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  • trunk/MagicSoft/TDAS-Extractor/Pedestal.tex

    r6286 r6301  
    11\section{Pedestal Extraction \label{sec:pedestals}}
    22
    3 \ldots {\it In this section, the distinction is made between:
    4 \begin{itemize}
    5 \item Defining the pedestal RMS as contribution
    6     to the extracted signal fluctuations (later used in the calibration)
    7 \item Defining the Pedestal Mean and RMS as the result of distributions obtained by
    8     applying the extractor to pedestal runs (yielding biases and modified widths).
    9 \item Deriving the correct probability for background fluctuations based on the extracted signal height.
    10   ( including biases and modified widths).
    11 \end{itemize}
    12 }
    13 
    143\subsection{Pedestal RMS}
    15 
    16 
    17 \vspace{1cm}
    18 \ldots {\it  Modified email by W. Wittek from 25 Oct 2004 and 10 Nov 2004}
    19 \vspace{1cm}
    204
    215The background $BG$ (Pedestal)
    … …  
    3115
    3216Consider a large number of signals (FADC spectra), all with the same
    33 integrated charge $ST$ (true signal). By applying some signal extractor
     17integrated charge $ST$ (true signal). By applying a signal extractor
    3418we obtain a distribution of extracted signals $SE$ (for fixed $ST$ and
    3519fixed background fluctuations $BG$). The distribution of the quantity
    … …  
    4327\begin{eqnarray}
    4428   B    &=& <X> \\
    45    R^2  &=& <(X-B)^2>
     29   R    &=& \sqrt{<(X-B)^2>}
    4630\end{eqnarray}
    4731
    48 One may also define
    49 
    50 \begin{equation}
    51    D^2 = <(SE-ST)^2> = <(SE-ST-B + B)^2> = B^2 + R^2
    52 \end{equation}
    53 
    54 The parameter $B$ can be called the bias of the pedestal extractor and $R$
    55 the RMS of the distribution of $X$ and $D$ is something
    56 like the (asymmetric) error of $SE$.
    57 The distribution of $X$, and thus the parameters $B$ and $R$,
     32The parameter $B$ can be called the {\textit{\bf bias}} of the pedestal extractor and $R$
     33the RMS of the distribution of $X$ which
    5834depend generally on the size of $ST$ and the size of the background fluctuations $BG$.
    5935
    … …  
    6339error $R$ should be known in order to calculate a correct background probability.
    6440\par
    65 Also for the model analysis $B$ and $R$ are needed if one wants to keep small
     41Also for the model analysis, $B$ and $R$ are needed if one wants to keep small
    6642signals.
    67 \par
    68 In the case of the calibration with the F-Factor methoid,
     43
     44\subsection{Pedestal Fluctuations as Contribution to the Signal Fluctuations}
     45
     46In case of the calibration with the F-Factor methoid,
    6947the basic relation is:
    7048
    … …  
    7452
    7553Here $\Delta ST$ is the fluctuation of the true signal $ST$ due to the
    76 fluctuation of the number of photo electrons. $ST$ is obtained from the
    77 measured fluctuations of $SE$  ($RMS_{SE}$) by subtracting those fluctuations of the
    78 extracted signal which are due to the fluctuation of the pedestal ($R$)\footnote{%
     54fluctuation of the number of photo-electrons. $ST$ is obtained from the
     55measured fluctuations of $SE$  ($RMS_{SE}$) subtracting those contributions to the
     56fluctuations of the
     57extracted signal which are due to the fluctuation of the pedestal\footnote{%
    7958A way to check whether the right RMS has been subtracted is to make the
    80 Razmick plot
     59``Razmick''-plot
    8160
    8261\begin{equation}
    … …  
    9877\end{equation}
    9978
    100 \subsection{How to Retrieve Bias $B$ and Error $R$}
     79If $R$ does not dependent on the signal height, (as it is the case
     80for the digital filter, eq.~\ref{eq:of_noise}), then one can retrieve $R$ by
     81applying the signal extractor on a {\textit{\bf fixed window}} of pedestal events.
     82
     83\subsection{Methods to Retrieve Bias $B$ and Errors $R$}
    10184
    10285$R$ is in general different from the pedestal RMS. It cannot be
    … …  
    10487for large signals (e.g. calibration signals).
    10588\par
    106 In the case of the optimum filter, $R$ is in theory independent from the
    107 signal amplitude $ST$ and depends only on the background $BG$ (eq.~\ref{of_noise}).
     89In the case of the digital filter, $R$ is in theory independent from the
     90signal amplitude $ST$ and depends only on the background $BG$ (eq.~\ref{eq:of_noise}).
    10891It can be obtained from the
    10992fitted error of the extracted signal ($\Delta(SE)_{fitted}$),
    … …  
    117100\item Determine $R$ by applying the signal extractor to a fixed window
    118101    of pedestal events. The background fluctuations can be simulated with different
    119     levels of night sky background and the continuous light, but no signal size
     102    levels of night sky background and the continuous light source, but no signal size
    120103    dependency can be retrieved with the method.
    121104\item Determine bias $B$ and resolution $R$ from MC events with and without added noise.
    … …  
    123106    get a dependency of both values from the size of the signal.
    124107\item Determine $R$ from the fitted error of $SE$, which is possible for the
    125     fit and the digital filter (eq.~\ref{of_noise}).
     108    fit and the digital filter (eq.~\ref{eq:of_noise}).
    126109    In prinicple, all dependencies can be retrieved with this method.
    127110\end{enumerate}
    … …  
    133116determine the parameter $R$ for the case of no signal ($ST = 0$). In the case of
    134117all extractors using a fixed window from the beginning (extractors nr. \#1 to \#22
    135 in section~\ref{sec:algorithms}), the results are thus the same by construction as calculating
    136 the mean and the RMS of a same (fixed) number of FADC slices (the conventional ``Pedestal
    137 Calculation'').
    138 \par
    139 In MARS, this possibility is implemented with a function-call to
    140 {\textit{\bf MJPedestal::SetExtractionWithExtractorRndm()}}.
    141 \par
    142 In the case of the amplitude extracting spline (extractor nr. \#23), we placed the
     118in section~\ref{sec:algorithms}), the results are by construction the same as calculating
     119the pedestal RMS.
     120\par
     121In MARS, this possibility is implemented with a function-call to: \\
     122
     123{\textit{\bf MJPedestal::SetExtractionWithExtractorRndm()}}. \\
     124
     125In the case of the {\textit{\bf amplitude extracting spline}} (extractor nr. \#23), we placed the
    143126spline maximum value (which determines the exact extraction window) at a random place
    144 within the digitizing binning resolution (0.01 FADC slices)
    145 of one central FADC slice.
    146 In the case of the digital filter (extractor nr. \#28), the time shift was 
    147 randomized for each event within one central FADC slice.
     127within the digitizing binning resolution of one central FADC slice.
     128In the case of the {\textit{\bf digital filter}} (extractor nr. \#28), the time shift was 
     129randomized for each event within a fixed global extraction window.
    148130
    149131\par
    … …  
    157139fluctuations than in usual observation conditions.
    158140\end{enumerate}
     141
     142
     143\begin{figure}[htp]
     144\centering
     145\includegraphics[height=0.29\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_00_18_02_14_Run_38993_Signal_Pixel200.eps}
     146\caption{MExtractTimeAndChargeDigitalFilter: Distribution of extracted "pedestals"  from pedestal run with
     147closed camera lids for one channel.}
     148\label{fig:df:distped:run38993}
     149\vspace{\floatsep}
     150\includegraphics[height=0.29\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_00_18_02_14_Run_38995_Signal_Pixel200.eps}
     151\caption{MExtractTimeAndChargeDigitalFilter: Distribution of extracted "pedestals"  from pedestal run with
     152extra-galactic star background for one channel.}
     153\label{fig:df:distped:run38995}
     154\vspace{\floatsep}
     155\includegraphics[height=0.29\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_00_18_02_14_Run_38996_Signal_Pixel200.eps}
     156\caption{MExtractTimeAndChargeDigitalFilter: Distribution of extracted "pedestals"  from run with
     157continuous light level 100 for one channel.}
     158\label{fig:df:distped:run38996}
     159\end{figure}
     160
     161\begin{figure}[htp]
     162\centering
     163\includegraphics[height=0.27\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38993_RelMean.eps}
     164\caption{MExtractTimeAndChargeDigitalFilter: Difference in mean pedestal (per FADC slice) from pedestal
     165run with closed camera lids (in photo-electrons)}
     166\label{fig:df:relmean:run38993}
     167\vspace{\floatsep}
     168\includegraphics[height=0.27\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38995_RelMean.eps}
     169\caption{MExtractTimeAndChargeDigitalFilter: Difference in mean pedestal (per FADC slice) from pedestal
     170run with extra-galactic star background (in photo-electrons)}
     171\label{fig:df:relmean:run38995}
     172\vspace{\floatsep}
     173\includegraphics[height=0.27\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38996_RelMean.eps}
     174\caption{MExtractTimeAndChargeDigitalFilter: Difference in mean pedestal (per FADC slice) from run
     175with continuous light level: 100 (in photo-electrons)}
     176\label{fig:df:relmean:run38996}
     177\end{figure}
     178
     179
     180\begin{figure}[htp]
     181\centering
     182\includegraphics[height=0.25\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38993_RMSDiff.eps}
     183\caption{MExtractTimeAndChargeDigitalFilter:  Difference pedestal RMS (per FADC slice) with extraction algorithm
     184appied on a fixed window, and simply summing up the same number of FADC slices.
     185Pedestal run
     186with closed camera lids for inner (left) and outer (right) pixels. }
     187\label{fig:df:relrms:run38993}
     188\vspace{\floatsep}
     189\includegraphics[height=0.25\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38995_RMSDiff.eps}
     190\caption{MExtractTimeAndChargeDigitalFilter:  Difference pedestal RMS (per FADC slice)  with extraction algorithm
     191appied on a fixed window, and simply summing up the same number of FADC slices.
     192 from pedestal run with extra-galactic star background for inner (left)
     193and outer (right) pixels. }
     194\label{fig:df:relrms:run38995}
     195\vspace{\floatsep}
     196\includegraphics[height=0.25\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38996_RMSDiff.eps}
     197\caption{MExtractTimeAndChargeDigitalFilter:  Difference pedestal RMS (per FADC slice)  with extraction algorithm
     198appied on a fixed window, and simply summing up the same number of FADC slices.
     199 from run with continuous light level: 100 for inner (left)
     200and outer (right) pixels. }
     201\label{fig:df:relrms:run38996}
     202\end{figure}
     203
     204
     205\begin{figure}[htp]
     206\centering
     207\includegraphics[height=0.29\textheight]{MExtractTimeAndChargeSpline_Amplitude_Range_00_10_04_11_Run_38993_Signal_Pixel200.eps}
     208\caption{MExtractTimeAndChargeSpline with amplitude: Distribution of extracted "pedestals"  from pedestal run
     209with closed camera lids for one channel.}
     210\label{fig:amp:distped:run38993}
     211\vspace{\floatsep}
     212\includegraphics[height=0.29\textheight]{MExtractTimeAndChargeSpline_Amplitude_Range_00_10_04_11_Run_38995_Signal_Pixel200.eps}
     213\caption{MExtractTimeAndChargeSpline with amplitude: Distribution of extracted "pedestals"  from pedestal run
     214with extra-galactic star background for one channel.}
     215\label{fig:amp:distped:run38995}
     216\vspace{\floatsep}
     217\includegraphics[height=0.29\textheight]{MExtractTimeAndChargeSpline_Amplitude_Range_00_10_04_11_Run_38996_Signal_Pixel200.eps}
     218\caption{MExtractTimeAndChargeSpline with amplitude: Distribution of extracted "pedestals"  from run with
     219continuous light level: 100 for one channel.}
     220\label{fig:amp:distped:run38996}
     221\end{figure}
     222
     223\begin{figure}[htp]
     224\centering
     225\includegraphics[height=0.27\textheight]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38993_RelMean.eps}
     226\caption{MExtractTimeAndChargeSpline with amplitude: Difference in mean pedestal (per FADC slice)  with extraction algorithm
     227appied on a fixed window, and simply summing up the same number of FADC slices.
     228Pedestal run with closed camera lids.}
     229\label{fig:amp:relmean:run38993}
     230\vspace{\floatsep}
     231\includegraphics[height=0.27\textheight]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38995_RelMean.eps}
     232\caption{MExtractTimeAndChargeSpline with amplitude: Difference in mean pedestal (per FADC slice)  with extraction algorithm
     233appied on a fixed window, and simply summing up the same number of FADC slices
     234Pedestal run with extra-galactic star background.}
     235\label{fig:amp:relmean:run38995}
     236\vspace{\floatsep}
     237\includegraphics[height=0.27\textheight]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38996_RelMean.eps}
     238\caption{MExtractTimeAndChargeSpline with amplitude: Difference in mean pedestal (per FADC slice)  with extraction algorithm
     239appied on a fixed window, and simply summing up the same number of FADC slices.
     240Pedestal run with continuous light level: 100}
     241\label{fig:amp:relmean:run38996}
     242\end{figure}
     243
     244
     245\begin{figure}[htp]
     246\centering
     247\includegraphics[height=0.25\textheight]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38993_RMSDiff.eps}
     248\caption{MExtractTimeAndChargeSpline with amplitude:  Difference pedestal RMS (per FADC slice)  with extraction
     249 algorithm appied on a fixed window, and simply summing up the same number of FADC slices.
     250Pedestal run
     251with closed camera lids for inner (left) and outer (right) pixels. }
     252\label{fig:amp:relrms:run38993}
     253\vspace{\floatsep}
     254\includegraphics[height=0.25\textheight]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38995_RMSDiff.eps}
     255\caption{MExtractTimeAndChargeSpline with amplitude:  Difference pedestal RMS (per FADC slice)  with extraction
     256algorithm appied on a fixed window, and simply summing up the same number of FADC slices.
     257Pedestal run with extra-galactic star background for inner (left)
     258and outer (right) pixels.}
     259\label{fig:amp:relrms:run38995}
     260\vspace{\floatsep}
     261\includegraphics[height=0.25\textheight]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38996_RMSDiff.eps}
     262\caption{MExtractTimeAndChargeSpline with amplitude:  Difference pedestal RMS (per FADC slice)  with extraction
     263algorithm appied on a fixed window, and simply summing up the same number of FADC slices.
     264Pedestal run with continuous light level: 100 for inner (left)
     265and outer (right) pixels.}
     266\label{fig:amp:relrms:run38996}
     267\end{figure}
    159268
    160269Figures~\ref{fig:df:distped:run38993},~\ref{fig:df:distped:run38995},~\ref{fig:df:distped:run38996},
    … …  
    189298normalized pedestal RMS.
    190299
    191 \begin{figure}[htp]
    192 \centering
    193 \includegraphics[height=0.29\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_00_18_02_14_Run_38993_Signal_Pixel200.eps}
    194 \caption{MExtractTimeAndChargeDigitalFilter: Distribution of extracted "pedestals"  from pedestal run with
    195 closed camera lids for one channel.}
    196 \label{fig:df:distped:run38993}
    197 \vspace{\floatsep}
    198 \includegraphics[height=0.29\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_00_18_02_14_Run_38995_Signal_Pixel200.eps}
    199 \caption{MExtractTimeAndChargeDigitalFilter: Distribution of extracted "pedestals"  from pedestal run with
    200 extra-galactic star background for one channel.}
    201 \label{fig:df:distped:run38995}
    202 \vspace{\floatsep}
    203 \includegraphics[height=0.29\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_00_18_02_14_Run_38996_Signal_Pixel200.eps}
    204 \caption{MExtractTimeAndChargeDigitalFilter: Distribution of extracted "pedestals"  from run with
    205 continuous light level 100 for one channel.}
    206 \label{fig:df:distped:run38996}
    207 \end{figure}
    208 
    209 \begin{figure}[htp]
    210 \centering
    211 \includegraphics[height=0.27\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38993_RelMean.eps}
    212 \caption{MExtractTimeAndChargeDigitalFilter: Difference in mean pedestal (per FADC slice) from pedestal
    213 run with closed camera lids (in photo-electrons)}
    214 \label{fig:df:relmean:run38993}
    215 \vspace{\floatsep}
    216 \includegraphics[height=0.27\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38995_RelMean.eps}
    217 \caption{MExtractTimeAndChargeDigitalFilter: Difference in mean pedestal (per FADC slice) from pedestal
    218 run with extra-galactic star background (in photo-electrons)}
    219 \label{fig:df:relmean:run38995}
    220 \vspace{\floatsep}
    221 \includegraphics[height=0.27\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38996_RelMean.eps}
    222 \caption{MExtractTimeAndChargeDigitalFilter: Difference in mean pedestal (per FADC slice) from run
    223 with continuous light level: 100 (in photo-electrons)}
    224 \label{fig:df:relmean:run38996}
    225 \end{figure}
    226 
    227 
    228 \begin{figure}[htp]
    229 \centering
    230 \includegraphics[height=0.25\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38993_RMSDiff.eps}
    231 \caption{MExtractTimeAndChargeDigitalFilter:  Difference pedestal RMS (per FADC slice) with extraction algorithm
    232 appied on a fixed window, and simply summing up the same number of FADC slices.
    233 Pedestal run
    234 with closed camera lids for inner (left) and outer (right) pixels. }
    235 \label{fig:df:relrms:run38993}
    236 \vspace{\floatsep}
    237 \includegraphics[height=0.25\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38995_RMSDiff.eps}
    238 \caption{MExtractTimeAndChargeDigitalFilter:  Difference pedestal RMS (per FADC slice)  with extraction algorithm
    239 appied on a fixed window, and simply summing up the same number of FADC slices.
    240  from pedestal run with extra-galactic star background for inner (left)
    241 and outer (right) pixels. }
    242 \label{fig:df:relrms:run38995}
    243 \vspace{\floatsep}
    244 \includegraphics[height=0.25\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38996_RMSDiff.eps}
    245 \caption{MExtractTimeAndChargeDigitalFilter:  Difference pedestal RMS (per FADC slice)  with extraction algorithm
    246 appied on a fixed window, and simply summing up the same number of FADC slices.
    247  from run with continuous light level: 100 for inner (left)
    248 and outer (right) pixels. }
    249 \label{fig:df:relrms:run38996}
    250 \end{figure}
    251 
    252 
    253 \begin{figure}[htp]
    254 \centering
    255 \includegraphics[height=0.29\textheight]{MExtractTimeAndChargeSpline_Amplitude_Range_00_10_04_11_Run_38993_Signal_Pixel200.eps}
    256 \caption{MExtractTimeAndChargeSpline with amplitude: Distribution of extracted "pedestals"  from pedestal run
    257 with closed camera lids for one channel.}
    258 \label{fig:amp:distped:run38993}
    259 \vspace{\floatsep}
    260 \includegraphics[height=0.29\textheight]{MExtractTimeAndChargeSpline_Amplitude_Range_00_10_04_11_Run_38995_Signal_Pixel200.eps}
    261 \caption{MExtractTimeAndChargeSpline with amplitude: Distribution of extracted "pedestals"  from pedestal run
    262 with extra-galactic star background for one channel.}
    263 \label{fig:amp:distped:run38995}
    264 \vspace{\floatsep}
    265 \includegraphics[height=0.29\textheight]{MExtractTimeAndChargeSpline_Amplitude_Range_00_10_04_11_Run_38996_Signal_Pixel200.eps}
    266 \caption{MExtractTimeAndChargeSpline with amplitude: Distribution of extracted "pedestals"  from run with
    267 continuous light level: 100 for one channel.}
    268 \label{fig:amp:distped:run38996}
    269 \end{figure}
    270 
    271 \begin{figure}[htp]
    272 \centering
    273 \includegraphics[height=0.27\textheight]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38993_RelMean.eps}
    274 \caption{MExtractTimeAndChargeSpline with amplitude: Difference in mean pedestal (per FADC slice)  with extraction algorithm
    275 appied on a fixed window, and simply summing up the same number of FADC slices.
    276 Pedestal run with closed camera lids.}
    277 \label{fig:amp:relmean:run38993}
    278 \vspace{\floatsep}
    279 \includegraphics[height=0.27\textheight]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38995_RelMean.eps}
    280 \caption{MExtractTimeAndChargeSpline with amplitude: Difference in mean pedestal (per FADC slice)  with extraction algorithm
    281 appied on a fixed window, and simply summing up the same number of FADC slices
    282 Pedestal run with extra-galactic star background.}
    283 \label{fig:amp:relmean:run38995}
    284 \vspace{\floatsep}
    285 \includegraphics[height=0.27\textheight]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38996_RelMean.eps}
    286 \caption{MExtractTimeAndChargeSpline with amplitude: Difference in mean pedestal (per FADC slice)  with extraction algorithm
    287 appied on a fixed window, and simply summing up the same number of FADC slices.
    288 Pedestal run with continuous light level: 100}
    289 \label{fig:amp:relmean:run38996}
    290 \end{figure}
    291 
    292 
    293 \begin{figure}[htp]
    294 \centering
    295 \includegraphics[height=0.25\textheight]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38993_RMSDiff.eps}
    296 \caption{MExtractTimeAndChargeSpline with amplitude:  Difference pedestal RMS (per FADC slice)  with extraction
    297  algorithm appied on a fixed window, and simply summing up the same number of FADC slices.
    298 Pedestal run
    299 with closed camera lids for inner (left) and outer (right) pixels. }
    300 \label{fig:amp:relrms:run38993}
    301 \vspace{\floatsep}
    302 \includegraphics[height=0.25\textheight]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38995_RMSDiff.eps}
    303 \caption{MExtractTimeAndChargeSpline with amplitude:  Difference pedestal RMS (per FADC slice)  with extraction
    304 algorithm appied on a fixed window, and simply summing up the same number of FADC slices.
    305 Pedestal run with extra-galactic star background for inner (left)
    306 and outer (right) pixels.}
    307 \label{fig:amp:relrms:run38995}
    308 \vspace{\floatsep}
    309 \includegraphics[height=0.25\textheight]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38996_RMSDiff.eps}
    310 \caption{MExtractTimeAndChargeSpline with amplitude:  Difference pedestal RMS (per FADC slice)  with extraction
    311 algorithm appied on a fixed window, and simply summing up the same number of FADC slices.
    312 Pedestal run with continuous light level: 100 for inner (left)
    313 and outer (right) pixels.}
    314 \label{fig:amp:relrms:run38996}
    315 \end{figure}
    316 
    317 
    318300
    319301\begin{figure}[htp]
    … …  
    360342of Pedestal Events}
    361343
    362 By applying the signal extractor to a sliding window of pedestal events, we
    363 determine the bias $B$ and the error $R$.
    364 \par
    365 In MARS, this possibility is implemented with a function-call to
     344In this section, we apply the signal extractor to a sliding window of pedestal events.
     345\par
     346In MARS, this possibility can be used with a call to
    366347{\textit{\bf MJPedestal::SetExtractionWithExtractor()}}.
     348\par
     349Because the background is determined by the single photo-electrons from the night-sky background,
     350the following possibilities can occur:
     351
     352\begin{enumerate}
     353\item There is no ``signal'' (photo-electron) in the extraction window and the extractor
     354finds only electronic noise.
     355Usually, the returned signal charge is then negative.
     356\item The extractor finds the signal from one photo-electron
     357\item The extractor finds an overlap of two or more photo-electrons.
     358\end{enumerate}
     359
     360Although the probability to find a certain number of photo-electrons in a fixed window follows a
     361Poisson distribution, the one for employing the sliding window is {\textit{not}} Poissonian. The extractor
     362will usually find one photo-electron even if more are present in the global search window, i.e. the
     363probability for two or more photo-electrons to occur in the global search window is much higher than
     364the probability for these photo-electrons to overlap in time such as to be recognized as a double
     365or triple photo-electron pulse by the extractor. This is especially true for small extraction windows
     366and for the digital filter.
     367
     368\par
     369
     370Given a global extraction window of size $WS$ and an average rate of photo-electrons from the night-sky
     371background $R$, we will now calculate the probability for the extractor to find zero photo-electrons in the
     372$WS$. The probability to find $k$ photo-electrons can be written as:
     373
     374\begin{equation}
     375P(k) = \frac{e^{-R\cdot WS} (R \cdot WS)^k}{k!}
     376\end{equation}
     377
     378and thus:
     379
     380\begin{equation}
     381P(0) = e^{-R\cdot WS}
     382\end{equation}
     383
     384The probability to find more than one photo-electron is then:
     385
     386\begin{equation}
     387P(>0) = 1 - e^{-R\cdot WS}
     388\end{equation}
     389
     390Figures~\ref{fig:sphe:sphespectrum:2.5} and~\ref{fig:sphe:sphespectrum:4.5} show spectra
     391obtained with the digital filter applied on two different global search windows.
     392One can clearly distinguish a pedestal peak (fitted to Gaussian with index 0),
     393corresponding to the case of  $P(0)$ and further
     394contributions of $P(1)$ and $P(2)$ (fitted to Gaussians with index 1 and 2).
     395One can also see that the contribution of $P(0)$ dimishes
     396with increasing global search window size.
     397
     398\begin{figure}
     399\centering
     400\includegraphics[height=0.3\textheight]{SinglePheSpectrum-28-Run38995-WS2.5.eps}
     401\caption{MExtractTimeAndChargeDigitalFilter: Signal spectrum obtained from the extraction
     402of a pedestal run using a sliding window of 6 FADC slices allowed to move within a window of
     4037 slices.
     404A pedestal run with galactic star background has been taken and one exemplary pixel (Nr. 100).
     405One can clearly see the pedestal contribution and a further part corresponding to one or more
     406photo-electrons.}
     407\label{fig:df:sphespectrum:2.5}
     408\vspace{\floatsep}
     409\includegraphics[height=0.3\textheight]{SinglePheSpectrum-28-Run38995-WS4.5.eps}
     410\caption{MExtractTimeAndChargeDigitalFilter: Signal spectrum obtained from the extraction
     411of a pedestal run using a sliding window of 6 FADC slices allowed to move within a window of
     4129 slices.
     413A pedestal run with galactic star background has been taken and one exemplary pixel (Nr. 100).
     414One can clearly see the pedestal contribution and a further part corresponding to one or more
     415photo-electrons.}
     416\label{fig:df:sphespectrum:4.5}
     417\end{figure}
     418
     419In the following, we will make a short consistency test: Assuming that the spectral peaks are
     420attributed correctly, one would expect the following relation:
     421
     422\begin{equation}
     423P(0) / P(>0) = \frac{e^{-R\cdot WS}}{1-e^{-R\cdot WS}}
     424\end{equation}
     425
     426We tested this relation assuming that the fitted area underneath the pedestal peak $Area_0$ is
     427proportional to $P(0)$ and the sum of the fitted areas underneath the single photo-electron peak
     428$Area_1$ and the double photo-electron peak $Area_2$ proportional to $P(>0)$. Thus, one expects:
     429
     430\begin{equation}
     431Area_0 / (Area_1 + Area+2 ) = \frac{e^{-R\cdot WS}}{1-e^{-R\cdot WS}}
     432\end{equation}
     433
     434We estimated the effective window size $WS$ as the sum of the range in which the digital filter
     435amplitude weights are greater than 0.5 (1.6 FADC slices) and the global search window minus the
     436size of the window size of the weights (which is 6 FADC slices). Figures~\ref{fig::df:ratiofit:run38995}
     437and~\ref{fig:df:ratiofit:run39258} show the result for two different levels of night-sky background.
     438
     439
     440
     441
     442\par
    367443
    368444\begin{figure}[htp]
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