Index: trunk/MagicSoft/TDAS-Extractor/Pedestal.tex =================================================================== --- trunk/MagicSoft/TDAS-Extractor/Pedestal.tex (revision 6300) +++ trunk/MagicSoft/TDAS-Extractor/Pedestal.tex (revision 6301) @@ -1,21 +1,5 @@ \section{Pedestal Extraction \label{sec:pedestals}} -\ldots {\it In this section, the distinction is made between: -\begin{itemize} -\item Defining the pedestal RMS as contribution - to the extracted signal fluctuations (later used in the calibration) -\item Defining the Pedestal Mean and RMS as the result of distributions obtained by - applying the extractor to pedestal runs (yielding biases and modified widths). -\item Deriving the correct probability for background fluctuations based on the extracted signal height. - ( including biases and modified widths). -\end{itemize} -} - \subsection{Pedestal RMS} - - -\vspace{1cm} -\ldots {\it Modified email by W. Wittek from 25 Oct 2004 and 10 Nov 2004} -\vspace{1cm} The background $BG$ (Pedestal) @@ -31,5 +15,5 @@ Consider a large number of signals (FADC spectra), all with the same -integrated charge $ST$ (true signal). By applying some signal extractor +integrated charge $ST$ (true signal). By applying a signal extractor we obtain a distribution of extracted signals $SE$ (for fixed $ST$ and fixed background fluctuations $BG$). The distribution of the quantity @@ -43,17 +27,9 @@ \begin{eqnarray} B &=& \\ - R^2 &=& <(X-B)^2> + R &=& \sqrt{<(X-B)^2>} \end{eqnarray} -One may also define - -\begin{equation} - D^2 = <(SE-ST)^2> = <(SE-ST-B + B)^2> = B^2 + R^2 -\end{equation} - -The parameter $B$ can be called the bias of the pedestal extractor and $R$ -the RMS of the distribution of $X$ and $D$ is something -like the (asymmetric) error of $SE$. -The distribution of $X$, and thus the parameters $B$ and $R$, +The parameter $B$ can be called the {\textit{\bf bias}} of the pedestal extractor and $R$ +the RMS of the distribution of $X$ which depend generally on the size of $ST$ and the size of the background fluctuations $BG$. @@ -63,8 +39,10 @@ error $R$ should be known in order to calculate a correct background probability. \par -Also for the model analysis $B$ and $R$ are needed if one wants to keep small +Also for the model analysis, $B$ and $R$ are needed if one wants to keep small signals. -\par -In the case of the calibration with the F-Factor methoid, + +\subsection{Pedestal Fluctuations as Contribution to the Signal Fluctuations} + +In case of the calibration with the F-Factor methoid, the basic relation is: @@ -74,9 +52,10 @@ Here $\Delta ST$ is the fluctuation of the true signal $ST$ due to the -fluctuation of the number of photo electrons. $ST$ is obtained from the -measured fluctuations of $SE$ ($RMS_{SE}$) by subtracting those fluctuations of the -extracted signal which are due to the fluctuation of the pedestal ($R$)\footnote{% +fluctuation of the number of photo-electrons. $ST$ is obtained from the +measured fluctuations of $SE$ ($RMS_{SE}$) subtracting those contributions to the +fluctuations of the +extracted signal which are due to the fluctuation of the pedestal\footnote{% A way to check whether the right RMS has been subtracted is to make the -Razmick plot +``Razmick''-plot \begin{equation} @@ -98,5 +77,9 @@ \end{equation} -\subsection{How to Retrieve Bias $B$ and Error $R$} +If $R$ does not dependent on the signal height, (as it is the case +for the digital filter, eq.~\ref{eq:of_noise}), then one can retrieve $R$ by +applying the signal extractor on a {\textit{\bf fixed window}} of pedestal events. + +\subsection{Methods to Retrieve Bias $B$ and Errors $R$} $R$ is in general different from the pedestal RMS. It cannot be @@ -104,6 +87,6 @@ for large signals (e.g. calibration signals). \par -In the case of the optimum filter, $R$ is in theory independent from the -signal amplitude $ST$ and depends only on the background $BG$ (eq.~\ref{of_noise}). +In the case of the digital filter, $R$ is in theory independent from the +signal amplitude $ST$ and depends only on the background $BG$ (eq.~\ref{eq:of_noise}). It can be obtained from the fitted error of the extracted signal ($\Delta(SE)_{fitted}$), @@ -117,5 +100,5 @@ \item Determine $R$ by applying the signal extractor to a fixed window of pedestal events. The background fluctuations can be simulated with different - levels of night sky background and the continuous light, but no signal size + levels of night sky background and the continuous light source, but no signal size dependency can be retrieved with the method. \item Determine bias $B$ and resolution $R$ from MC events with and without added noise. @@ -123,5 +106,5 @@ get a dependency of both values from the size of the signal. \item Determine $R$ from the fitted error of $SE$, which is possible for the - fit and the digital filter (eq.~\ref{of_noise}). + fit and the digital filter (eq.~\ref{eq:of_noise}). In prinicple, all dependencies can be retrieved with this method. \end{enumerate} @@ -133,17 +116,16 @@ determine the parameter $R$ for the case of no signal ($ST = 0$). In the case of all extractors using a fixed window from the beginning (extractors nr. \#1 to \#22 -in section~\ref{sec:algorithms}), the results are thus the same by construction as calculating -the mean and the RMS of a same (fixed) number of FADC slices (the conventional ``Pedestal -Calculation''). -\par -In MARS, this possibility is implemented with a function-call to -{\textit{\bf MJPedestal::SetExtractionWithExtractorRndm()}}. -\par -In the case of the amplitude extracting spline (extractor nr. \#23), we placed the +in section~\ref{sec:algorithms}), the results are by construction the same as calculating +the pedestal RMS. +\par +In MARS, this possibility is implemented with a function-call to: \\ + +{\textit{\bf MJPedestal::SetExtractionWithExtractorRndm()}}. \\ + +In the case of the {\textit{\bf amplitude extracting spline}} (extractor nr. \#23), we placed the spline maximum value (which determines the exact extraction window) at a random place -within the digitizing binning resolution (0.01 FADC slices) -of one central FADC slice. -In the case of the digital filter (extractor nr. \#28), the time shift was -randomized for each event within one central FADC slice. +within the digitizing binning resolution of one central FADC slice. +In the case of the {\textit{\bf digital filter}} (extractor nr. \#28), the time shift was +randomized for each event within a fixed global extraction window. \par @@ -157,4 +139,131 @@ fluctuations than in usual observation conditions. \end{enumerate} + + +\begin{figure}[htp] +\centering +\includegraphics[height=0.29\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_00_18_02_14_Run_38993_Signal_Pixel200.eps} +\caption{MExtractTimeAndChargeDigitalFilter: Distribution of extracted "pedestals" from pedestal run with +closed camera lids for one channel.} +\label{fig:df:distped:run38993} +\vspace{\floatsep} +\includegraphics[height=0.29\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_00_18_02_14_Run_38995_Signal_Pixel200.eps} +\caption{MExtractTimeAndChargeDigitalFilter: Distribution of extracted "pedestals" from pedestal run with +extra-galactic star background for one channel.} +\label{fig:df:distped:run38995} +\vspace{\floatsep} +\includegraphics[height=0.29\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_00_18_02_14_Run_38996_Signal_Pixel200.eps} +\caption{MExtractTimeAndChargeDigitalFilter: Distribution of extracted "pedestals" from run with +continuous light level 100 for one channel.} +\label{fig:df:distped:run38996} +\end{figure} + +\begin{figure}[htp] +\centering +\includegraphics[height=0.27\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38993_RelMean.eps} +\caption{MExtractTimeAndChargeDigitalFilter: Difference in mean pedestal (per FADC slice) from pedestal +run with closed camera lids (in photo-electrons)} +\label{fig:df:relmean:run38993} +\vspace{\floatsep} +\includegraphics[height=0.27\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38995_RelMean.eps} +\caption{MExtractTimeAndChargeDigitalFilter: Difference in mean pedestal (per FADC slice) from pedestal +run with extra-galactic star background (in photo-electrons)} +\label{fig:df:relmean:run38995} +\vspace{\floatsep} +\includegraphics[height=0.27\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38996_RelMean.eps} +\caption{MExtractTimeAndChargeDigitalFilter: Difference in mean pedestal (per FADC slice) from run +with continuous light level: 100 (in photo-electrons)} +\label{fig:df:relmean:run38996} +\end{figure} + + +\begin{figure}[htp] +\centering +\includegraphics[height=0.25\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38993_RMSDiff.eps} +\caption{MExtractTimeAndChargeDigitalFilter: Difference pedestal RMS (per FADC slice) with extraction algorithm +appied on a fixed window, and simply summing up the same number of FADC slices. +Pedestal run +with closed camera lids for inner (left) and outer (right) pixels. } +\label{fig:df:relrms:run38993} +\vspace{\floatsep} +\includegraphics[height=0.25\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38995_RMSDiff.eps} +\caption{MExtractTimeAndChargeDigitalFilter: Difference pedestal RMS (per FADC slice) with extraction algorithm +appied on a fixed window, and simply summing up the same number of FADC slices. + from pedestal run with extra-galactic star background for inner (left) +and outer (right) pixels. } +\label{fig:df:relrms:run38995} +\vspace{\floatsep} +\includegraphics[height=0.25\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38996_RMSDiff.eps} +\caption{MExtractTimeAndChargeDigitalFilter: Difference pedestal RMS (per FADC slice) with extraction algorithm +appied on a fixed window, and simply summing up the same number of FADC slices. + from run with continuous light level: 100 for inner (left) +and outer (right) pixels. } +\label{fig:df:relrms:run38996} +\end{figure} + + +\begin{figure}[htp] +\centering +\includegraphics[height=0.29\textheight]{MExtractTimeAndChargeSpline_Amplitude_Range_00_10_04_11_Run_38993_Signal_Pixel200.eps} +\caption{MExtractTimeAndChargeSpline with amplitude: Distribution of extracted "pedestals" from pedestal run +with closed camera lids for one channel.} +\label{fig:amp:distped:run38993} +\vspace{\floatsep} +\includegraphics[height=0.29\textheight]{MExtractTimeAndChargeSpline_Amplitude_Range_00_10_04_11_Run_38995_Signal_Pixel200.eps} +\caption{MExtractTimeAndChargeSpline with amplitude: Distribution of extracted "pedestals" from pedestal run +with extra-galactic star background for one channel.} +\label{fig:amp:distped:run38995} +\vspace{\floatsep} +\includegraphics[height=0.29\textheight]{MExtractTimeAndChargeSpline_Amplitude_Range_00_10_04_11_Run_38996_Signal_Pixel200.eps} +\caption{MExtractTimeAndChargeSpline with amplitude: Distribution of extracted "pedestals" from run with +continuous light level: 100 for one channel.} +\label{fig:amp:distped:run38996} +\end{figure} + +\begin{figure}[htp] +\centering +\includegraphics[height=0.27\textheight]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38993_RelMean.eps} +\caption{MExtractTimeAndChargeSpline with amplitude: Difference in mean pedestal (per FADC slice) with extraction algorithm +appied on a fixed window, and simply summing up the same number of FADC slices. +Pedestal run with closed camera lids.} +\label{fig:amp:relmean:run38993} +\vspace{\floatsep} +\includegraphics[height=0.27\textheight]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38995_RelMean.eps} +\caption{MExtractTimeAndChargeSpline with amplitude: Difference in mean pedestal (per FADC slice) with extraction algorithm +appied on a fixed window, and simply summing up the same number of FADC slices +Pedestal run with extra-galactic star background.} +\label{fig:amp:relmean:run38995} +\vspace{\floatsep} +\includegraphics[height=0.27\textheight]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38996_RelMean.eps} +\caption{MExtractTimeAndChargeSpline with amplitude: Difference in mean pedestal (per FADC slice) with extraction algorithm +appied on a fixed window, and simply summing up the same number of FADC slices. +Pedestal run with continuous light level: 100} +\label{fig:amp:relmean:run38996} +\end{figure} + + +\begin{figure}[htp] +\centering +\includegraphics[height=0.25\textheight]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38993_RMSDiff.eps} +\caption{MExtractTimeAndChargeSpline with amplitude: Difference pedestal RMS (per FADC slice) with extraction + algorithm appied on a fixed window, and simply summing up the same number of FADC slices. +Pedestal run +with closed camera lids for inner (left) and outer (right) pixels. } +\label{fig:amp:relrms:run38993} +\vspace{\floatsep} +\includegraphics[height=0.25\textheight]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38995_RMSDiff.eps} +\caption{MExtractTimeAndChargeSpline with amplitude: Difference pedestal RMS (per FADC slice) with extraction +algorithm appied on a fixed window, and simply summing up the same number of FADC slices. +Pedestal run with extra-galactic star background for inner (left) +and outer (right) pixels.} +\label{fig:amp:relrms:run38995} +\vspace{\floatsep} +\includegraphics[height=0.25\textheight]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38996_RMSDiff.eps} +\caption{MExtractTimeAndChargeSpline with amplitude: Difference pedestal RMS (per FADC slice) with extraction +algorithm appied on a fixed window, and simply summing up the same number of FADC slices. +Pedestal run with continuous light level: 100 for inner (left) +and outer (right) pixels.} +\label{fig:amp:relrms:run38996} +\end{figure} Figures~\ref{fig:df:distped:run38993},~\ref{fig:df:distped:run38995},~\ref{fig:df:distped:run38996}, @@ -189,131 +298,4 @@ normalized pedestal RMS. -\begin{figure}[htp] -\centering -\includegraphics[height=0.29\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_00_18_02_14_Run_38993_Signal_Pixel200.eps} -\caption{MExtractTimeAndChargeDigitalFilter: Distribution of extracted "pedestals" from pedestal run with -closed camera lids for one channel.} -\label{fig:df:distped:run38993} -\vspace{\floatsep} -\includegraphics[height=0.29\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_00_18_02_14_Run_38995_Signal_Pixel200.eps} -\caption{MExtractTimeAndChargeDigitalFilter: Distribution of extracted "pedestals" from pedestal run with -extra-galactic star background for one channel.} -\label{fig:df:distped:run38995} -\vspace{\floatsep} -\includegraphics[height=0.29\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_00_18_02_14_Run_38996_Signal_Pixel200.eps} -\caption{MExtractTimeAndChargeDigitalFilter: Distribution of extracted "pedestals" from run with -continuous light level 100 for one channel.} -\label{fig:df:distped:run38996} -\end{figure} - -\begin{figure}[htp] -\centering -\includegraphics[height=0.27\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38993_RelMean.eps} -\caption{MExtractTimeAndChargeDigitalFilter: Difference in mean pedestal (per FADC slice) from pedestal -run with closed camera lids (in photo-electrons)} -\label{fig:df:relmean:run38993} -\vspace{\floatsep} -\includegraphics[height=0.27\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38995_RelMean.eps} -\caption{MExtractTimeAndChargeDigitalFilter: Difference in mean pedestal (per FADC slice) from pedestal -run with extra-galactic star background (in photo-electrons)} -\label{fig:df:relmean:run38995} -\vspace{\floatsep} -\includegraphics[height=0.27\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38996_RelMean.eps} -\caption{MExtractTimeAndChargeDigitalFilter: Difference in mean pedestal (per FADC slice) from run -with continuous light level: 100 (in photo-electrons)} -\label{fig:df:relmean:run38996} -\end{figure} - - -\begin{figure}[htp] -\centering -\includegraphics[height=0.25\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38993_RMSDiff.eps} -\caption{MExtractTimeAndChargeDigitalFilter: Difference pedestal RMS (per FADC slice) with extraction algorithm -appied on a fixed window, and simply summing up the same number of FADC slices. -Pedestal run -with closed camera lids for inner (left) and outer (right) pixels. } -\label{fig:df:relrms:run38993} -\vspace{\floatsep} -\includegraphics[height=0.25\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38995_RMSDiff.eps} -\caption{MExtractTimeAndChargeDigitalFilter: Difference pedestal RMS (per FADC slice) with extraction algorithm -appied on a fixed window, and simply summing up the same number of FADC slices. - from pedestal run with extra-galactic star background for inner (left) -and outer (right) pixels. } -\label{fig:df:relrms:run38995} -\vspace{\floatsep} -\includegraphics[height=0.25\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38996_RMSDiff.eps} -\caption{MExtractTimeAndChargeDigitalFilter: Difference pedestal RMS (per FADC slice) with extraction algorithm -appied on a fixed window, and simply summing up the same number of FADC slices. - from run with continuous light level: 100 for inner (left) -and outer (right) pixels. } -\label{fig:df:relrms:run38996} -\end{figure} - - -\begin{figure}[htp] -\centering -\includegraphics[height=0.29\textheight]{MExtractTimeAndChargeSpline_Amplitude_Range_00_10_04_11_Run_38993_Signal_Pixel200.eps} -\caption{MExtractTimeAndChargeSpline with amplitude: Distribution of extracted "pedestals" from pedestal run -with closed camera lids for one channel.} -\label{fig:amp:distped:run38993} -\vspace{\floatsep} -\includegraphics[height=0.29\textheight]{MExtractTimeAndChargeSpline_Amplitude_Range_00_10_04_11_Run_38995_Signal_Pixel200.eps} -\caption{MExtractTimeAndChargeSpline with amplitude: Distribution of extracted "pedestals" from pedestal run -with extra-galactic star background for one channel.} -\label{fig:amp:distped:run38995} -\vspace{\floatsep} -\includegraphics[height=0.29\textheight]{MExtractTimeAndChargeSpline_Amplitude_Range_00_10_04_11_Run_38996_Signal_Pixel200.eps} -\caption{MExtractTimeAndChargeSpline with amplitude: Distribution of extracted "pedestals" from run with -continuous light level: 100 for one channel.} -\label{fig:amp:distped:run38996} -\end{figure} - -\begin{figure}[htp] -\centering -\includegraphics[height=0.27\textheight]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38993_RelMean.eps} -\caption{MExtractTimeAndChargeSpline with amplitude: Difference in mean pedestal (per FADC slice) with extraction algorithm -appied on a fixed window, and simply summing up the same number of FADC slices. -Pedestal run with closed camera lids.} -\label{fig:amp:relmean:run38993} -\vspace{\floatsep} -\includegraphics[height=0.27\textheight]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38995_RelMean.eps} -\caption{MExtractTimeAndChargeSpline with amplitude: Difference in mean pedestal (per FADC slice) with extraction algorithm -appied on a fixed window, and simply summing up the same number of FADC slices -Pedestal run with extra-galactic star background.} -\label{fig:amp:relmean:run38995} -\vspace{\floatsep} -\includegraphics[height=0.27\textheight]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38996_RelMean.eps} -\caption{MExtractTimeAndChargeSpline with amplitude: Difference in mean pedestal (per FADC slice) with extraction algorithm -appied on a fixed window, and simply summing up the same number of FADC slices. -Pedestal run with continuous light level: 100} -\label{fig:amp:relmean:run38996} -\end{figure} - - -\begin{figure}[htp] -\centering -\includegraphics[height=0.25\textheight]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38993_RMSDiff.eps} -\caption{MExtractTimeAndChargeSpline with amplitude: Difference pedestal RMS (per FADC slice) with extraction - algorithm appied on a fixed window, and simply summing up the same number of FADC slices. -Pedestal run -with closed camera lids for inner (left) and outer (right) pixels. } -\label{fig:amp:relrms:run38993} -\vspace{\floatsep} -\includegraphics[height=0.25\textheight]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38995_RMSDiff.eps} -\caption{MExtractTimeAndChargeSpline with amplitude: Difference pedestal RMS (per FADC slice) with extraction -algorithm appied on a fixed window, and simply summing up the same number of FADC slices. -Pedestal run with extra-galactic star background for inner (left) -and outer (right) pixels.} -\label{fig:amp:relrms:run38995} -\vspace{\floatsep} -\includegraphics[height=0.25\textheight]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38996_RMSDiff.eps} -\caption{MExtractTimeAndChargeSpline with amplitude: Difference pedestal RMS (per FADC slice) with extraction -algorithm appied on a fixed window, and simply summing up the same number of FADC slices. -Pedestal run with continuous light level: 100 for inner (left) -and outer (right) pixels.} -\label{fig:amp:relrms:run38996} -\end{figure} - - \begin{figure}[htp] @@ -360,9 +342,103 @@ of Pedestal Events} -By applying the signal extractor to a sliding window of pedestal events, we -determine the bias $B$ and the error $R$. -\par -In MARS, this possibility is implemented with a function-call to +In this section, we apply the signal extractor to a sliding window of pedestal events. +\par +In MARS, this possibility can be used with a call to {\textit{\bf MJPedestal::SetExtractionWithExtractor()}}. +\par +Because the background is determined by the single photo-electrons from the night-sky background, +the following possibilities can occur: + +\begin{enumerate} +\item There is no ``signal'' (photo-electron) in the extraction window and the extractor +finds only electronic noise. +Usually, the returned signal charge is then negative. +\item The extractor finds the signal from one photo-electron +\item The extractor finds an overlap of two or more photo-electrons. +\end{enumerate} + +Although the probability to find a certain number of photo-electrons in a fixed window follows a +Poisson distribution, the one for employing the sliding window is {\textit{not}} Poissonian. The extractor +will usually find one photo-electron even if more are present in the global search window, i.e. the +probability for two or more photo-electrons to occur in the global search window is much higher than +the probability for these photo-electrons to overlap in time such as to be recognized as a double +or triple photo-electron pulse by the extractor. This is especially true for small extraction windows +and for the digital filter. + +\par + +Given a global extraction window of size $WS$ and an average rate of photo-electrons from the night-sky +background $R$, we will now calculate the probability for the extractor to find zero photo-electrons in the +$WS$. The probability to find $k$ photo-electrons can be written as: + +\begin{equation} +P(k) = \frac{e^{-R\cdot WS} (R \cdot WS)^k}{k!} +\end{equation} + +and thus: + +\begin{equation} +P(0) = e^{-R\cdot WS} +\end{equation} + +The probability to find more than one photo-electron is then: + +\begin{equation} +P(>0) = 1 - e^{-R\cdot WS} +\end{equation} + +Figures~\ref{fig:sphe:sphespectrum:2.5} and~\ref{fig:sphe:sphespectrum:4.5} show spectra +obtained with the digital filter applied on two different global search windows. +One can clearly distinguish a pedestal peak (fitted to Gaussian with index 0), +corresponding to the case of  $P(0)$ and further +contributions of $P(1)$ and $P(2)$ (fitted to Gaussians with index 1 and 2). +One can also see that the contribution of $P(0)$ dimishes +with increasing global search window size. + +\begin{figure} +\centering +\includegraphics[height=0.3\textheight]{SinglePheSpectrum-28-Run38995-WS2.5.eps} +\caption{MExtractTimeAndChargeDigitalFilter: Signal spectrum obtained from the extraction +of a pedestal run using a sliding window of 6 FADC slices allowed to move within a window of +7 slices. +A pedestal run with galactic star background has been taken and one exemplary pixel (Nr. 100). +One can clearly see the pedestal contribution and a further part corresponding to one or more +photo-electrons.} +\label{fig:df:sphespectrum:2.5} +\vspace{\floatsep} +\includegraphics[height=0.3\textheight]{SinglePheSpectrum-28-Run38995-WS4.5.eps} +\caption{MExtractTimeAndChargeDigitalFilter: Signal spectrum obtained from the extraction +of a pedestal run using a sliding window of 6 FADC slices allowed to move within a window of +9 slices. +A pedestal run with galactic star background has been taken and one exemplary pixel (Nr. 100). +One can clearly see the pedestal contribution and a further part corresponding to one or more +photo-electrons.} +\label{fig:df:sphespectrum:4.5} +\end{figure} + +In the following, we will make a short consistency test: Assuming that the spectral peaks are +attributed correctly, one would expect the following relation: + +\begin{equation} +P(0) / P(>0) = \frac{e^{-R\cdot WS}}{1-e^{-R\cdot WS}} +\end{equation} + +We tested this relation assuming that the fitted area underneath the pedestal peak $Area_0$ is +proportional to $P(0)$ and the sum of the fitted areas underneath the single photo-electron peak +$Area_1$ and the double photo-electron peak $Area_2$ proportional to $P(>0)$. Thus, one expects: + +\begin{equation} +Area_0 / (Area_1 + Area+2 ) = \frac{e^{-R\cdot WS}}{1-e^{-R\cdot WS}} +\end{equation} + +We estimated the effective window size $WS$ as the sum of the range in which the digital filter +amplitude weights are greater than 0.5 (1.6 FADC slices) and the global search window minus the +size of the window size of the weights (which is 6 FADC slices). Figures~\ref{fig::df:ratiofit:run38995} +and~\ref{fig:df:ratiofit:run39258} show the result for two different levels of night-sky background. + + + + +\par \begin{figure}[htp]