Index: /trunk/MagicSoft/TDAS-Extractor/Pedestal.tex =================================================================== --- /trunk/MagicSoft/TDAS-Extractor/Pedestal.tex (revision 6621) +++ /trunk/MagicSoft/TDAS-Extractor/Pedestal.tex (revision 6622) @@ -6,9 +6,9 @@ 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. +where the square root of the diagonal elements give what is usually denoted as the ``Pedestal RMS''. +\par + +By definition, $\boldsymbol{B}$ and thus the ``pedestal RMS'' +is independent of the signal extractor. \subsection{Pedestal Fluctuations as Contribution to the Signal Fluctuations \label{sec:ffactor}} @@ -31,47 +31,48 @@ \end{eqnarray} +Only in the case that the intrinsic extractor resolution $R$ at fixed background $BG$ does not depend on the signal +intensity\footnote{Theoretically, this is the case for the digital filter, eq.~\ref{eq:of_noise}.}, $Var[Q]$ can be obtained from: \begin{eqnarray} -Var(Q) &\approx& Var(\widehat{Q}) - Var(\widehat{Q}=0) +Var(Q) &\approx& Var(\widehat{Q}) - Var(\widehat{Q})\,\vline_{\,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 $R$ +%\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 $/$.} + + 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 $Var[\widehat{Q}=0]$. +bias vanishes and measure $Var(\widehat{Q})\,\vline_{\,Q=0}$. \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: +It can be obtained by applying the signal extractor to pedestal events yielding the bias and +the resolution $R$. +\par +In the case of the digital filter, $R$ is expected to be independent of the +signal amplitude $S$ and dependent 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 the bias and Mean-squared error, we proceed in the following ways: \begin{enumerate} \item Determine $R$ by applying the signal extractor to a fixed window @@ -79,10 +80,13 @@ 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 +\item Determine $B$ and $MSE$ from MC events with added noise. +% Assuming that $MSE$ and $B$ are negligible for the events without noise, one can + With this method, one can get a dependency of both values from the size of the signal, + although the MC might contain systematic differences with respect to the real data. +\item Determine $MSE$ from the error retrieved from the fit results 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. + In principle, all dependencies can be retrieved with this method, although some systematic errors are not taken into account + with this method: Deviations of the real pulse from the fitted one, errors in the noise auto-correlation matrix and numerical +precision issues. All these systematic effects add an additional contribution to the true resolution proportional to the signal strength. \end{enumerate} @@ -147,23 +151,25 @@ 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 +determine the parameter $R$ for the case of no signal ($Q = 0$)\footnote{% +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. +as calculating the RMS of the sum of a fixed number of FADC slice, traditionally +named ``pedestal RMS'' in MARS.}. \par In MARS, this functionality is implemented with a function-call to: \\ -{\textit{\bf MJPedestal::SetExtractionWithExtractorRndm()}} and/or \\ +{\textit{\bf MJPedestal::SetExtractionWithExtractorRndm()}} including \\ {\textit{\bf MExtractPedestal::SetRandomCalculation()}}\\ Besides fixing the global extraction window, additionally the following steps are undertaken -in order to assure that the bias vanishes: +in order to assure an un-biased resolution. \begin{description} \item[\textit{MExtractTimeAndChargeSpline}:\xspace] The spline -maximum position -- which determines the exact extraction window -- is placed arbitrarily +maximum position -- which determines the exact extraction window -- is placed 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. +offset $\tau$ (eq.~\ref{eq:offsettau}) is chosen randomly for each event. \end{description} @@ -181,7 +187,7 @@ 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. +and for the different levels of (night-sky) background applied to 1000 pedestal events. +One can see that the bias vanishes to an accuracy of better than 2\% of a photo-electron +makefor the extractors which are used in this TDAS. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%1 @@ -218,5 +224,5 @@ Every entry corresponds to one pixel.} -\label{fig:amp:relrms} +\label{fig:int:relrms} \end{figure} @@ -242,8 +248,10 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -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. +Figures~\ref{fig:amp:relrms} through~\ref{fig:df:relrms} show the +differences in $R$ between the RMS of simply summing up the FADC slices over the extraction window +(in MARS called: ``Fundamental Pedestal RMS'') and +the one obtained by applying the extractor to the same extraction window +(in MARS called: ``Pedestal RMS with Extractor Rndm''). One entry of each histogram 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 @@ -265,59 +273,75 @@ \par -Table~\ref{tab:bias} shows bias, resolution and mean-square error for all extractors using +Table~\ref{tab:bias} shows the bias, the resolution and the 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. +value of 2.5 and 3 photo-electrons RMS for an extra-galactic and a galactic star-field, respectively. \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 +with increasing window size, while the error $R$ increases with +increasing window size, except for the digital filter. 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 +The mean-squared error has an optimum somewhere in 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 +at 4 slices. The global winners is extractor~\#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. - +\par +The important information for the image cleaning is the number of photo-electrons above which the probability for obtaining +a noise fluctuation is smaller than 0.3\% (3$\sigma$). We approximated that number with the formula: + +\begin{equation} +N_{\mathrm{phe}}^{\mathrm{thres.}} \approx B + 3\cdot R +\end{equation} + +Table~\ref{tab:bias} shows that most of the sliding window algorithms yield a smaller signal threshold than the fixed window ones, +although 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 +for the galactic star-field is obtained by the digital filter fitting 4 FADC slices (extractor~\%29). +This is almost a factor 2 lower than the fixed window results. Also the spline integrating 1 FADC slice (extractor~\%24) yields almost +comparable results. + +\begin{landscape} +%\rotatebox{90}{% \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) \\ +\vspace{3cm} +\scriptsize{% +\centering +\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} +\hline +\hline +\multicolumn{16}{|c|}{Statistical Parameters for $S=0$} \\ +\hline +\hline + & & \multicolumn{4}{|c|}{Closed camera} & \multicolumn{5}{|c|}{Extra-galactic NSB} & \multicolumn{5}{|c|}{Galactic NSB} \\ +\hline +\hline +Nr. & Name & $R$ & $R$ & $B$ & $\sqrt{MSE}$ & $R$ &$R$ & $B$ & $\sqrt{MSE}$ & $B+3R$ & $R$ & $R$& $B$ & $\sqrt{MSE}$ & $B+3R$ \\ + & & (FW) & (SW)& (SW)& (SW) & (FW) &(SW) & (SW)& (SW) & (99.7\% prob.) & (FW)&(SW) & (SW)&(SW) & (99.7\% prob.) \\ \hline \hline -4 & Fixed Win. 8 & 1.2 & -- & 0.0 & 1.2 & 2.5 & -- & 0.0 & 2.5 & 3.0 & -- & 0.0 & 3.0 \\ +4 & 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 \\ \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 \\ +-- & 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 \\ +17 & 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 \\ +18 & 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 \\ +20 & 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 \\ +21 & 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 \\ \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 \\ +23 & 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 \\ +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 & 4.6 & 1.3 & \textcolor{red}{\bf 1.3} & 1.3 & 1.8 & 5.2 \\ +25 & 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 \\ +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 & 5.3 & 2.0 & 2.0 & 1.0 & 2.2 & 7.0 \\ +27 & 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 \\ \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} \\ +28 & 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 \\ +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} & \textcolor{red}{\bf 4.2} & 1.0 & \textcolor{red}{\bf 1.3} & 1.1 & \textcolor{red}{\bf 1.7} & \textcolor{red}{\bf 5.0 }\\ \hline \hline \end{tabular} -} +\vspace{1cm} \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 @@ -329,5 +353,10 @@ (MExtractTimeAndChargeDigitalFilter with an integration window size of 4 slices)} \label{tab:bias} +} \end{table} +%} +\end{landscape} + +\clearpage Figures~\ref{fig:sw:distped} through~\ref{fig:df4:distped} show the @@ -419,8 +448,10 @@ \end{figure} +\clearpage + \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. +Figure~\ref{fig:df:sphespectrum} shows spectra +obtained with the digital filter applied on three different global search windows. One can clearly distinguish a pedestal peak (fitted to Gaussian with index 0) and further, positive contributions. @@ -434,6 +465,6 @@ 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 There are more than one photo-electron in the extraction window, but separated by more than +two FADC slices whereupon the extractor finds the one with the highest charge (upward fluctuation) of both. \item The extractor finds an overlap of two or more photo-electrons. \end{enumerate} @@ -449,10 +480,10 @@ \par -Given a global extraction window of size $WS$ and an average rate of photo-electrons from the night-sky +Given a global extraction window of size $\mathrm{\it WS}$ 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: +$\mathrm{\it WS}$. 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!} +P(k) = \frac{e^{-R\cdot \mathrm{\it WS}} (R \cdot \mathrm{\it WS})^k}{k!} \end{equation} @@ -460,5 +491,5 @@ \begin{equation} -P(0) = e^{-R\cdot WS} +P(0) = e^{-R\cdot \mathrm{\it WS}} \end{equation} @@ -466,5 +497,5 @@ \begin{equation} -P(>0) = 1 - e^{-R\cdot WS} +P(>0) = 1 - e^{-R\cdot \mathrm{\it WS}} \end{equation} @@ -499,18 +530,19 @@ \end{equation} -We tested this relation assuming that the fitted area underneath the pedestal peak Area$_0$ is +We tested this relation assuming that the fitted area underneath the pedestal peak $\mathrm{\it Area}_0$ 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: +$\mathrm{\it Area}_1$ and the double photo-electron peak $\mathrm{\it Area}_2$ proportional to $P(>0)$. We assumed +that the probability for a triple photo-electron to occur is negligible. Thus, one expects: \begin{equation} -\mathrm{Area}_0 / (\mathrm{Area}_1 + \mathrm{Area}_2 ) = \frac{e^{-R\cdot WS}}{1-e^{-R\cdot WS}} +\mathrm{\it Area}_0 / (\mathrm{\it Area}_1 + \mathrm{\it 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 +We estimated the effective window size $\mathrm{\it WS}$ 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 +size of the window size of the weights (which is 6 FADC slices). Figure~\ref{fig:df:ratiofit} +shows 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\% lower than those obtained +from 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 @@ -518,5 +550,5 @@ 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. +single photo-electrons are separated from the pure electronics noise. \par @@ -542,15 +574,17 @@ \end{eqnarray} -where $\mu_0$ is the mean position of the pedestal peak and $\mu_1$ the mean position of the (assumed) +where $\mu_0$ denotes 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, +with the assumption that the digital filter finds the most upward fluctuating pulse out of several. Therefore, +$\mu_1$ is biased against higher values. The F-Factor is also systematically low (however with huge error bars), +which is also consistent +with the assumption that the spacing between $\mu_1$ and $\mu_0$ is artificially high. +Unfortunately, the error bars are too high for a ``calibration'' of the F-Factor. +\par +In conclusion, 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. +resolution is too poor. These limitations might be overcome if a higher sampling speed is used and the artificial +pulse shaping removed. We expect to improve this method considerably with the new 2\,GSamples/s~FADC readout of MAGIC. \begin{figure}[htp]