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1\section{Pedestal Extraction \label{sec:pedestals}}
2
3\subsection{Pedestal RMS}
4
5The background $BG$ (Pedestal)
6can be completely described by the noise-autocorrelation matrix $\boldsymbol{B}$
7(eq.~\ref{eq:autocorr}),
8where the square root of the diagonal elements give what is usually denoted as the ``pedestal RMS''.
9\par
10
11By definition, $\boldsymbol{B}$ and thus the ``pedestal RMS''
12is independent of the signal extractor.
13
14\subsection{Pedestal Fluctuations as Contribution to the Signal Fluctuations \label{sec:ffactor}}
15
16A photo-multiplier signal yields, to a very good approximation, the
17following relation:
18
19\begin{equation}
20\frac{Var[Q]}{<Q>^2} = \frac{1}{<n_{\mathrm{phe}}>} * F^2
21\end{equation}
22
23Here, $Q$ is the signal due to a number $n_{\mathrm{phe}}$ of signal photo-electrons
24(equiv. to the signal $S$) after subtraction of the pedestal. $Var[Q]$ is the fluctuation of the true signal $Q$
25due to the Poisson fluctuations of the number of photo-electrons. Because of:
26
27\begin{eqnarray}
28\widehat{Q} &=& Q + X \\
29Var[\widehat{Q}] &=& Var[Q] + Var[X] \\
30Var[Q] &=& Var[\widehat{Q}] - Var[X]
31\end{eqnarray}
32
33Here, $Var[X]$ is the fluctuation due to the signal extraction, mainly as a result of the background fluctuations and
34the numerical precision of the extraction algorithm.
35\par
36Only in the case that the intrinsic extractor resolution $R$ at fixed background $BG$ does not depend on the signal
37intensity\footnote{Theoretically, this is the case for the digital filter, eq.~\ref{eq:of_noise}.},
38$Var[Q]$ can be obtained from:
39
40\begin{eqnarray}
41Var[Q] &\approx& Var[\widehat{Q}] - Var[\widehat{Q}]\,\vline_{\,Q=0}
42\label{eq:rmssubtraction}
43\end{eqnarray}
44
45%\footnote{%
46%A way to check whether the right RMS has been subtracted is to make the
47%``Razmick''-plot
48%
49%\begin{equation}
50% \frac{Var[\widehat{Q}]}{<\widehat{Q}>^2} \quad \textit{vs.} \quad \frac{1}{<\widehat{Q}>}
51%\end{equation}
52%
53%This should give a straight line passing through the origin. The slope of
54%the line is equal to
55%
56%\begin{equation}
57% c * F^2
58%\end{equation}
59%
60%where $c$ is the photon/ADC conversion factor $<Q>/<m_{pe}>$.}
61
62 One can determine $R$ by applying the signal extractor with a {\textit{\bf fixed window}} to pedestal events, where the
63bias vanishes and measure $Var(\widehat{Q})\,\vline_{\,Q=0}$.
64
65\subsection{Methods to Retrieve Bias and Mean-Squared Error}
66
67In general, the extracted signal variance $R$ is different from the pedestal RMS.
68It can be obtained by applying the signal extractor to pedestal events yielding the bias and
69the resolution $R$.
70\par
71In the case of the digital filter, $R$ is expected to be independent of the
72signal amplitude $S$ and dependent only on the background $BG$ (eq.~\ref{eq:of_noise}).
73%It can then be obtained from the calculation of the variance $Var[\widehat{Q}]$
74%by applying the extractor with a fixed window to pure background events (``pedestal events'').
75\par
76
77In order to calculate the statistical parameters, we proceed in the following ways:
78\begin{enumerate}
79\item Determine $R$ by applying the signal extractor to a fixed window
80 of pedestal events. The background fluctuations can be simulated with different
81 levels of night sky background and the continuous light source, but no signal size
82 dependence can be retrieved by this method.
83\item Determine $B$ and $MSE$ from MC events with added noise.
84% Assuming that $MSE$ and $B$ are negligible for the events without noise, one can
85 With this method, one can get a dependence of both values on the size of the signal,
86 although the MC might contain systematic differences with respect to the real data.
87\item Determine $MSE$ from the error retrieved from the fit results of $\widehat{S}$, which is possible for the
88 fit and the digital filter (eq.~\ref{eq:of_noise}).
89 In principle, all dependencies can be retrieved with this method, although some systematic errors are not taken into account
90 with this method: Deviations of the real pulse from the fitted one, errors in the noise auto-correlation matrix and numerical
91precision issues. All these systematic effects add an additional contribution to the true resolution proportional to the signal strength.
92\end{enumerate}
93
94
95%\begin{figure}[htp]
96%\centering
97%\includegraphics[width=0.3\linewidth]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38993_RelMean.eps}
98%\vspace{\floatsep}
99%\includegraphics[width=0.3\linewidth]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38995_RelMean.eps}
100%\vspace{\floatsep}
101%\includegraphics[width=0.3\linewidth]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38996_RelMean.eps}
102%\caption{MExtractTimeAndChargeSpline with amplitude extraction:
103%Difference in mean pedestal (per FADC slice) between extraction algorithm
104%applied on a fixed window of 1 FADC slice (``extractor random'') and a simple addition of
105%2 fixed FADC slices (``fundamental''). On the left, a run with closed camera has been taken, in the center
106% an opened camera observing an extra-galactic star field and on the right, an open camera being
107%illuminated by the continuous light of the calibration (level: 100). Every entry corresponds to one
108%pixel.}
109%\label{fig:amp:relmean}
110%\end{figure}
111
112%\begin{figure}[htp]
113%\centering
114%\includegraphics[width=0.3\linewidth]{MExtractTimeAndChargeSpline_Rise-and-Fall-Time_0.5_1.5_Range_01_10_02_12_Run_38993_RelMean.eps}
115%\vspace{\floatsep}
116%\includegraphics[width=0.3\linewidth]{MExtractTimeAndChargeSpline_Rise-and-Fall-Time_0.5_1.5_Range_01_10_02_12_Run_38995_RelMean.eps}
117%\vspace{\floatsep}
118%\includegraphics[width=0.3\linewidth]{MExtractTimeAndChargeSpline_Rise-and-Fall-Time_0.5_1.5_Range_01_10_02_12_Run_38996_RelMean.eps}
119%\caption{MExtractTimeAndChargeSpline with integral over 2 slices:
120%Difference in mean pedestal (per FADC slice) between extraction algorithm
121%applied on a fixed window of 2 FADC slices (``extractor random'') and a simple addition of
122%2 FADC fixed slices (``fundamental''). On the left, a run with closed camera has been taken, in the center
123% an opened camera observing an extra-galactic star field and on the right, an open camera being
124%illuminated by the continuous light of the calibration (level: 100). Every entry corresponds to one
125%pixel.}
126%\label{fig:int:relmean}
127%\end{figure}
128
129%\begin{figure}[htp]
130%\centering
131%\vspace{\floatsep}
132%\includegraphics[width=0.3\linewidth]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38993_RelMean.eps}
133%\vspace{\floatsep}
134%\includegraphics[width=0.3\linewidth]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38995_RelMean.eps}
135%\vspace{\floatsep}
136%\includegraphics[width=0.3\linewidth]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38996_RelMean.eps}
137%\caption{MExtractTimeAndChargeDigitalFilter:
138%Difference in mean pedestal (per FADC slice) between extraction algorithm
139%applied on a fixed window of 6 FADC slices and time-randomized weights (``extractor random'')
140%and a simple addition of
141%6 FADC fixed slices (``fundamental''). On the left, a run with closed camera has been taken, in the center
142% an opened camera observing an extra-galactic star field and on the right, an open camera being
143%illuminated by the continuous light of the calibration (level: 100). Every entry corresponds to one
144%pixel.}
145%\label{fig:df:relmean}
146%\end{figure}
147
148%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
149
150\subsubsection{ \label{sec:ped:fixedwindow} Application of the Signal Extractor to a Fixed Window
151of Pedestal Events}
152
153By applying the signal extractor with a fixed window to pedestal events, we
154determine the parameter $R$ for the case of no signal ($Q = 0$)\footnote{%
155In the case of
156extractors using a fixed window (extractors nr. \#1 to \#22
157in section~\ref{sec:algorithms}), the results are the same by construction
158as calculating the RMS of the sum of a fixed number of FADC slice, traditionally
159named ``pedestal RMS'' in MARS.}.
160\par
161In MARS, this functionality is implemented with a function-call to: \\
162
163{\textit{\bf MJPedestal::SetExtractionWithExtractorRndm()}} including \\
164{\textit{\bf MExtractPedestal::SetRandomCalculation()}}\\
165
166Besides fixing the global extraction window, additionally the following steps are undertaken
167in order to assure an un-biased resolution.
168
169\begin{description}
170\item[\textit{MExtractTimeAndChargeSpline}:\xspace] The spline
171maximum position -- which determines the exact extraction window -- is placed
172at a random place within the digitizing binning resolution of one central FADC slice.
173\item[\textit{MExtractTimeAndChargeDigitalFilter}:\xspace] The second step timing
174offset $\tau$ (eq.~\ref{eq:offsettau}) is chosen randomly for each event.
175\end{description}
176
177\par
178
179%The following figures~\ref{fig:amp:relmean} through~\ref{fig:df:relrms} show results
180%obtained with the second method for three background intensities:
181
182%\begin{enumerate}
183%\item Closed camera and no (Poissonian) fluctuation due to photons from the night sky background
184%\item The camera pointing to an extra-galactic region with stars in the field of view
185%\item The camera illuminated by a continuous light source of intensity 100.
186%\end{enumerate}
187
188The calculated biases obtained with this method for all pixels in the camera
189and for the different levels of (night-sky) background applied vanish
190to an accuracy of better than 2\% of a photo-electron
191for the extractors which are used in this TDAS.
192\par
193Table~\ref{tab::ped:fw} shows the resolutions $R$ obtained
194by applying an extractor to a fixed extraction window,
195for the inner and outer pixels, respectively, for four different camera illumination conditions:
196Closed camera (run \#38993), star-field of an extra-galactic source observation (run~\#38995),
197star-field of the Crab-Nebula observation (run~\#39258) and observation with the almost fully
198illuminated moon at an angular distance of about~60$^\circ$ from the telescope pointing position
199(run~\#46471). In the first three cases, the RMS of the values has been calculated while in the
200fourth case, the high-end side of the signal distributions have been fitted to a Gaussian.
201\par
202The entries belonging to the rows denoted as ``Slid. Win.'' are by construction identical to those
203obtained by simply summing up the FADC slices (the ``fundamental Pedestal RMS'').
204Note that the digital filter yields much smaller values of $R$ than the ``sliding windows'' of
205a same window size. This characteristic shows the
206``filter''--capacity of that algorithm. It ``filters out'' up to 50\% of the night sky
207background photo-electrons.
208\par
209One can see that the ratio between the pedestal RMS of outer and inner pixels is around a factor~3
210for the closed camera and then 1.6--1.9 for the other conditions.
211
212%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
213
214
215%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%1
216
217%\begin{figure}[htp]
218%\centering
219%\includegraphics[width=0.47\linewidth]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38993_RMSDiff.eps}
220%\vspace{\floatsep}
221%\includegraphics[width=0.47\linewidth]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38995_RMSDiff.eps}
222%\vspace{\floatsep}
223%\includegraphics[width=0.47\linewidth]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38996_RMSDiff.eps}
224%\caption{MExtractTimeAndChargeSpline with amplitude:
225%Difference in RMS (per FADC slice) between extraction algorithm
226%applied on a fixed window and the corresponding pedestal RMS.
227%Closed camera (left), open camera observing extra-galactic star field (right) and
228%camera being illuminated by the continuous light (bottom).
229%Every entry corresponds to one pixel.}
230%\label{fig:amp:relrms}
231%\end{figure}
232
233
234%\begin{figure}[htp]
235%\centering
236%\includegraphics[width=0.47\linewidth]{MExtractTimeAndChargeSpline_Rise-and-Fall-Time_0.5_1.5_Range_01_10_02_12_Run_38993_RMSDiff.eps}
237%\vspace{\floatsep}
238%\includegraphics[width=0.47\linewidth]{MExtractTimeAndChargeSpline_Rise-and-Fall-Time_0.5_1.5_Range_01_10_02_12_Run_38995_RMSDiff.eps}
239%\vspace{\floatsep}
240%\includegraphics[width=0.47\linewidth]{MExtractTimeAndChargeSpline_Rise-and-Fall-Time_0.5_1.5_Range_01_10_02_12_Run_38996_RMSDiff.eps}
241%\caption{MExtractTimeAndChargeSpline with integral over 2 slices:
242%Difference in RMS (per FADC slice) between extraction algorithm
243%applied on a fixed window and the corresponding pedestal RMS.
244%Closed camera (left), open camera observing extra-galactic star field (right) and
245%camera being illuminated by the continuous light (bottom).
246%Every entry corresponds to one
247%pixel.}
248%\label{fig:int:relrms}
249%\end{figure}
250
251
252%\begin{figure}[htp]
253%\centering
254%\includegraphics[width=0.47\linewidth]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38993_RMSDiff.eps}
255%\vspace{\floatsep}
256%\includegraphics[width=0.47\linewidth]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38995_RMSDiff.eps}
257%\vspace{\floatsep}
258%\includegraphics[width=0.47\linewidth]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38996_RMSDiff.eps}
259%\caption{MExtractTimeAndChargeDigitalFilter:
260%Difference in RMS (per FADC slice) between extraction algorithm
261%applied on a fixed window and the corresponding pedestal RMS.
262%Closed camera (left), open camera observing extra-galactic star field (right) and
263%camera being illuminated by the continuous light (bottom).
264%Every entry corresponds to one pixel.}
265%\label{fig:df:relrms}
266%\end{figure}
267
268%\begin{landscape}
269%\rotatebox{90}{%
270\begin{table}[htp]
271\vspace{3cm}
272\small{%
273\centering
274\begin{tabular}{|c|c||c|c||c|c||c|c||c|c|}
275\hline
276\hline
277\multicolumn{10}{|c|}{Resolution for $S=0$ and fixed window (units in $N_{\mathrm{phe}}$) \rule{0mm}{6mm} \rule[-2mm]{0mm}{6mm} \hspace{-3mm}} \\
278\hline
279\hline
280 & & \multicolumn{2}{|c|}{Closed camera} & \multicolumn{2}{|c|}{Extra-gal. NSB} & \multicolumn{2}{|c|}{Galactic NSB} & \multicolumn{2}{|c|}{Moon} \\
281\hline
282\hline
283Nr. & Name & $R$ & $R$ & $R$ & $R$ & $R$ & $R$ & $R$ & $R$ \\
284 & & inner & outer & inner & outer & inner & outer & inner & outer \\
285\hline
28617 & Slid. Win. 2 & 0.3 & 0.9 & 1.2 & 2.0 & 1.5 & 2.4 & 3.0 & 5.3 \\
28718 & Slid. Win. 4 & 0.4 & 1.2 & 1.6 & 2.7 & 2.0 & 3.3 & 3.9 & 7.3 \\
28820 & Slid. Win. 6 & 0.5 & 1.6 & 2.0 & 3.5 & 2.4 & 4.3 & 4.7 & 9.0 \\
28921 & Slid. Win. 8 & 0.6 & 2.0 & 2.3 & 4.1 & 2.9 & 5.0 & 5.3 & 10.1 \\
290\hline
29123 & Spline Amp. & 0.3 & 0.8 & 1.0 & 1.8 & 1.2 & 2.2 & 2.5 & 4.9 \\
29224 & Spline Int. 1 & 0.3 & 0.7 & 0.9 & 1.6 & 1.1 & 1.9 & 2.5 & 4.6 \\
29325 & Spline Int. 2 & 0.3 & 0.9 & 1.2 & 2.0 & 1.5 & 2.4 & 3.0 & 5.3 \\
29426 & Spline Int. 4 & 0.4 & 1.2 & 1.6 & 2.8 & 1.9 & 3.4 & 3.6 & 7.1 \\
29527 & Spline Int. 6 & 0.5 & 1.6 & 1.9 & 3.6 & 2.4 & 4.2 & 4.3 & 8.7 \\
296\hline
29728 & Dig. Filt. 6 & 0.3 & 0.8 & 1.0 & 1.6 & 1.2 & 1.9 & 2.8 & 4.3 \\
29829 & Dig. Filt. 4 & 0.3 & 0.7 & 0.9 & 1.6 & 1.1 & 1.9 & 2.5 & 4.3 \\
299\hline
300\hline
301\end{tabular}
302\vspace{1cm}
303\caption{The mean resolution $R$ for different extractors applied to a fixed window of pedestal events.
304Four different conditions of night sky background are shown:
305Closed camera, extra-galactic star-field, galactic star-field and almost full moon at 60$^\circ$ angular
306distance from the pointing position. With the first three conditions, a simple RMS of the extracted
307signals has been calculated while in the fourth case, a Gauss fit to the high part of the distribution
308has been made.
309The obtained values can typically vary by up to 10\% for different channels of the camera readout.}
310\label{tab:ped:fw}
311}
312\end{table}
313%}
314%\end{landscape}
315
316
317%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
318
319
320
321\subsubsection{ \label{sec:ped:slidingwindow} Application of the Signal Extractor to a Sliding Window
322of Pedestal Events}
323
324By applying the signal extractor with a global extraction window to pedestal events, allowing
325it to ``slide'' and maximize the encountered signal, we
326determine the bias $B$ and the mean-squared error $MSE$ for the case of no signal ($S=0$).
327\par
328In MARS, this functionality is implemented with a function-call to: \\
329
330{\textit{\bf MJPedestal::SetExtractionWithExtractor()}} \\
331
332\par
333Table~\ref{tab:bias} shows the bias, the resolution and the mean-square error for all extractors using
334a sliding window. In this sample, every extractor had the freedom to move 5 slices,
335i.e. the global window size was fixed to five plus the extractor window size. This first line
336shows the resolution of the smallest existing robust fixed window algorithm in order to give the reference
337value of 2.5 and 3 photo-electrons RMS for an extra-galactic and a galactic star-field, respectively.
338\par
339One can see that the bias $B$ typically decreases
340with increasing window size, while the error $R$ increases with
341increasing window size, except for the digital filter. There is also a small difference between the obtained error
342on a fixed window extraction and the one obtained from a sliding window extraction in the case of the spline and digital
343filter algorithms.
344The mean-squared error has an optimum somewhere in between: In the case of the
345sliding window and the spline at the lowest window size, in the case of the digital filter
346at 4 slices. The global winners is extractor~\#29
347(digital filter with integration of 4 slices). All sliding window extractors -- except \#21 --
348have a smaller mean-square error than the resolution of the fixed window reference extractor (row\ 1,\#4). This means
349that the global error of the sliding window extractors is smaller than the one of the fixed window extractors
350with 8~FADC slices even if the first have a bias.
351\par
352The important information for the image cleaning is the number of photo-electrons above which the probability for obtaining
353a noise fluctuation is smaller than 0.3\% (3$\sigma$). We approximated that number with the formula:
354
355\begin{equation}
356N_{\mathrm{phe}}^{\mathrm{thres.}} \approx B + 3\cdot R
357\end{equation}
358
359Table~\ref{tab:bias} shows that most of the sliding window algorithms yield a smaller signal threshold than the fixed window ones,
360although the first have a bias. The lowest threshold of only 4.2~photo-electrons for the extra-galactic star-field and 5.0~photo-electrons
361for the galactic star-field is obtained by the digital filter fitting 4 FADC slices (extractor~\%29).
362This is almost a factor 2 lower than the fixed window results. Also the spline integrating 1 FADC slice (extractor~\%24) yields almost
363comparable results.
364
365\begin{landscape}
366%\rotatebox{90}{%
367\begin{table}[htp]
368\vspace{3cm}
369\scriptsize{%
370\centering
371\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
372\hline
373\hline
374\multicolumn{16}{|c|}{Statistical Parameters for $S=0$ (units in $N_{\mathrm{phe}}$) \rule{0mm}{6mm} \rule[-2mm]{0mm}{6mm} \hspace{-3mm}} \\
375\hline
376\hline
377 & & \multicolumn{4}{|c|}{Closed camera} & \multicolumn{5}{|c|}{Extra-galactic NSB} & \multicolumn{5}{|c|}{Galactic NSB} \\
378\hline
379\hline
380Nr. & Name & $R$ & $R$ & $B$ & $\sqrt{MSE}$ & $R$ &$R$ & $B$ & $\sqrt{MSE}$ & $B+3R$ & $R$ & $R$& $B$ & $\sqrt{MSE}$ & $B+3R$ \\
381 & & (FW) & (SW)& (SW)& (SW) & (FW) &(SW) & (SW)& (SW) & (99.7\% prob.) & (FW)&(SW) & (SW)&(SW) & (99.7\% prob.) \\
382\hline
383\hline
3844 & Fixed Win. 8 & 1.2 & -- & 0.0 & 1.2 & 2.5 & -- & 0.0 & 2.5 & 7.5 & 3.0 & -- & 0.0 & 3.0 & 9.0 \\
385\hline
386-- & Slid. Win. 1 & 0.4 & 0.4 & 0.4 & 0.6 & 1.2 & 1.2 & 1.3 & 1.8 & 4.9 & 1.4 & 1.4 & 1.5 & 2.0 & 5.7 \\
38717 & Slid. Win. 2 & 0.5 & 0.5 & 0.4 & 0.6 & 1.4 & 1.4 & 1.2 & 1.8 & 5.4 & 1.6 & 1.6 & 1.5 & 2.2 & 6.1 \\
38818 & Slid. Win. 4 & 0.8 & 0.8 & 0.5 & 0.9 & 1.9 & 1.9 & 1.2 & 2.2 & 6.9 & 2.2 & 2.3 & 1.6 & 2.8 & 7.5 \\
38920 & Slid. Win. 6 & 1.0 & 1.0 & 0.4 & 1.1 & 2.2 & 2.2 & 1.1 & 2.5 & 7.7 & 2.6 & 2.7 & 1.4 & 3.0 & 9.5 \\
39021 & Slid. Win. 8 & 1.2 & 1.3 & 0.4 & 1.4 & 2.5 & 2.5 & 1.0 & 2.7 & 8.5 & 3.0 & 3.2 & 1.4 & 3.5 & 10.0 \\
391\hline
39223 & Spline Amp. & 0.4 & \textcolor{red}{\bf 0.4} & 0.4 & 0.6 & 1.1 & 1.2 & 1.3 & 1.8 & 4.9 & 1.3 & 1.4 & 1.6 & 2.1 & 5.8 \\
39324 & \textcolor{red}{\bf Spline Int. 1} & 0.4 & \textcolor{red}{\bf 0.4} & 0.3 & \textcolor{red}{\bf 0.5} & 1.0 & 1.2 & 1.0 & 1.6 & 4.6 & 1.3 & \textcolor{red}{\bf 1.3} & 1.3 & 1.8 & 5.2 \\
39425 & Spline Int. 2 & 0.5 & 0.5 & 0.3 & 0.6 & 1.3 & 1.4 & 0.9 & 1.7 & 5.1 & 1.7 & 1.6 & 1.2 & 2.0 & 6.0 \\
39526 & Spline Int. 4 & 0.7 & 0.7 & \textcolor{red}{\bf 0.2 } & 0.7 & 1.5 & 1.7 & \textcolor{red}{\bf 0.8} & 1.9 & 5.3 & 2.0 & 2.0 & 1.0 & 2.2 & 7.0 \\
39627 & Spline Int. 6 & 1.0 & 1.0 & 0.3 & 1.0 & 2.0 & 2.0 & \textcolor{red}{\bf 0.8} & 2.2 & 6.8 & 2.6 & 2.5 & \textcolor{red}{\bf 0.9} & 2.7 & 8.4 \\
397\hline
39828 & Dig. Filt. 6 & 0.4 & 0.5 & 0.4 & 0.6 & 1.1 & 1.3 & 1.3 & 1.8 & 5.2 & 1.3 & 1.5 & 1.5 & 2.1 & 6.0 \\
39929 & \textcolor{red}{\bf Dig. Filt. 4} & 0.3 & \textcolor{red}{\bf 0.4} & 0.3 & \textcolor{red}{\bf 0.5} & 0.9 & \textcolor{red}{\bf 1.1} & 0.9 & \textcolor{red}{\bf 1.4} & \textcolor{red}{\bf 4.2} & 1.0 & \textcolor{red}{\bf 1.3} & 1.1 & \textcolor{red}{\bf 1.7} & \textcolor{red}{\bf 5.0 }\\
400\hline
401\hline
402\end{tabular}
403\vspace{1cm}
404\caption{The statistical parameters bias, resolution and mean error for the algorithms which can be applied to sliding
405windows (SW) and/or fixed windows (FW) of pedestal events.
406The first line displays the resolution of the smallest existing robust fixed--window extractor
407for reference. All units in equiv.
408photo-electrons, uncertainty: 0.1 phes. All extractors were allowed to move 5 FADC slices plus
409their window size. The ``winners'' for each column are marked in red. Global winners (within the given
410uncertainty) are the extractors Nr. \#24 (MExtractTimeAndChargeSpline with an integration window of
4111 FADC slice) and Nr.\#29
412(MExtractTimeAndChargeDigitalFilter with an integration window size of 4 slices)}
413\label{tab:bias}
414}
415\end{table}
416%}
417\end{landscape}
418
419\clearpage
420
421Figures~\ref{fig:sw:distped} through~\ref{fig:df4:distped} show the
422extracted pedestal distributions for some selected extractors (\#18, \#23, \#25, \#28 and \#29)
423 for one typical channel (pixel 100) and two background situations: Closed camera with only electronic
424noise and open camera pointing to an extra-galactic source.
425One can see the (asymmetric) Poisson behaviour of the
426night sky background photons for the distributions with open camera.
427
428\begin{figure}[htp]
429\centering
430\includegraphics[height=0.43\textheight]{PedestalSpectrum-18-Run38993.eps}
431\vspace{\floatsep}
432\includegraphics[height=0.43\textheight]{PedestalSpectrum-18-Run38995.eps}
433\caption{MExtractTimeAndChargeSlidingWindow with extraction window of 4 FADC slices:
434Distribution of extracted "pedestals" from pedestal run with
435closed camera (top) and open camera observing an extra-galactic star field (bottom) for one channel
436(pixel 100). The result obtained from a simple addition of 4 FADC
437slice contents (``fundamental'') is displayed as red histogram, the one obtained from the application of
438the algorithm on
439a fixed window of 4 FADC slices as blue histogram (``extractor random'') and the one obtained from the
440full algorithm allowed to slide within a global window of 12 slices. The obtained histogram means and
441RMSs have been converted to equiv. photo-electrons.}
442\label{fig:sw:distped}
443\end{figure}
444
445
446\begin{figure}[htp]
447\centering
448\includegraphics[height=0.43\textheight]{PedestalSpectrum-23-Run38993.eps}
449\vspace{\floatsep}
450\includegraphics[height=0.43\textheight]{PedestalSpectrum-23-Run38995.eps}
451\caption{MExtractTimeAndChargeSpline with amplitude extraction:
452Spectrum of extracted "pedestals" from pedestal run with
453closed camera lids (top) and open lids observing an extra-galactic star field (bottom) for one channel
454(pixel 100). The result obtained from a simple addition of 2 FADC
455slice contents (``fundamental'') is displayed as red histogram, the one obtained from the application
456of the algorithm on a fixed window of 1 FADC slice as blue histogram (``extractor random'')
457and the one obtained from the
458full algorithm allowed to slide within a global window of 12 slices. The obtained histogram means and
459RMSs have been converted to equiv. photo-electrons.}
460\label{fig:amp:distped}
461\end{figure}
462
463\begin{figure}[htp]
464\centering
465\includegraphics[height=0.43\textheight]{PedestalSpectrum-25-Run38993.eps}
466\vspace{\floatsep}
467\includegraphics[height=0.43\textheight]{PedestalSpectrum-25-Run38995.eps}
468\caption{MExtractTimeAndChargeSpline with integral extraction over 2 FADC slices:
469Distribution of extracted "pedestals" from pedestal run with
470closed camera lids (top) and open lids observing an extra-galactic star field (bottom) for one channel
471(pixel 100). The result obtained from a simple addition of 2 FADC
472slice contents (``fundamental'') is displayed as red histogram, the one obtained from the application
473of time-randomized weights on a fixed window of 2 FADC slices as blue histogram and the one obtained from the
474full algorithm allowed to slide within a global window of 12 slices. The obtained histogram means and
475RMSs have been converted to equiv. photo-electrons.}
476\label{fig:int:distped}
477\end{figure}
478
479\begin{figure}[htp]
480\centering
481\includegraphics[height=0.43\textheight]{PedestalSpectrum-28-Run38993.eps}
482\vspace{\floatsep}
483\includegraphics[height=0.43\textheight]{PedestalSpectrum-28-Run38995.eps}
484\caption{MExtractTimeAndChargeDigitalFilter: Spectrum of extracted "pedestals" from pedestal run with
485closed camera lids (top) and open lids observing an extra-galactic star field (bottom) for one channel
486(pixel 100). The result obtained from a simple addition of 6 FADC
487slice contents (``fundamental'') is displayed as red histogram, the one obtained from the application
488of time-randomized weights on a fixed window of 6 slices as blue histogram and the one obtained from the
489full algorithm allowed to slide within a global window of 12 slices. The obtained histogram means and
490RMSs have been converted to equiv. photo-electrons.}
491\label{fig:df6:distped}
492\end{figure}
493
494\begin{figure}[htp]
495\centering
496\includegraphics[height=0.43\textheight]{PedestalSpectrum-29-Run38993.eps}
497\vspace{\floatsep}
498\includegraphics[height=0.43\textheight]{PedestalSpectrum-29-Run38995.eps}
499\caption{MExtractTimeAndChargeDigitalFilter: Spectrum of extracted "pedestals" from pedestal run with
500closed camera lids (top) and open lids observing an extra-galactic star field (bottom) for one channel
501(pixel 100). The result obtained from a simple addition of 4 FADC
502slice contents (``fundamental'') is displayed as red histogram, the one obtained from the application
503of time-randomized weights on a fixed window of 4 slices as blue histogram and the one obtained from the
504full algorithm allowed to slide within a global window of 10 slices. The obtained histogram means and
505RMSs have been converted to equiv. photo-electrons.}
506\label{fig:df4:distped}
507\end{figure}
508
509\clearpage
510
511\subsection{ \label{sec:ped:singlephe} Single Photo-Electron Extraction with the Digital Filter}
512
513Figure~\ref{fig:df:sphespectrum} shows spectra
514obtained with the digital filter applied on three different global search windows.
515One can clearly distinguish a pedestal peak (fitted to Gaussian with index 0)
516and further, positive contributions.
517\par
518Because the background is determined by the single photo-electrons from the night-sky background,
519the following possibilities can occur:
520
521\begin{enumerate}
522\item There is no ``signal'' (photo-electron) in the extraction window and the extractor
523finds only electronic noise.
524Usually, the returned signal charge is then negative.
525\item There is one photo-electron in the extraction window and the extractor finds it.
526\item There are more than one photo-electron in the extraction window, but separated by more than
527two FADC slices whereupon the extractor finds the one with the highest charge (upward fluctuation) of both.
528\item The extractor finds an overlap of two or more photo-electrons.
529\end{enumerate}
530
531Although the probability to find a certain number of photo-electrons in a fixed window follows a
532Poisson distribution, the one for employing the sliding window is {\textit{not}} Poissonian. The extractor
533will usually find one photo-electron even if more are present in the global search window, i.e. the
534probability for two or more photo-electrons to occur in the global search window is much higher than
535the probability for these photo-electrons to overlap in time such as to be recognized as a double
536or triple photo-electron pulse by the extractor. This is especially true for small extraction windows
537and for the digital filter.
538
539\par
540
541Given a global extraction window of size $\mathrm{\it WS}$ and an average rate of photo-electrons from the night-sky
542background $R$, we will now calculate the probability for the extractor to find zero photo-electrons in the
543$\mathrm{\it WS}$. The probability to find any number of $k$ photo-electrons can be written as:
544
545\begin{equation}
546P(k) = \frac{e^{-R\cdot \mathrm{\it WS}} (R \cdot \mathrm{\it WS})^k}{k!}
547\end{equation}
548
549and thus:
550
551\begin{equation}
552P(0) = e^{-R\cdot \mathrm{\it WS}}
553\end{equation}
554
555The probability to find one or more photo-electrons is then:
556
557\begin{equation}
558P(>0) = 1 - e^{-R\cdot \mathrm{\it WS}}
559\end{equation}
560
561In figures~\ref{fig:df:sphespectrum},
562one can clearly distinguish the pedestal peak (fitted to Gaussian with index 0),
563corresponding to the case of  $P(0)$ and further
564contributions of $P(1)$ and $P(2)$ (fitted to Gaussians with index 1 and 2).
565One can also see that the contribution of $P(0)$ dimishes
566with increasing global search window size.
567
568\begin{figure}
569\centering
570\includegraphics[height=0.3\textheight]{SinglePheSpectrum-28-Run38995-WS2.5.eps}
571\vspace{\floatsep}
572\includegraphics[height=0.3\textheight]{SinglePheSpectrum-28-Run38995-WS4.5.eps}
573\vspace{\floatsep}
574\includegraphics[height=0.3\textheight]{SinglePheSpectrum-28-Run38995-WS8.5.eps}
575\caption{MExtractTimeAndChargeDigitalFilter: Spectrum obtained from the extraction
576of a pedestal run using a sliding window of 6 FADC slices allowed to move within a window of
5777 (top), 9 (center) and 13 slices.
578A pedestal run with galactic star background has been taken and one typical pixel (Nr. 100).
579One can clearly see the pedestal contribution and a further part corresponding to one or more
580photo-electrons.}
581\label{fig:df:sphespectrum}
582\end{figure}
583
584In the following, we will make a short consistency test: Assuming that the spectral peaks are
585attributed correctly, one would expect the following relation:
586
587\begin{equation}
588P(0) / P(>0) = \frac{e^{-R\cdot WS}}{1-e^{-R\cdot WS}}
589\end{equation}
590
591We tested this relation assuming that the fitted area underneath the pedestal peak $\mathrm{\it Area}_0$ is
592proportional to $P(0)$ and the sum of the fitted areas underneath the single photo-electron peak
593$\mathrm{\it Area}_1$ and the double photo-electron peak $\mathrm{\it Area}_2$ proportional to $P(>0)$. We assumed
594that the probability for a triple photo-electron to occur is negligible. Thus, one expects:
595
596\begin{equation}
597\mathrm{\it Area}_0 / (\mathrm{\it Area}_1 + \mathrm{\it Area}_2 ) = \frac{e^{-R\cdot WS}}{1-e^{-R\cdot WS}}
598\end{equation}
599
600We estimated the effective window size $\mathrm{\it WS}$ as the sum of the range in which the digital filter
601amplitude weights are greater than 0.5 (1.5 FADC slices) and the global search window minus the
602size of the window size of the weights (which is 6 FADC slices). Figure~\ref{fig:df:ratiofit}
603shows the result for two different levels of night-sky background. The fitted rates deliver
6040.08 and 0.1 phes/ns, respectively. These rates are about 50\% lower than those obtained
605from the November 2004 test campaign. However, we should take into account that the method is at
606the limit of distinguishing single photo-electrons. It may occur often that a single photo-electron
607signal is too low in order to get recognized as such. We tried various pixels and found that
608some of them do not permit to apply this method at all. The ones which succeed, however, yield about
609the same fitted rates. To conclude, one may say that there is consistency within the double-peak
610structure of the pedestal spectrum found by the digital filter which can be explained by the fact that
611single photo-electrons are separated from the pure electronics noise.
612\par
613
614\begin{figure}[htp]
615\centering
616\includegraphics[height=0.4\textheight]{SinglePheRatio-28-Run38995.eps}
617\vspace{\floatsep}
618\includegraphics[height=0.4\textheight]{SinglePheRatio-28-Run39258.eps}
619\caption{MExtractTimeAndChargeDigitalFilter: Fit to the ratio of the area beneath the pedestal peak and
620the single and double photo-electron(s) peak(s) with the extraction algorithm
621applied on a sliding window of different sizes.
622In the top plot, a pedestal run with extra-galactic star background has been taken and in the bottom,
623a galactic star background. An typical pixel (Nr. 100) has been used.
624Above, a rate of 0.08 phe/ns and below, a rate of 0.1 phe/ns has been obtained.}
625\label{fig:df:ratiofit}
626\end{figure}
627
628Figure~\ref{fig:df:convfit} shows the obtained ``conversion factors'' and ``F-Factor'' computed as~\cite{MAGIC-calibration}:
629
630\begin{eqnarray}
631c_{phe} &=& \frac{1}{\mu_1 - \mu_0} \\
632F_{phe} &=& \sqrt{1 + \frac{\sigma_1^2 - \sigma_0^2}{(\mu_1 - \mu_0)^2} }
633\end{eqnarray}
634
635where $\mu_0$ denotes the mean position of the pedestal peak and $\mu_1$ the mean position of the (assumed)
636single photo-electron peak. The obtained conversion factors are systematically lower than the ones
637obtained from the standard calibration and decrease with increasing window size. This is consistent
638with the assumption that the digital filter finds the most upward fluctuating pulse out of several. Therefore,
639$\mu_1$ is biased against higher values. The F-Factor is also systematically low (however with huge error bars),
640which is also consistent
641with the assumption that the spacing between $\mu_1$ and $\mu_0$ is artificially high.
642Unfortunately, the error bars are too high for a ``calibration'' of the F-Factor.
643\par
644In conclusion, the digital filter is at the edge of being able to see single photo-electrons,
645however a single photo-electron calibration cannot yet be done with the current FADC system because the
646resolution is too poor. These limitations might be overcome if a higher sampling speed is used and the artificial
647pulse shaping removed. We expect to improve this method considerably with the new 2\,GSamples/s~FADC readout of MAGIC.
648
649\begin{figure}[htp]
650\centering
651\includegraphics[height=0.4\textheight]{ConvFactor-28-Run38995.eps}
652\vspace{\floatsep}
653\includegraphics[height=0.4\textheight]{FFactor-28-Run38995.eps}
654\caption{MExtractTimeAndChargeDigitalFilter: Obtained conversion factors (top) and F-Factors (bottom)
655from the position and width of
656the fitted Gaussian mean of the single photo-electron peak and the pedestal peak depending on
657the applied global extraction window sizes.
658A pedestal run with extra-galactic star background has been taken and
659an typical pixel (Nr. 100) used. The conversion factor obtained from the
660standard calibration is shown as a reference line. The obtained conversion factors are systematically
661lower than the reference one.}
662\label{fig:df:convfit}
663\end{figure}
664
665
666
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