centering
358\includegraphics[width=0.99\linewidth]{PheVsCharge-4.eps}
359\caption{Conversion factor $c_{phe}$ for three exemplary inner pixels (upper plots)
360and three exemplary outer ones (lower plots) obtained with the extractor
361{\textit{MExtractFixedWindow}} on a window size of 8 high-gain and 8 low-gain slices
362(extractor \#4). }
363\label{fig:linear:phevscharge4}
364\end{figure}
365
366In this section, we test the linearity of the conversion factors FADC counts to photo-electrons:
367
368\begin{equation}
369c_{phe} =\ <N_{phe}> / <\widehat{S}>
370\end{equation}
371
372As the photo-multiplier and the subsequent
373optical transmission devices~\cite{david} is a linear device over a
374wide dynamic range, the number of photo-electrons per charge has to remain constant over the tested
375linearity region.
376\par
377A first test concerns the stability of the conversion factor: mean number of averaged photo-electrons
378per FADC counts over the tested intensity region. This test includes all systematic uncertainties
379in the calculation of the number of photo-electrons and the computation of the mean signal.
380A more detailed investigation of the linearity will be shown in a
381separate TDAS~\cite{tdas-calibration}, although there, the number of photo-electrons will be calculated
382in a more independent way.
383
384\par
385Figure~\ref{fig:linear:phevscharge4} shows the conversion factor $c_{phe}$ obtained for different light intensities
386and colours for three exemplary inner and three exemplary outer pixels using a fixed window on
3878 FADC slices. The conversion factor seems to be linear to a good approximation,
388except for two cases:
389\begin{itemize}
390\item The green pulses yield systematically low conversion factors
391\item Some of the pixels show a difference
392between the high-gain ($<$100\ phes for the inner, $<$300\ phes for the outer pixels) and the low-gain
393($>$100\ phes for the inner, $>$300\ phes for the outer pixels) region and
394a rather good stability of $c_{phe}$ for each region separately.
395\end{itemize}
396
397We conclude that, apart from the two reasons above,
398the fixed window extractor \#4 is a linear extractor for both high-gain
399and low-gain regions, separately.
400\par
401
402Figures~\ref{fig:linear:phevscharge9} and~\ref{fig:linear:phevscharge15} show the conversion factors
403using an integrated spline and a fixed window with global peak search, respectively, over
404an extraction window of 8 FADC slices. The same behaviour is obtained as before. These extractors are
405linear to a good approximation, except for the two cases mentionned above.
406\par
407
408\begin{figure}[h!]
409\centering
410\includegraphics[width=0.99\linewidth]{PheVsCharge-9.eps}
411\caption{Conversion factor $c_{phe}$ for three exemplary inner pixels (upper plots)
412and three exemplary outer ones (lower plots) obtained with the extractor
413{\textit{MExtractFixedWindowSpline}}
414on a window size of 8 high-gain and 8 low-gain slices (extractor \#9). }
415\label{fig:linear:phevscharge9}
416\end{figure}
417
418\begin{figure}[h!]
419\centering
420\includegraphics[width=0.99\linewidth]{PheVsCharge-15.eps}
421\caption{Conversion factor $c_{phe}$ for three exemplary inner pixels (upper plots)
422and three exemplary outer ones (lower plots) obtained with the extractor
423{\textit{MExtractFixedWindowPeakSearch}} on a window size of 8 high-gain and 8 low-gain slices
424(extractor \#15). }
425\label{fig:linear:phevscharge15}
426\end{figure}
427
428\begin{figure}[h!]
429\centering
430\includegraphics[width=0.99\linewidth]{PheVsCharge-11.eps}
431\caption{Example of a the development of the conversion factor FADC counts to photo-electrons for three
432exemplary inner pixels (upper plots) and three exemplary outer ones (lower plots) obtained with the extractor
433{\textit{MExtractFixedWindowPeakSearch}}
434on a window size of 2 high-gain and 2 low-gain slices (extractor \#11). }
435\label{fig:linear:phevscharge11}
436\end{figure}
437
438Figure~\ref{fig:linear:phevscharge11} shows the conversion factors using a fixed window with global peak search
439integrating a window of 2 FADC slices. One can see that the linearity is completely lost! Especially in the low-gain,
440the reconstructed number of photo-electrons is much too low and the conversion factors bend down. A similiar behaviour can
441be found for all extractors with window sizes smaller than 6 FADC slices, especially in the low-gain region. (This behaviour
442was already visible in the investigations on the number of photo-electrons in the previous section~\ref{sec:photo-electrons}).
443\par
444Figure~\ref{fig:linear:phevscharge20} shows the conversion factors using a sliding window of 6 FADC slices.
445The linearity is maintained like in the previous examples, except for the smallest signals the effect
446of the bias is already visible.
447\par
448
449\begin{figure}[h!]
450\centering
451\includegraphics[width=0.99\linewidth]{PheVsCharge-20.eps}
452\caption{Example of a the development of the conversion factor FADC counts to photo-electrons for three
453exemplary inner pixels (upper plots) and three exemplary outer ones (lower plots) obtained with the extractor
454{\textit{MExtractTimeAndChargeSlidingWindow}}
455on a window size of 6 high-gain and 6 low-gain slices (extractor \#20). }
456\label{fig:linear:phevscharge20}
457\end{figure}
458
459Figure~\ref{fig:linear:phevscharge23} shows the conversion factors using the amplitude-extracting spline
460(extractor \#23).
461Here, the linearity worse than in the previous sample. A very clear difference between high-gain and
462low-gain regions can be seen as well as a bigger general spread in conversion factors. In order to investigate
463if there is a common, systematic effect of the extractor, we show the averaged conversion factors over all
464inner and outer pixels in figure~\ref{fig:linear:phevschargearea23}. Both characteristics are maintained
465there. Although the differences between high-gain and low-gain can be easily corrected for, we conclude
466that extractor \#23 is still unstable against the linearity tests.
467\par
468
469\begin{figure}[h!]
470\centering
471\includegraphics[width=0.99\linewidth]{PheVsCharge-23.eps}
472\caption{Conversion factor $c_{phe}$ for three exemplary inner pixels (upper plots)
473and three exemplary outer ones (lower plots) obtained with the extractor
474{\textit{MExtractTimeAndChargeSpline}} with amplitude extraction (extractor \#23). }
475\label{fig:linear:phevscharge23}
476\vspace{\floatsep}
477\includegraphics[width=0.9\linewidth]{PheVsCharge-Area-23.eps}
478\caption{Conversion factor $c_{phe}$ averaged over all inner (left) and all outer (right) pixels
479obtained with the extractor
480{\textit{MExtractTimeAndChargeSpline}} with amplitude extraction (extractor \#23). }
481\label{fig:linear:phevschargearea23}
482\end{figure}
483
484Figure~\ref{fig:linear:phevscharge24} shows the conversion factors using a spline integrating over
485one effective FADC slice in the high-gain and 1.5 effective FADC slices in the low-gain (extractor \#24).
486The same problems are found as with extractor \#23, however to a much lower extent.
487The difference between high-gain and low-gain regions is less pronounced and the spread
488in conversion factors is smaller.
489Figure~\ref{fig:linear:phevschargearea24} shows already rather good stability except for the two
490lowest intensity pulses in green and blue. We conclude that extractor \#24 is still un-stable, but
491preferable to amplitude extractor.
492\par
493
494\begin{figure}[h!]
495\centering
496\includegraphics[width=0.99\linewidth]{PheVsCharge-24.eps}
497\caption{Conversion factor $c_{phe}$ for three exemplary inner pixels (upper plots)
498and three exemplary outer ones (lower plots) obtained with the extractor
499{\textit{MExtractTimeAndChargeSpline}} with window size of 1 high-gain and 2 low-gain slices
500(extractor \#24). }
501\label{fig:linear:phevscharge24}
502\vspace{\floatsep}
503\includegraphics[width=0.9\linewidth]{PheVsCharge-Area-24.eps}
504\caption{Conversion factor $c_{phe}$ averaged over all inner (left) and all outer (right) pixels
505obtained with the extractor
506{\textit{MExtractTimeAndChargeSpline}} with window size of 1 high-gain and 2 low-gain slices
507(extractor \#24). }
508\label{fig:linear:phevschargearea24}
509\end{figure}
510
511Looking at figure~\ref{fig:linear:phevscharge25}, one can see that raising the integration window
512to two effective FADC slices in the high-gain and three effective FADC slices in the low-gain
513(extractor \#25), the stability is completely resumed, except for
514a small systematic increase of the conversion factor in the low-gain range. This effect
515is not very significant, however it can be seen in five out of the
516six tested channels. We conclude that extractor \#25 is almost as stable as the fixed window extractors.
517\par
518
519\begin{figure}[htp]
520\centering
521\includegraphics[width=0.99\linewidth]{PheVsCharge-25.eps}
522\caption{Conversion factor $c_{phe}$ for three exemplary inner pixels (upper plots)
523and three exemplary outer ones (lower plots) obtained with the extractor
524{\textit{MExtractTimeAndChargeSpline}} with window size of 2 high-gain and 3 low-gain slices
525(extractor \#25). }
526\label{fig:linear:phevscharge25}
527\vspace{\floatsep}
528\includegraphics[width=0.9\linewidth]{PheVsCharge-Area-25.eps}
529\caption{Conversion factor $c_{phe}$ averaged over all inner (left) and all outer (right) pixels
530obtained with the extractor
531{\textit{MExtractTimeAndChargeSpline}} with window size of 2 high-gain and 3 low-gain slices
532(extractor \#25). }
533\label{fig:linear:phevschargearea25}
534\end{figure}
535
536Figure~\ref{fig:linear:phevscharge30} and~\ref{fig:linear:phevscharge31} show the conversion factors using a digital filter,
537applied on 6 FADC slices and respectively 4 FADC slices with weights calculated from the UV-calibration pulse.
538One can see that one or two blue calibration pulses at low and intermediate intensity fall
539out of the linear region, moreover there is a small systematic offset between the high-gain and low-gain region.
540It seems that the digital filter does not pass this test if the pulse form changes for more than 2\,ns from the
541expected one. The effect is not as problematic as it may appear here, because the actual calibration
542will not calculate the number of photo-electrons (with the F-Factor method) for every signal intensity.
543Thus, one possible reason for the instability falls away in the cosmics analysis. However, the limits
544of this extraction are clearly visible here and have to be monitored further.
545
546\par
547
548\begin{figure}[htp]
549\centering
550\includegraphics[width=0.99\linewidth]{PheVsCharge-30.eps}
551\caption{Conversion factor $c_{phe}$ for three exemplary inner pixels (upper plots)
552and three exemplary outer ones (lower plots) obtained with the extractor
553{\textit{MExtractTimeAndChargeDigitalFilter}}
554using a window size of 6 high-gain and 6 low-gain slices with UV-weights (extractor \#30). }
555\label{fig:linear:phevscharge30}
556\vspace{\floatsep}
557\includegraphics[width=0.9\linewidth]{PheVsCharge-Area-30.eps}
558\caption{Conversion factor $c_{phe}$ averaged over all inner (left) and all outer (right) pixels
559obtained with the extractor
560{\textit{MExtractTimeAndChargeDigitalFilter}} with window size of 6 high-gain and 6 low-gain slices and UV-weight
561(extractor \#30). }
562\label{fig:linear:phevschargearea30}
563\end{figure}
564
565
566\begin{figure}[htp]
567\centering
568\includegraphics[width=0.99\linewidth]{PheVsCharge-31.eps}
569\caption{Conversion factor $c_{phe}$ for three exemplary inner pixels (upper plots)
570and three exemplary outer ones (lower plots) obtained with the extractor
571{\textit{MExtractTimeAndChargeDigitalFilter}} using a window size of
5724 high-gain and 4 low-gain slices (extractor \#31). }
573\label{fig:linear:phevscharge31}
574\vspace{\floatsep}
575\includegraphics[width=0.9\linewidth]{PheVsCharge-Area-31.eps}
576\caption{Conversion factor $c_{phe}$ averaged over all inner (left) and all outer (right) pixels
577obtained with the extractor
578{\textit{MExtractTimeAndChargeDigitalFilter}} with window size of 6 high-gain and 6 low-gain slices and blue weights
579(extractor \#31). }
580\label{fig:linear:phevschargearea3}
581\end{figure}
582
583\clearpage
584
585\subsection{Relative Arrival Time Calibration}
586
587The extractors \#17--33 are able to compute the arrival time of each pulse. The calibration LEDs
588deliver a fast-rising pulses, uniform over the camera in signal size and time.
589We estimate the time-uniformity to better
590than 300\,ps, a limit due to the different travel times of the light between inner and outer parts of the
591camera. Since the calibration does not permit a precise measurement of the absolute arrival time, we measure
592the relative arrival time for every channel with respect to a reference channel (usually pixel Nr.\,1):
593
594\begin{equation}
595\delta t_i = t_i - t_1
596\end{equation}
597
598where $t_i$ denotes the reconstructed arrival time of pixel number $i$ and $t_1$ the reconstructed
599arrival time of the reference pixel nr. 1 (software numbering). In one calibration run, one can then fill
600histograms of $\delta t_i$ and fit them to the expected Gaussian distribution. The fits
601yield a mean $\mu(\delta t_i)$, comparable to
602systematic delays in the signal travel time, and a sigma $\sigma(\delta t_i)$, a measure of the
603combined time resolutions of pixel $i$ and pixel 1. Assuming that the PMTs and readout channels are
604of a same kind, we obtain an approximate time resolution of pixel $i$:
605
606\begin{equation}
607t^{res}_i \approx \sigma(\delta t_i)/\sqrt(2)
608\end{equation}
609
610Figures~\ref{fig:reltimesinnerleduv} show distributions of $\delta t_i$
611for a typical inner pixel and a non-saturating calibration pulse of UV-light,
612obtained with six different extractors.
613One can see that all of them yield acceptable Gaussian distributions,
614except for the sliding window extracting 2 slices which shows a three-peak structure and cannot be fitted.
615We discarded that particular extractor from the further studies of this section.
616
617\begin{figure}[htp]
618\centering
619\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor17.eps}
620\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor18.eps}
621\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor23.eps}
622\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor24.eps}
623\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor30.eps}
624\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor31.eps}
625\caption{Examples of a distributions of relative arrival times $\delta t_i$ of an inner pixel (Nr. 100) \protect\\
626Top: {\textit{\bf MExtractTimeAndChargeSlidingWindow}} over 2 slices (\#17) and 4 slices (\#18) \protect\\
627Center: {\textit{\bf MExtractTimeAndChargeSpline}} with maximum (\#23) and half-maximum pos. (\#24) \protect\\
628Bottom: {\textit{\bf MExtractTimeAndChargeDigitalFilter}} fitted to a UV-calibration pulse over 6 slices (\#30) and 4 slices (\#31) \protect\\
629A medium sized UV-pulse (5\,Leds UV) has been used which does not saturate the high-gain readout channel.}
630\label{fig:reltimesinnerleduv}
631\end{figure}
632
633Figures~\ref{fig:reltimesinnerledblue1} and~\ref{fig:reltimesinnerledblue2} show
634the distributions of $\delta t_i$ for a typical inner pixel and an intense, high-gain-saturating calibration
635pulse of blue light.
636One can see that the sliding window extractors yield double Gaussian structures, except for the
637largest window sizes of 8 and 10 FADC slices. Even then, the distributions are not exactly Gaussian.
638The maximum position extracting spline also yields distributions which are not exactly Gaussian and seems
639to miss the exact arrival time in some events. Only the position of the half-maximum gives the
640expected result of a single Gaussian distribution.
641A similiar problem occurs in the case of the digital filter: If one takes the correct weights
642(fig.~\ref{fig:reltimesinnerledblue2} bottom), the distribution is perfectly Gaussian and the resolution good,
643however a rather slight change from the blue calibration pulse weights to cosmics pulses weights (top)
644adds a secondary peak of events with mis-reconstructed arrival times.
645
646\begin{figure}[htp]
647\centering
648\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor18_logain.eps}
649\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor19_logain.eps}
650\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor21_logain.eps}
651\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor22_logain.eps}
652\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor23_logain.eps}
653\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor24_logain.eps}
654\caption{Examples of a distributions of relative arrival times $\delta t_i$ of an inner pixel (Nr. 100) \protect\\
655Top: {\textit{\bf MExtractTimeAndChargeSlidingWindow}} over 4 slices (\#18) and 6 slices (\#19) \protect\\
656Center: {\textit{\bf MExtractTimeAndChargeSlidingWindow}} over 8 slices (\#20) and 10 slices (\#21)\protect\\
657Bottom: {\textit{\bf MExtractTimeAndChargeSpline}} with maximum (\#23) and half-maximum pos. (\#24) \protect\\
658A strong Blue pulse (23\,Leds Blue) has been used which does not saturate the high-gain readout channel.}
659\label{fig:reltimesinnerledblue1}
660\end{figure}
661
662\begin{figure}[htp]
663\centering
664\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor30_logain.eps}
665\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor31_logain.eps}
666\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor32_logain.eps}
667\includegraphics[width=0.45\linewidth]{RelTime_100_Extractor33_logain.eps}
668\caption{Examples of a distributions of relative arrival times $\delta t_i$ of an inner pixel (Nr. 100) \protect\\
669Top: {\textit{\bf MExtractTimeAndChargeDigitalFilter}}
670fitted to cosmics pulses over 6 slices (\#30) and 4 slices (\#31) \protect\\
671Bottom: {\textit{\bf MExtractTimeAndChargeDigitalFilter}} fitted to the correct blue calibration pulse over 6 slices (\#30) and 4 slices (\#31)
672A strong Blue pulse (23\,Leds Blue) has been used which does not saturate the high-gain readout channel.}
673\label{fig:reltimesinnerledblue2}
674\end{figure}
675
676%\begin{figure}[htp]
677%\centering
678%\includegraphics[width=0.31\linewidth]{RelArrTime_Pixel400_10LedUV_Extractor32.eps}
679%\includegraphics[width=0.31\linewidth]{RelArrTime_Pixel400_10LedUV_Extractor23.eps}
680%\includegraphics[width=0.31\linewidth]{RelArrTime_Pixel400_10LedUV_Extractor17.eps}
681%\caption{Example of a two distributions of relative arrival times of an outer pixel with respect to
682%the arrival time of the reference pixel Nr. 1. The left plot shows the result using the digital filter
683% (extractor \#32), the central plot shows the result obtained with the half-maximum of the spline and the
684%right plot the result of the sliding window with a window size of 2 slices (extractor \#17). A
685%medium sized UV-pulse (10Leds UV) has been used which does not saturate the high-gain readout channel.}
686%\label{fig:reltimesouter10leduv}
687%\end{figure}
688
689%\begin{figure}[htp]
690%\centering
691%\includegraphics[width=0.31\linewidth]{RelArrTime_Pixel97_10LedBlue_Extractor23.eps}
692%\includegraphics[width=0.31\linewidth]{RelArrTime_Pixel97_10LedBlue_Extractor32.eps}
693%\caption{Example of a two distributions of relative arrival times of an inner pixel with respect to
694%the arrival time of the reference pixel Nr. 1. The left plot shows the result using the half-maximum of the spline (extractor \#23), the right plot shows the result obtained with the digital filter
695%(extractor \#32). A
696%medium sized Blue-pulse (10Leds Blue) has been used which saturates the high-gain readout channel.}
697%\label{fig:reltimesinner10ledsblue}
698%\end{figure}
699
700
701
702%\begin{figure}[htp]
703%\centering
704%\includegraphics[width=0.31\linewidth]{RelArrTime_Pixel400_10LedBlue_Extractor23.eps}
705%\includegraphics[width=0.31\linewidth]{RelArrTime_Pixel400_10LedBlue_Extractor32.eps}
706%\caption{Example of a two distributions of relative arrival times of an outer pixel with respect to
707%the arrival time of the reference pixel Nr. 1. The left plot shows the result using the half-maximum of the spline (extractor \#23), the right plot shows the result obtained with the digital filter
708%(extractor \#32). A
709%medium sized Blue-pulse (10Leds Blue) has been used which saturates the high-gain readout channel.}
710%\label{fig:reltimesouter10ledsblue}
711%\end{figure}
712
713%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
714
715\subsection{Number of Outliers}
716
717As in section~\ref{sec:uncalibrated}, we tested the number of outliers from the Gaussian distribution
718in order to count how many times the extractor has failed to reconstruct the correct arrival time.
719
720\begin{figure}[htp]
721\centering
722\includegraphics[width=0.95\linewidth]{UnsuitTimeVsExtractor-5LedsUV-Colour-12.eps}
723\caption{Reconstructed arrival time resolutions from a typical, not saturating calibration pulse
724of colour UV, reconstructed with each of the tested arrival time extractors.
725The first plots shows the time resolutions obtained for the inner pixels, the second one
726for the outer pixels. Points
727denote the mean of all not-excluded pixels, the error bars their RMS.}
728\label{fig:time:5ledsuv}
729\end{figure}
730
731\begin{figure}[htp]
732\centering
733\includegraphics[width=0.95\linewidth]{UnsuitTimeVsExtractor-1LedUV-Colour-04.eps}
734\caption{Reconstructed arrival time resolutions from the lowest intensity calibration pulse
735of colour UV (carrying a mean number of 4 photo-electrons),
736reconstructed with each of the tested arrival time extractors.
737The first plots shows the time resolutions obtained for the inner pixels, the second one
738for the outer pixels. Points
739denote the mean of all not-excluded pixels, the error bars their RMS.}
740\label{fig:time:1leduv}
741\end{figure}
742
743\begin{figure}[htp]
744\centering
745\includegraphics[width=0.95\linewidth]{UnsuitTimeVsExtractor-2LedsGreen-Colour-02.eps}
746\caption{Reconstructed arrival time resolutions from a typical, not saturating calibration pulse
747of colour Green, reconstructed with each of the tested arrival time extractors.
748The first plots shows the time resolutions obtained for the inner pixels, the second one
749for the outer pixels. Points
750denote the mean of all not-excluded pixels, the error bars their RMS.}
751\label{fig:time:2ledsgreen}
752\end{figure}
753
754\begin{figure}[htp]
755\centering
756\includegraphics[width=0.95\linewidth]{UnsuitTimeVsExtractor-23LedsBlue-Colour-00.eps}
757\caption{Reconstructed arrival time resolutions from the highest intensity calibration pulse
758of colour blue, reconstructed with each of the tested arrival time extractors.
759The first plots shows the time resolutions obtained for the inner pixels, the second one
760for the outer pixels. Points
761denote the mean of all not-excluded pixels, the error bars their RMS.}
762\label{fig:time:23ledsblue}
763\end{figure}
764
765%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
766
767\subsection{Time Resolution}
768
769\begin{figure}[htp]
770\centering
771\includegraphics[width=0.95\linewidth]{TimeResExtractor-5LedsUV-Colour-12.eps}
772\caption{Reconstructed arrival time resolutions from a typical, not saturating calibration pulse
773of colour UV, reconstructed with each of the tested arrival time extractors.
774The first plots shows the time resolutions obtained for the inner pixels, the second one
775for the outer pixels. Points
776denote the mean of all not-excluded pixels, the error bars their RMS.}
777\label{fig:time:5ledsuv}
778\end{figure}
779
780\begin{figure}[htp]
781\centering
782\includegraphics[width=0.95\linewidth]{TimeResExtractor-1LedUV-Colour-04.eps}
783\caption{Reconstructed arrival time resolutions from the lowest intensity calibration pulse
784of colour UV (carrying a mean number of 4 photo-electrons),
785reconstructed with each of the tested arrival time extractors.
786The first plots shows the time resolutions obtained for the inner pixels, the second one
787for the outer pixels. Points
788denote the mean of all not-excluded pixels, the error bars their RMS.}
789\label{fig:time:1leduv}
790\end{figure}
791
792\begin{figure}[htp]
793\centering
794\includegraphics[width=0.95\linewidth]{TimeResExtractor-2LedsGreen-Colour-02.eps}
795\caption{Reconstructed arrival time resolutions from a typical, not saturating calibration pulse
796of colour Green, reconstructed with each of the tested arrival time extractors.
797The first plots shows the time resolutions obtained for the inner pixels, the second one
798for the outer pixels. Points
799denote the mean of all not-excluded pixels, the error bars their RMS.}
800\label{fig:time:2ledsgreen}
801\end{figure}
802
803\begin{figure}[htp]
804\centering
805\includegraphics[width=0.95\linewidth]{TimeResExtractor-23LedsBlue-Colour-00.eps}
806\caption{Reconstructed arrival time resolutions from the highest intensity calibration pulse
807of colour blue, reconstructed with each of the tested arrival time extractors.
808The first plots shows the time resolutions obtained for the inner pixels, the second one
809for the outer pixels. Points
810denote the mean of all not-excluded pixels, the error bars their RMS.}
811\label{fig:time:23ledsblue}
812\end{figure}
813
814
815\begin{figure}[htp]
816\centering
817\includegraphics[width=0.47\linewidth]{TimeResVsCharge-Area-21.eps}
818\includegraphics[width=0.47\linewidth]{TimeResVsCharge-Area-24.eps}
819\vspace{\floatsep}
820\includegraphics[width=0.47\linewidth]{TimeResVsCharge-Area-30.eps}
821\includegraphics[width=0.47\linewidth]{TimeResVsCharge-Area-31.eps}
822\caption{Reconstructed mean arrival time resolutions as a function of the extracted mean number of
823photo-electrons for the weighted sliding window with a window size of 8 slices (extractor \#21, top left),
824the half-maximum searching spline (extractor \#24, top right),
825the digital filter with UV calibration-pulse weights over 6 slices (extractor \#30, bottom left)
826and the digital filter with UV calibration-pulse weights over 4 slices (extractor \#31, bottom rigth).
827Error bars denote the spread (RMS) of time resolutions of the investigated channels.
828The marker colours show the applied
829pulser colour, except for the last (green) point where all three colours were used.}
830\label{fig:time:dep}
831\end{figure}
832
833
834\begin{figure}[htp]
835\centering
836\includegraphics[width=0.32\linewidth]{TimeResVsSqrtPhe-Area-24.eps}
837\includegraphics[width=0.32\linewidth]{TimeResVsSqrtPhe-Area-30.eps}
838\includegraphics[width=0.32\linewidth]{TimeResVsSqrtPhe-Area-31.eps}
839\caption{Reconstructed arrival time resolutions as a function of the square root of the
840extimated number of photo-electrons for the half-maximum searching spline (extractor \#24, left) a
841and the digital filter with the calibration pulse weigths fitted to UV pulses over 6 FADC slices (extractor \#30, center)
842and the digital filter with the calibration pulse weigths fitted to UV pulses over 4 FADC slices (extractor \#31, right).
843The time resolutions have been fitted from
844The marker colours show the applied
845pulser colour, except for the last (green) point where all three colours were used.}
846\label{fig:time:fit2430}
847\end{figure}
848
849
850%%% Local Variables:
851%%% mode: latex
852%%% TeX-master: "MAGIC_signal_reco"
853%%% End:
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