\section{Pedestal Extraction \label{sec:pedestals}} \subsection{Pedestal RMS} The background $BG$ (Pedestal) can be completely described by the noise-autocorrelation matrix $\boldsymbol{B}$ (eq.~\ref{eq:autocorr}), where the diagonal elements give what is usually denoted as the ``Pedestal RMS''. \par By definition, the $\boldsymbol{B}$ and thus the ``pedestal RMS'' is independent from the signal extractor. \subsection{Bias and Mean-squared Error} Consider a large number of same signals $S$. By applying a signal extractor we obtain a distribution of estimated signals $\widehat{S}$ (for fixed $S$ and fixed background fluctuations $BG$). The distribution of the quantity \begin{equation} X = \widehat{S}-S \end{equation} has the mean $B$ and the Variance $MSE$ defined as: \begin{eqnarray} B \ \ \ \ = \ \ \ \ \ \ \ \ \ \ \ &=& \ \ <\widehat{S}> \ -\ S\\ R \ \ \ \ = \ <(X-B)^2> &=& \ Var[\widehat{S}]\\ MSE \ = \ \ \ \ \ \ \ \ \ &=& \ Var[\widehat{S}] +\ B^2 \end{eqnarray} The parameter $B$ is also called the {\textit{\bf BIAS}} of the estimator and $MSE$ the {\textit{\bf MEAN-SQUARED ERROR}} which combines the variance of $\widehat{S}$ and the bias. Both depend generally on the size of $S$ and the background fluctuations $BG$, thus: $B = B(S,BG)$ and $MSE = MSE(S,BG)$. \par Usually, one measures easily the parameter $R$, but needs the $MSE$ for statistical analysis (e.g. in the image cleaning). However, only in case of a vanishing bias $B$, the two numbers are equal. Otherwise, the bias $B$ has to be known beforehand. Note that every sliding window extractor has a bias, especially at low or vanishing signals $S$. \subsection{Pedestal Fluctuations as Contribution to the Signal Fluctuations \label{sec:ffactor}} A photo-multiplier signal yields, to a very good approximation, the following relation: \begin{equation} \frac{Var[Q]}{^2} = \frac{1}{} * F^2 \end{equation} Here, $Q$ is the signal fluctuation due to the number of signal photo-electrons (equiv. to the signal $S$), and $Var[Q]$ the fluctuations of the true signal $Q$ due to the Poisson fluctuations of the number of photo-electrons. Because of: \begin{eqnarray} \widehat{Q} &=& Q + X \\ Var(\widehat{Q}) &=& Var(Q) + Var(X) \\ Var(Q) &=& Var(\widehat{Q}) - Var(X) \end{eqnarray} $Var[Q]$ can be obtained from: \begin{eqnarray} Var(Q) &\approx& Var(\widehat{Q}) - Var(\widehat{Q}=0) \label{eq:rmssubtraction} \end{eqnarray} In the last line of eq.~\ref{eq:rmssubtraction}, it is assumed that $R$ does not dependent on the signal height\footnote{% A way to check whether the right RMS has been subtracted is to make the ``Razmick''-plot \begin{equation} \frac{Var[\widehat{Q}]}{<\widehat{Q}>^2} \quad \textit{vs.} \quad \frac{1}{<\widehat{Q}>} \end{equation} This should give a straight line passing through the origin. The slope of the line is equal to \begin{equation} c * F^2 \end{equation} where $c$ is the photon/ADC conversion factor $/$.} (as is the case for the digital filter, eq.~\ref{eq:of_noise}). One can then retrieve $R$ by applying the signal extractor to a {\textit{\bf fixed window}} of pedestal events, where the bias vanishes and measure $Var[\widehat{Q}=0]$. \subsection{Methods to Retrieve Bias and Mean-Squared Error} In general, the extracted signal variance $R$ is different from the pedestal RMS. It cannot be obtained by applying the signal extractor to pedestal events, because of the (unknown) bias. \par In the case of the digital filter, $R$ is expected to be independent from the signal amplitude $S$ and depends only on the background $BG$ (eq.~\ref{eq:of_noise}). It can then be obtained from the calculation of the variance $Var[\widehat{Q}]$ by applying the extractor to a fixed window of pure background events (``pedestal events'') and get rid of the bias in that way. \par In order to calculate bias and Mean-squared error, we proceeded in the following ways: \begin{enumerate} \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 source, but no signal size dependency can be retrieved with this method. \item Determine $B$ and $MSE$ from MC events with and without added noise. Assuming that $MSE$ and $B$ are negligible for the events without noise, one can get a dependency of both values from the size of the signal. \item Determine $MSE$ from the fitted error of $\widehat{S}$, which is possible for the fit and the digital filter (eq.~\ref{eq:of_noise}). In principle, all dependencies can be retrieved with this method. \end{enumerate} \begin{figure}[htp] \centering \includegraphics[width=0.3\linewidth]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38993_RelMean.eps} \vspace{\floatsep} \includegraphics[width=0.3\linewidth]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38995_RelMean.eps} \vspace{\floatsep} \includegraphics[width=0.3\linewidth]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38996_RelMean.eps} \caption{MExtractTimeAndChargeSpline with amplitude extraction: Difference in mean pedestal (per FADC slice) between extraction algorithm applied on a fixed window of 1 FADC slice (``extractor random'') and a simple addition of 2 FADC slices (``fundamental''). On the left, a run with closed camera has been taken, in the center an opened camera observing an extra-galactic star field and on the right, an open camera being illuminated by the continuous light of the calibration (level: 100). Every entry corresponds to one pixel.} \label{fig:amp:relmean} \end{figure} \begin{figure}[htp] \centering \includegraphics[width=0.3\linewidth]{MExtractTimeAndChargeSpline_Rise-and-Fall-Time_0.5_1.5_Range_01_10_02_12_Run_38993_RelMean.eps} \vspace{\floatsep} \includegraphics[width=0.3\linewidth]{MExtractTimeAndChargeSpline_Rise-and-Fall-Time_0.5_1.5_Range_01_10_02_12_Run_38995_RelMean.eps} \vspace{\floatsep} \includegraphics[width=0.3\linewidth]{MExtractTimeAndChargeSpline_Rise-and-Fall-Time_0.5_1.5_Range_01_10_02_12_Run_38996_RelMean.eps} \caption{MExtractTimeAndChargeSpline with integral over 2 slices: Difference in mean pedestal (per FADC slice) between extraction algorithm applied on a fixed window of 2 FADC slices (``extractor random'') and a simple addition of 2 FADC slices (``fundamental''). On the left, a run with closed camera has been taken, in the center an opened camera observing an extra-galactic star field and on the right, an open camera being illuminated by the continuous light of the calibration (level: 100). Every entry corresponds to one pixel.} \label{fig:int:relmean} \end{figure} \begin{figure}[htp] \centering \vspace{\floatsep} \includegraphics[width=0.3\linewidth]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38993_RelMean.eps} \vspace{\floatsep} \includegraphics[width=0.3\linewidth]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38995_RelMean.eps} \vspace{\floatsep} \includegraphics[width=0.3\linewidth]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38996_RelMean.eps} \caption{MExtractTimeAndChargeDigitalFilter: Difference in mean pedestal (per FADC slice) between extraction algorithm applied on a fixed window of 6 FADC slices and time-randomized weights (``extractor random'') and a simple addition of 6 FADC slices (``fundamental''). On the left, a run with closed camera has been taken, in the center an opened camera observing an extra-galactic star field and on the right, an open camera being illuminated by the continuous light of the calibration (level: 100). Every entry corresponds to one pixel.} \label{fig:df:relmean} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{ \label{sec:ped:fixedwindow} Application of the Signal Extractor to a Fixed Window of Pedestal Events} By applying the signal extractor to a fixed window of pedestal events, we determine the parameter $R$ for the case of no signal ($Q = 0$). In the case of extractors using a fixed window (extractors nr. \#1 to \#22 in section~\ref{sec:algorithms}), the results are the same by construction as calculating the pedestal RMS. \par In MARS, this functionality is implemented with a function-call to: \\ {\textit{\bf MJPedestal::SetExtractionWithExtractorRndm()}} and/or \\ {\textit{\bf MExtractPedestal::SetRandomCalculation()}}\\ Besides fixing the global extraction window, additionally the following steps are undertaken in order to assure that the bias vanishes: \begin{description} \item[\textit{MExtractTimeAndChargeSpline}:\xspace] The spline maximum position -- which determines the exact extraction window -- is placed arbitrarily at a random place within the digitizing binning resolution of one central FADC slice. \item[\textit{MExtractTimeAndChargeDigitalFilter}:\xspace] The second step timing offset $\tau$ (eq.~\ref{eq:offsettau}) gets randomized for each event. \end{description} \par The following figures~\ref{fig:amp:relmean} through~\ref{fig:df:relrms} show results obtained with the second method for three background intensities: \begin{enumerate} \item Closed camera and no (Poissonian) fluctuation due to photons from the night sky background \item The camera pointing to an extra-galactic region with stars in the field of view \item The camera illuminated by a continuous light source of intensity 100. \end{enumerate} Figures~\ref{fig:amp:relmean} through~\ref{fig:df:relmean} show the calculated biases obtained with this method for all pixels in the camera and for the different levels of (night-sky) background. One can see that the bias vanishes to an accuracy of better than 1\% for the extractors which are used in this TDAS. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%1 \begin{figure}[htp] \centering \includegraphics[width=0.47\linewidth]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38993_RMSDiff.eps} \vspace{\floatsep} \includegraphics[width=0.47\linewidth]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38995_RMSDiff.eps} \vspace{\floatsep} \includegraphics[width=0.47\linewidth]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38996_RMSDiff.eps} \caption{MExtractTimeAndChargeSpline with amplitude: Difference in RMS (per FADC slice) between extraction algorithm applied on a fixed window and the corresponding pedestal RMS. Closed camera (left), open camera observing extra-galactic star field (right) and camera being illuminated by the continuous light (bottom). Every entry corresponds to one pixel.} \label{fig:amp:relrms} \end{figure} \begin{figure}[htp] \centering \includegraphics[width=0.47\linewidth]{MExtractTimeAndChargeSpline_Rise-and-Fall-Time_0.5_1.5_Range_01_10_02_12_Run_38993_RMSDiff.eps} \vspace{\floatsep} \includegraphics[width=0.47\linewidth]{MExtractTimeAndChargeSpline_Rise-and-Fall-Time_0.5_1.5_Range_01_10_02_12_Run_38995_RMSDiff.eps} \vspace{\floatsep} \includegraphics[width=0.47\linewidth]{MExtractTimeAndChargeSpline_Rise-and-Fall-Time_0.5_1.5_Range_01_10_02_12_Run_38996_RMSDiff.eps} \caption{MExtractTimeAndChargeSpline with integral over 2 slices: Difference in RMS (per FADC slice) between extraction algorithm applied on a fixed window and the corresponding pedestal RMS. Closed camera (left), open camera observing extra-galactic star field (right) and camera being illuminated by the continuous light (bottom). Every entry corresponds to one pixel.} \label{fig:amp:relrms} \end{figure} \begin{figure}[htp] \centering \includegraphics[width=0.47\linewidth]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38993_RMSDiff.eps} \vspace{\floatsep} \includegraphics[width=0.47\linewidth]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38995_RMSDiff.eps} \vspace{\floatsep} \includegraphics[width=0.47\linewidth]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38996_RMSDiff.eps} \caption{MExtractTimeAndChargeDigitalFilter: Difference in RMS (per FADC slice) between extraction algorithm applied on a fixed window and the corresponding pedestal RMS. Closed camera (left), open camera observing extra-galactic star field (right) and camera being illuminated by the continuous light (bottom). Every entry corresponds to one pixel.} \label{fig:df:relrms} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Figures~\ref{fig:amp:relrms} through~\ref{fig:amp:relrms} show the differences in $R$ between the calculated pedestal RMS and the one obtained by applying the extractor, converted to equivalent photo-electrons. One entry corresponds to one pixel of the camera. The distributions have a negative mean in the case of the digital filter showing the ``filter'' capacity of that algorithm. It ``filters out'' between 0.12 photo-electrons night sky background for the extra-galactic star-field until 0.2 photo-electrons for the continuous light. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{ \label{sec:ped:slidingwindow} Application of the Signal Extractor to a Sliding Window of Pedestal Events} By applying the signal extractor to a global extraction window of pedestal events, allowing it to ``slide'' and maximize the encountered signal, we determine the bias $B$ and the mean-squared error $MSE$ for the case of no signal ($S=0$). \par In MARS, this functionality is implemented with a function-call to: \\ {\textit{\bf MJPedestal::SetExtractionWithExtractor()}} \\ \par Table~\ref{tab:bias} shows bias, resolution and mean-square error for all extractors using a sliding window. In this sample, every extractor had the freedom to move 5 slices, i.e. the global window size was fixed to five plus the extractor window size. This first line shows the resolution of the smallest existing robust fixed window algorithm in order to give the reference value of 2.5 and 3 photo-electrons RMS. \par One can see that the bias $B$ typically decreases with increasing window size (except for the digital filter), while the error $R$ increases with increasing window size. There is also a small difference between the obtained error on a fixed window extraction and the one obtained from a sliding window extraction in the case of the spline and digital filter algorithms. The mean-squared error has an optimum somewhere between: In the case of the sliding window and the spline at the lowest window size, in the case of the digital filter at 4 slices. The global winners are extractors \#25 (spline with integration of 1 slice) and \#29 (digital filter with integration of 4 slices). All sliding window extractors -- except \#21 -- have a smaller mean-square error than the resolution of the fixed window reference extractor. This means that the global error of the sliding window extractors is smaller than the one of the fixed window extractors even if the first have a bias. \begin{table}[htp] \centering \scriptsize{ \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline \hline \multicolumn{14}{|c|}{Statistical Parameters for $S=0$} \\ \hline \hline & & \multicolumn{4}{|c|}{Closed camera} & \multicolumn{4}{|c|}{Extra-galactic NSB} & \multicolumn{4}{|c|}{Galactic NSB} \\ \hline \hline Nr. & Name & $R$ & $R$ & $B$ & $\sqrt{MSE}$ & $R$ &$R$ & $B$ & $\sqrt{MSE}$& $R$ & $R$& $B$ & $\sqrt{MSE}$ \\ & & (FW) & (SW)& (SW)& (SW) & (FW) &(SW) & (SW)& (SW) & (FW)&(SW) & (SW)&(SW) \\ \hline \hline 4 & Fixed Win. 8 & 1.2 & -- & 0.0 & 1.2 & 2.5 & -- & 0.0 & 2.5 & 3.0 & -- & 0.0 & 3.0 \\ \hline -- & Slid. Win. 1 & 0.4 & 0.4 & 0.4 & 0.6 & 1.2 & 1.2 & 1.3 & 1.8 & 1.4 & 1.4 & 1.5 & 2.0 \\ 17 & Slid. Win. 2 & 0.5 & 0.5 & 0.4 & 0.6 & 1.4 & 1.4 & 1.2 & 1.8 & 1.6 & 1.6 & 1.5 & 2.2 \\ 18 & Slid. Win. 4 & 0.8 & 0.8 & 0.5 & 0.9 & 1.9 & 1.9 & 1.2 & 2.2 & 2.2 & 2.3 & 1.6 & 2.8 \\ 20 & Slid. Win. 6 & 1.0 & 1.0 & 0.4 & 1.1 & 2.2 & 2.2 & 1.1 & 2.5 & 2.6 & 2.7 & 1.4 & 3.0 \\ 21 & Slid. Win. 8 & 1.2 & 1.3 & 0.4 & 1.4 & 2.5 & 2.5 & 1.0 & 2.7 & 3.0 & 3.2 & 1.4 & 3.5 \\ \hline 23 & Spline Amp. & 0.4 & \textcolor{red}{\bf 0.4} & 0.4 & 0.6 & 1.1 & 1.2 & 1.3 & 1.8 & 1.3 & 1.4 & 1.6 & 2.1 \\ 24 & \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 & 1.3 & \textcolor{red}{\bf 1.3} & 1.3 & 1.8 \\ 25 & Spline Int. 2 & 0.5 & 0.5 & 0.3 & 0.6 & 1.3 & 1.4 & 0.9 & 1.7 & 1.7 & 1.6 & 1.2 & 2.0 \\ 26 & 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 & 2.0 & 2.0 & 1.0 & 2.2 \\ 27 & Spline Int. 6 & 1.0 & 1.0 & 0.3 & 1.0 & 2.0 & 2.0 & \textcolor{red}{\bf 0.8} & 2.2 & 2.6 & 2.5 & \textcolor{red}{\bf 0.9} & 2.7 \\ \hline 28 & Dig. Filt. 6 & 0.4 & 0.5 & 0.4 & 0.6 & 1.1 & 1.3 & 1.3 & 1.8 & 1.3 & 1.5 & 1.5 & 2.1 \\ 29 & \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} & 1.0 & 1.3 & 1.1 & \textcolor{red}{\bf 1.7} \\ \hline \hline \end{tabular} } \caption{The statistical parameters bias, resolution and mean error for the sliding window algorithm. The first line displays the resolution of the smallest existing robust fixed--window extractor for reference. All units in equiv. photo-electrons, uncertainty: 0.1 phes. All extractors were allowed to move 5 FADC slices plus their window size. The ``winners'' for each row are marked in red. Global winners (within the given uncertainty) are the extractors Nr. \#24 (MExtractTimeAndChargeSpline with an integration window of 1 FADC slice) and Nr.\#29 (MExtractTimeAndChargeDigitalFilter with an integration window size of 4 slices)} \label{tab:bias} \end{table} Figures~\ref{fig:sw:distped} through~\ref{fig:df4:distped} show the extracted pedestal distributions for some selected extractors (\#18, \#23, \#25, \#28 and \#29) for one exemplary channel (pixel 100) and two background situations: Closed camera with only electronic noise and open camera pointing to an extra-galactic source. One can see the (asymmetric) Poisson behaviour of the night sky background photons for the distributions with open camera. \begin{figure}[htp] \centering \includegraphics[height=0.43\textheight]{PedestalSpectrum-18-Run38993.eps} \vspace{\floatsep} \includegraphics[height=0.43\textheight]{PedestalSpectrum-18-Run38995.eps} \caption{MExtractTimeAndChargeSlidingWindow with extraction window of 4 FADC slices: Distribution of extracted "pedestals" from pedestal run with closed camera (top) and open camera observing an extra-galactic star field (bottom) for one channel (pixel 100). The result obtained from a simple addition of 4 FADC slice contents (``fundamental'') is displayed as red histogram, the one obtained from the application of the algorithm on a fixed window of 4 FADC slices as blue histogram (``extractor random'') and the one obtained from the full algorithm allowed to slide within a global window of 12 slices. The obtained histogram means and RMSs have been converted to equiv. photo-electrons.} \label{fig:sw:distped} \end{figure} \begin{figure}[htp] \centering \includegraphics[height=0.43\textheight]{PedestalSpectrum-23-Run38993.eps} \vspace{\floatsep} \includegraphics[height=0.43\textheight]{PedestalSpectrum-23-Run38995.eps} \caption{MExtractTimeAndChargeSpline with amplitude extraction: Spectrum of extracted "pedestals" from pedestal run with closed camera lids (top) and open lids observing an extra-galactic star field (bottom) for one channel (pixel 100). The result obtained from a simple addition of 2 FADC slice contents (``fundamental'') is displayed as red histogram, the one obtained from the application of the algorithm on a fixed window of 1 FADC slice as blue histogram (``extractor random'') and the one obtained from the full algorithm allowed to slide within a global window of 12 slices. The obtained histogram means and RMSs have been converted to equiv. photo-electrons.} \label{fig:amp:distped} \end{figure} \begin{figure}[htp] \centering \includegraphics[height=0.43\textheight]{PedestalSpectrum-25-Run38993.eps} \vspace{\floatsep} \includegraphics[height=0.43\textheight]{PedestalSpectrum-25-Run38995.eps} \caption{MExtractTimeAndChargeSpline with integral extraction over 2 FADC slices: Distribution of extracted "pedestals" from pedestal run with closed camera lids (top) and open lids observing an extra-galactic star field (bottom) for one channel (pixel 100). The result obtained from a simple addition of 2 FADC slice contents (``fundamental'') is displayed as red histogram, the one obtained from the application of time-randomized weights on a fixed window of 2 FADC slices as blue histogram and the one obtained from the full algorithm allowed to slide within a global window of 12 slices. The obtained histogram means and RMSs have been converted to equiv. photo-electrons.} \label{fig:int:distped} \end{figure} \begin{figure}[htp] \centering \includegraphics[height=0.43\textheight]{PedestalSpectrum-28-Run38993.eps} \vspace{\floatsep} \includegraphics[height=0.43\textheight]{PedestalSpectrum-28-Run38995.eps} \caption{MExtractTimeAndChargeDigitalFilter: Spectrum of extracted "pedestals" from pedestal run with closed camera lids (top) and open lids observing an extra-galactic star field (bottom) for one channel (pixel 100). The result obtained from a simple addition of 6 FADC slice contents (``fundamental'') is displayed as red histogram, the one obtained from the application of time-randomized weights on a fixed window of 6 slices as blue histogram and the one obtained from the full algorithm allowed to slide within a global window of 12 slices. The obtained histogram means and RMSs have been converted to equiv. photo-electrons.} \label{fig:df6:distped} \end{figure} \begin{figure}[htp] \centering \includegraphics[height=0.43\textheight]{PedestalSpectrum-29-Run38993.eps} \vspace{\floatsep} \includegraphics[height=0.43\textheight]{PedestalSpectrum-29-Run38995.eps} \caption{MExtractTimeAndChargeDigitalFilter: Spectrum of extracted "pedestals" from pedestal run with closed camera lids (top) and open lids observing an extra-galactic star field (bottom) for one channel (pixel 100). The result obtained from a simple addition of 4 FADC slice contents (``fundamental'') is displayed as red histogram, the one obtained from the application of time-randomized weights on a fixed window of 4 slices as blue histogram and the one obtained from the full algorithm allowed to slide within a global window of 10 slices. The obtained histogram means and RMSs have been converted to equiv. photo-electrons.} \label{fig:df4:distped} \end{figure} \subsection{ \label{sec:ped:singlephe} Single Photo-Electron Extraction with the Digital Filter} Figures~\ref{fig:df:sphespectrum} 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) and further, positive contributions. \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 There is one photo-electron in the extraction window and the extractor finds it. \item There are more than on photo-electrons in the extraction window, but separated by more than two FADC slices whereupon the extractor finds the one with the highest charge (upward fluctuation). \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 any number of $k$ 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 one or more photo-electrons is then: \begin{equation} P(>0) = 1 - e^{-R\cdot WS} \end{equation} In figures~\ref{fig:df:sphespectrum}, one can clearly distinguish the 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} \vspace{\floatsep} \includegraphics[height=0.3\textheight]{SinglePheSpectrum-28-Run38995-WS4.5.eps} \vspace{\floatsep} \includegraphics[height=0.3\textheight]{SinglePheSpectrum-28-Run38995-WS8.5.eps} \caption{MExtractTimeAndChargeDigitalFilter: 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 (top), 9 (center) and 13 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} \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} \mathrm{Area}_0 / (\mathrm{Area}_1 + \mathrm{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 (1.5 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} show the result for two different levels of night-sky background. The fitted rates deliver 0.08 and 0.1 phes/ns, respectively. These rates are about 50\% too low compared to the results obtained in the November 2004 test campaign. However, we should take into account that the method is at the limit of distinguishing single photo-electrons. It may occur often that a single photo-electron signal is too low in order to get recognized as such. We tried various pixels and found that some of them do not permit to apply this method at all. The ones which succeed, however, yield about the same fitted rates. To conclude, one may say that there is consistency within the double-peak structure of the pedestal spectrum found by the digital filter which can be explained by the fact that single photo-electrons are found. \par \begin{figure}[htp] \centering \includegraphics[height=0.4\textheight]{SinglePheRatio-28-Run38995.eps} \vspace{\floatsep} \includegraphics[height=0.4\textheight]{SinglePheRatio-28-Run39258.eps} \caption{MExtractTimeAndChargeDigitalFilter: Fit to the ratio of the area beneath the pedestal peak and the single and double photo-electron(s) peak(s) with the extraction algorithm applied on a sliding window of different sizes. In the top plot, a pedestal run with extra-galactic star background has been taken and in the bottom, a galatic star background. An exemplary pixel (Nr. 100) has been used. Above, a rate of 0.08 phe/ns and below, a rate of 0.1 phe/ns has been obtained.} \label{fig:df:ratiofit} \end{figure} Figure~\ref{fig:df:convfit} shows the obtained ``conversion factors'' and ``F-Factor'' computed as: \begin{eqnarray} c_{phe} &=& \frac{1}{\mu_1 - \mu_0} \\ F_{phe} &=& \sqrt{1 + \frac{\sigma_1^2 - \sigma_0^2}{(\mu_1 - \mu_0)^2} } \end{eqnarray} where $\mu_0$ is the mean position of the pedestal peak and $\mu_1$ the mean position of the (assumed) single photo-electron peak. The obtained conversion factors are systematically lower than the ones obtained from the standard calibration and decrease with increasing window size. This is consistent with the assumption that the digital filter finds the upward fluctuating pulse out of several. Therefore, $\mu_1$ is biased against higher values. The F-Factor is also systematically low, which is also consistent with the assumption that the spacing between $\mu_1$ and $\mu_0$ is artificially high. One can also see that the error bars are too high for a ``calibration'' of the F-Factor. \par In conclusion, one can say that the digital filter is at the edge of being able to see single photo-electrons, however a single photo-electron calibration cannot yet be done with the current FADC system because the resolution is too poor. \begin{figure}[htp] \centering \includegraphics[height=0.4\textheight]{ConvFactor-28-Run38995.eps} \vspace{\floatsep} \includegraphics[height=0.4\textheight]{FFactor-28-Run38995.eps} \caption{MExtractTimeAndChargeDigitalFilter: Obtained conversion factors (top) and F-Factors (bottom) from the position and width of the fitted Gaussian mean of the single photo-electron peak and the pedestal peak depending on the applied global extraction window sizes. A pedestal run with extra-galactic star background has been taken and an exemplary pixel (Nr. 100) used. The conversion factor obtained from the standard calibration is shown as a reference line. The obtained conversion factors are systematically lower than the reference one.} \label{fig:df:convfit} \end{figure} %%% Local Variables: %%% mode: latex %%% TeX-master: "MAGIC_signal_reco" %%% TeX-master: "MAGIC_signal_reco" %%% TeX-master: "MAGIC_signal_reco" %%% End: