Changeset 6622


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Timestamp:
02/19/05 12:14:36 (20 years ago)
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gaug
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  • trunk/MagicSoft/TDAS-Extractor/Pedestal.tex

    r6559 r6622  
    66can be completely described by the noise-autocorrelation matrix $\boldsymbol{B}$
    77(eq.~\ref{eq:autocorr}),
    8 where the diagonal elements give what is usually denoted as the ``Pedestal RMS''.
    9 \par
    10 
    11 By definition, the $\boldsymbol{B}$ and thus the ``pedestal RMS''
    12 is independent from the signal extractor.
     8where the square root of the diagonal elements give what is usually denoted as the ``Pedestal RMS''.
     9\par
     10
     11By definition, $\boldsymbol{B}$ and thus the ``pedestal RMS''
     12is independent of the signal extractor.
    1313
    1414\subsection{Pedestal Fluctuations as Contribution to the Signal Fluctuations \label{sec:ffactor}}
    … …  
    3131\end{eqnarray}
    3232
     33Only in the case that the intrinsic extractor resolution $R$ at fixed background $BG$ does not depend on the signal
     34intensity\footnote{Theoretically, this is the case for the digital filter, eq.~\ref{eq:of_noise}.},
    3335$Var[Q]$ can be obtained from:
    3436
    3537\begin{eqnarray}
    36 Var(Q) &\approx& Var(\widehat{Q}) - Var(\widehat{Q}=0)
     38Var(Q) &\approx& Var(\widehat{Q}) - Var(\widehat{Q})\,\vline_{\,Q=0}
    3739\label{eq:rmssubtraction}
    3840\end{eqnarray}
    3941
    40 In the last line of eq.~\ref{eq:rmssubtraction}, it is assumed that $R$ does not dependent
    41 on the signal height\footnote{%
    42 A way to check whether the right RMS has been subtracted is to make the
    43 ``Razmick''-plot
    44 
    45 \begin{equation}
    46     \frac{Var[\widehat{Q}]}{<\widehat{Q}>^2} \quad \textit{vs.} \quad \frac{1}{<\widehat{Q}>}
    47 \end{equation}
    48 
    49 This should give a straight line passing through the origin. The slope of
    50 the line is equal to
    51 
    52 \begin{equation}
    53     c * F^2
    54 \end{equation}
    55 
    56 where $c$ is the photon/ADC conversion factor  $<Q>/<m_{pe}>$.}
    57 (as is the case
    58 for the digital filter, eq.~\ref{eq:of_noise}). One can then retrieve $R$
     42%\footnote{%
     43%A way to check whether the right RMS has been subtracted is to make the
     44%``Razmick''-plot
     45%
     46%\begin{equation}
     47%    \frac{Var[\widehat{Q}]}{<\widehat{Q}>^2} \quad \textit{vs.} \quad \frac{1}{<\widehat{Q}>}
     48%\end{equation}
     49%
     50%This should give a straight line passing through the origin. The slope of
     51%the line is equal to
     52%
     53%\begin{equation}
     54%    c * F^2
     55%\end{equation}
     56%
     57%where $c$ is the photon/ADC conversion factor  $<Q>/<m_{pe}>$.}
     58
     59 One can then retrieve $R$
    5960by applying the signal extractor to a {\textit{\bf fixed window}} of pedestal events, where the
    60 bias vanishes and measure $Var[\widehat{Q}=0]$.
     61bias vanishes and measure $Var(\widehat{Q})\,\vline_{\,Q=0}$.
    6162
    6263\subsection{Methods to Retrieve Bias and Mean-Squared Error}
    6364
    6465In general, the extracted signal variance $R$ is different from the pedestal RMS.
    65 It cannot be obtained by applying the signal extractor to pedestal events, because of the
    66 (unknown) bias.
    67 \par
    68 In the case of the digital filter, $R$ is expected to be independent from the
    69 signal amplitude $S$ and depends only on the background $BG$ (eq.~\ref{eq:of_noise}).
    70 It can then be obtained from the calculation of the variance $Var[\widehat{Q}]$
    71 by applying the extractor to a fixed window of pure background events (``pedestal events'')
    72 and get rid of the bias in that way.
    73 \par
    74 
    75 In order to calculate bias and Mean-squared error, we proceeded in the following ways:
     66It can be obtained by applying the signal extractor to pedestal events yielding the bias and
     67the resolution $R$.
     68\par
     69In the case of the digital filter, $R$ is expected to be independent of the
     70signal amplitude $S$ and dependent only on the background $BG$ (eq.~\ref{eq:of_noise}).
     71%It can then be obtained from the calculation of the variance $Var[\widehat{Q}]$
     72%by applying the extractor to a fixed window of pure background events (``pedestal events'')
     73%and get rid of the bias in that way.
     74\par
     75
     76In order to calculate the bias and Mean-squared error, we proceed in the following ways:
    7677\begin{enumerate}
    7778\item Determine $R$ by applying the signal extractor to a fixed window
    … …  
    7980    levels of night sky background and the continuous light source, but no signal size
    8081    dependency can be retrieved with this method.
    81 \item Determine $B$ and $MSE$ from MC events with and without added noise.
    82     Assuming that $MSE$ and $B$ are negligible for the events without noise, one can
    83     get a dependency of both values from the size of the signal.
    84 \item Determine $MSE$ from the fitted error of $\widehat{S}$, which is possible for the
     82\item Determine $B$ and $MSE$ from MC events with added noise.
     83%    Assuming that $MSE$ and $B$ are negligible for the events without noise, one can
     84        With this method, one can get a dependency of both values from the size of the signal,
     85        although the MC might contain systematic differences with respect to the real data.
     86\item Determine $MSE$ from the error retrieved from the fit results of $\widehat{S}$, which is possible for the
    8587    fit and the digital filter (eq.~\ref{eq:of_noise}).
    86     In principle, all dependencies can be retrieved with this method.
     88    In principle, all dependencies can be retrieved with this method, although some systematic errors are not taken into account
     89   with this method: Deviations of the real pulse from the fitted one, errors in the noise auto-correlation matrix and numerical
     90precision issues. All these systematic effects add an additional contribution to the true resolution proportional to the signal strength.
    8791\end{enumerate}
    8892
    … …  
    147151
    148152By applying the signal extractor to a fixed window of pedestal events, we
    149 determine the parameter $R$ for the case of no signal ($Q = 0$). In the case of
     153determine the parameter $R$ for the case of no signal ($Q = 0$)\footnote{%
     154In the case of
    150155extractors using a fixed window (extractors nr. \#1 to \#22
    151156in section~\ref{sec:algorithms}), the results are the same by construction
    152 as calculating the pedestal RMS.
     157as calculating the RMS of the sum of a fixed number of FADC slice, traditionally
     158named ``pedestal RMS'' in MARS.}.
    153159\par
    154160In MARS, this functionality is implemented with a function-call to: \\
    155161
    156 {\textit{\bf MJPedestal::SetExtractionWithExtractorRndm()}} and/or \\
     162{\textit{\bf MJPedestal::SetExtractionWithExtractorRndm()}} including \\
    157163{\textit{\bf MExtractPedestal::SetRandomCalculation()}}\\
    158164
    159165Besides fixing the global extraction window, additionally the following steps are undertaken
    160 in order to assure that the bias vanishes:
     166in order to assure an un-biased resolution.
    161167
    162168\begin{description}
    163169\item[\textit{MExtractTimeAndChargeSpline}:\xspace] The spline
    164 maximum position -- which determines the exact extraction window -- is placed arbitrarily
     170maximum position -- which determines the exact extraction window -- is placed
    165171at a random place within the digitizing binning resolution of one central FADC slice.
    166172\item[\textit{MExtractTimeAndChargeDigitalFilter}:\xspace] The second step timing
    167 offset $\tau$ (eq.~\ref{eq:offsettau}) gets randomized for each event.
     173offset $\tau$ (eq.~\ref{eq:offsettau}) is chosen randomly for each event.
    168174\end{description}
    169175
    … …  
    181187Figures~\ref{fig:amp:relmean} through~\ref{fig:df:relmean}
    182188show the calculated biases obtained with this method for all pixels in the camera
    183 and for the different levels of (night-sky) background.
    184 One can see that the bias vanishes to an accuracy of better than 1\%
    185 for the extractors which are used in this TDAS.
     189and for the different levels of (night-sky) background applied to 1000 pedestal events.
     190One can see that the bias vanishes to an accuracy of better than 2\% of a photo-electron
     191makefor the extractors which are used in this TDAS.
    186192
    187193%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%1
    … …  
    218224Every entry corresponds to one
    219225pixel.}
    220 \label{fig:amp:relrms}
     226\label{fig:int:relrms}
    221227\end{figure}
    222228
    … …  
    242248%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    243249
    244 Figures~\ref{fig:amp:relrms} through~\ref{fig:amp:relrms} show the
    245 differences in $R$ between the calculated pedestal RMS and
    246 the one obtained by applying the extractor, converted to equivalent photo-electrons. One entry
    247 corresponds to one pixel of the camera.
     250Figures~\ref{fig:amp:relrms} through~\ref{fig:df:relrms} show the
     251differences in $R$ between the RMS of simply summing up the FADC slices over the extraction window
     252(in MARS called: ``Fundamental Pedestal RMS'') and
     253the one obtained by applying the extractor to the same extraction window
     254(in MARS called: ``Pedestal RMS with Extractor Rndm''). One entry of each histogram corresponds to one
     255pixel of the camera.
    248256The distributions have a negative mean in the case of the digital filter showing the
    249257``filter'' capacity of that algorithm. It ``filters out'' between 0.12 photo-electrons night sky
    … …  
    265273
    266274\par
    267 Table~\ref{tab:bias} shows bias, resolution and mean-square error for all extractors using
     275Table~\ref{tab:bias} shows the bias, the resolution and the mean-square error for all extractors using
    268276a sliding window. In this sample, every extractor had the freedom to move 5 slices,
    269277i.e. the global window size was fixed to five plus the extractor window size. This first line
    270278shows the resolution of the smallest existing robust fixed window algorithm in order to give the reference
    271 value of 2.5 and 3 photo-electrons RMS.
     279value of 2.5 and 3 photo-electrons RMS for an extra-galactic and a galactic star-field, respectively.
    272280\par
    273281One can see that the bias $B$ typically decreases
    274 with increasing window size (except for the digital filter), while the error $R$ increases with
    275 increasing window size. There is also a small difference between the obtained error on a fixed window
    276 extraction and the one obtained from a sliding window extraction in the case of the spline and digital
     282with increasing window size, while the error $R$ increases with
     283increasing window size, except for the digital filter. There is also a small difference between the obtained error
     284on a fixed window extraction and the one obtained from a sliding window extraction in the case of the spline and digital
    277285filter algorithms.
    278 The mean-squared error has an optimum somewhere between: In the case of the
     286The mean-squared error has an optimum somewhere in between: In the case of the
    279287sliding window and the spline at the lowest window size, in the case of the digital filter
    280 at 4 slices. The global winners are extractors \#25 (spline with integration of 1 slice) and \#29
     288at 4 slices. The global winners is extractor~\#29
    281289(digital filter with integration of 4 slices). All sliding window extractors -- except \#21 --
    282290have a smaller mean-square error than the resolution of the fixed window reference extractor. This means
    283291that the global error of the sliding window extractors is smaller than the one of the fixed window extractors
    284292even if the first have a bias.
    285 
     293\par
     294The important information for the image cleaning is the number of photo-electrons above which the probability for obtaining
     295a noise fluctuation is smaller than 0.3\% (3$\sigma$). We approximated that number with the formula:
     296
     297\begin{equation}
     298N_{\mathrm{phe}}^{\mathrm{thres.}} \approx B + 3\cdot R
     299\end{equation}
     300
     301Table~\ref{tab:bias} shows that most of the sliding window algorithms yield a smaller signal threshold than the fixed window ones,
     302although 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
     303for the galactic star-field is obtained by the digital filter fitting 4 FADC slices (extractor~\%29).
     304This is almost a factor 2 lower than the fixed window results. Also the spline integrating 1 FADC slice (extractor~\%24) yields almost
     305comparable results.
     306
     307\begin{landscape}
     308%\rotatebox{90}{%
    286309\begin{table}[htp]
    287 \centering
    288 \scriptsize{
    289 \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
    290 \hline
    291 \hline
    292 \multicolumn{14}{|c|}{Statistical Parameters for $S=0$} \\
    293 \hline
    294 \hline
    295  & & \multicolumn{4}{|c|}{Closed camera} & \multicolumn{4}{|c|}{Extra-galactic NSB}  & \multicolumn{4}{|c|}{Galactic NSB} \\
    296 \hline
    297 \hline
    298 Nr. & Name         &  $R$  & $R$ & $B$ & $\sqrt{MSE}$ &  $R$ &$R$  & $B$ & $\sqrt{MSE}$& $R$ &  $R$& $B$ & $\sqrt{MSE}$ \\
    299     &              &  (FW) & (SW)& (SW)& (SW) & (FW) &(SW) & (SW)& (SW) & (FW)&(SW) & (SW)&(SW) \\
     310\vspace{3cm}
     311\scriptsize{%
     312\centering
     313\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
     314\hline
     315\hline
     316\multicolumn{16}{|c|}{Statistical Parameters for $S=0$} \\
     317\hline
     318\hline
     319 & & \multicolumn{4}{|c|}{Closed camera} & \multicolumn{5}{|c|}{Extra-galactic NSB}  & \multicolumn{5}{|c|}{Galactic NSB} \\
     320\hline
     321\hline
     322Nr. & Name         &  $R$  & $R$ & $B$ & $\sqrt{MSE}$ &  $R$ &$R$  & $B$ & $\sqrt{MSE}$ & $B+3R$ & $R$ &  $R$& $B$ & $\sqrt{MSE}$ & $B+3R$  \\
     323    &              &  (FW) & (SW)& (SW)& (SW) & (FW) &(SW) & (SW)& (SW) & (99.7\% prob.) & (FW)&(SW) & (SW)&(SW) & (99.7\% prob.) \\
    300324\hline                                                     
    301325\hline                                                     
    302 4   & Fixed Win. 8  & 1.2  & --  & 0.0 & 1.2  & 2.5  & --  & 0.0 &  2.5 & 3.0 &  -- & 0.0 & 3.0 \\   
     3264   & 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 \\   
    303327\hline                                                     
    304 --  & 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 \\
    305 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 \\
    306 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 \\
    307 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 \\
    308 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 \\
     328--  & 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 \\
     32917  & 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 \\
     33018  & 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 \\
     33120  & 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 \\
     33221  & 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 \\
    309333\hline                                                                             
    310 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 \\
    311 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 \\
    312 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 \\
    313 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 \\
    314 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 \\
     33423  & 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 \\
     33524  & \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 \\
     33625  & 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 \\
     33726  & 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 \\
     33827  & 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 \\
    315339\hline                                                                             
    316 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 \\
    317 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} \\
     34028  & 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 \\
     34129  & \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 }\\
    318342\hline
    319343\hline
    320344\end{tabular}
    321 }
     345\vspace{1cm}
    322346\caption{The statistical parameters bias, resolution and mean error for the sliding window
    323347algorithm. The first line displays the resolution of the smallest existing robust fixed--window extractor
    … …  
    329353(MExtractTimeAndChargeDigitalFilter with an integration window size of 4 slices)}
    330354\label{tab:bias}
     355}
    331356\end{table}
     357%}
     358\end{landscape}
     359
     360\clearpage
    332361
    333362Figures~\ref{fig:sw:distped} through~\ref{fig:df4:distped} show the
    … …  
    419448\end{figure}
    420449
     450\clearpage
     451
    421452\subsection{ \label{sec:ped:singlephe} Single Photo-Electron Extraction with the Digital Filter}
    422453
    423 Figures~\ref{fig:df:sphespectrum} show spectra
    424 obtained with the digital filter applied on two different global search windows.
     454Figure~\ref{fig:df:sphespectrum} shows spectra
     455obtained with the digital filter applied on three different global search windows.
    425456One can clearly distinguish a pedestal peak (fitted to Gaussian with index 0)
    426457and further, positive contributions.
    … …  
    434465Usually, the returned signal charge is then negative.
    435466\item There is one photo-electron in the extraction window and the extractor finds it.
    436 \item There are more than on photo-electrons in the extraction window, but separated by more than
    437 two FADC slices whereupon the extractor finds the one with the highest charge (upward fluctuation).
     467\item There are more than one photo-electron in the extraction window, but separated by more than
     468two FADC slices whereupon the extractor finds the one with the highest charge (upward fluctuation) of both.
    438469\item The extractor finds an overlap of two or more photo-electrons.
    439470\end{enumerate}
    … …  
    449480\par
    450481
    451 Given a global extraction window of size $WS$ and an average rate of photo-electrons from the night-sky
     482Given a global extraction window of size $\mathrm{\it WS}$ and an average rate of photo-electrons from the night-sky
    452483background $R$, we will now calculate the probability for the extractor to find zero photo-electrons in the
    453 $WS$. The probability to find any number of $k$ photo-electrons can be written as:
     484$\mathrm{\it WS}$. The probability to find any number of $k$ photo-electrons can be written as:
    454485
    455486\begin{equation}
    456 P(k) = \frac{e^{-R\cdot WS} (R \cdot WS)^k}{k!}
     487P(k) = \frac{e^{-R\cdot \mathrm{\it WS}} (R \cdot \mathrm{\it WS})^k}{k!}
    457488\end{equation}
    458489
    … …  
    460491
    461492\begin{equation}
    462 P(0) = e^{-R\cdot WS}
     493P(0) = e^{-R\cdot \mathrm{\it WS}}
    463494\end{equation}
    464495
    … …  
    466497
    467498\begin{equation}
    468 P(>0) = 1 - e^{-R\cdot WS}
     499P(>0) = 1 - e^{-R\cdot \mathrm{\it WS}}
    469500\end{equation}
    470501
    … …  
    499530\end{equation}
    500531
    501 We tested this relation assuming that the fitted area underneath the pedestal peak Area$_0$ is
     532We tested this relation assuming that the fitted area underneath the pedestal peak $\mathrm{\it Area}_0$ is
    502533proportional to $P(0)$ and the sum of the fitted areas underneath the single photo-electron peak
    503 Area$_1$ and the double photo-electron peak Area$_2$ proportional to $P(>0)$. Thus, one expects:
     534$\mathrm{\it Area}_1$ and the double photo-electron peak $\mathrm{\it Area}_2$ proportional to $P(>0)$. We assumed
     535that the probability for a triple photo-electron to occur is negligible. Thus, one expects:
    504536
    505537\begin{equation}
    506 \mathrm{Area}_0 / (\mathrm{Area}_1 + \mathrm{Area}_2 ) = \frac{e^{-R\cdot WS}}{1-e^{-R\cdot WS}}
     538\mathrm{\it Area}_0 / (\mathrm{\it Area}_1 + \mathrm{\it Area}_2 ) = \frac{e^{-R\cdot WS}}{1-e^{-R\cdot WS}}
    507539\end{equation}
    508540
    509 We estimated the effective window size $WS$ as the sum of the range in which the digital filter
     541We estimated the effective window size $\mathrm{\it WS}$ as the sum of the range in which the digital filter
    510542amplitude weights are greater than 0.5 (1.5 FADC slices) and the global search window minus the
    511 size of the window size of the weights (which is 6 FADC slices). Figures~\ref{fig:df:ratiofit}
    512 show the result for two different levels of night-sky background. The fitted rates deliver
    513 0.08 and 0.1 phes/ns, respectively. These rates are about 50\% too low compared to the results obtained
    514 in the November 2004 test campaign. However, we should take into account that the method is at
     543size of the window size of the weights (which is 6 FADC slices). Figure~\ref{fig:df:ratiofit}
     544shows the result for two different levels of night-sky background. The fitted rates deliver
     5450.08 and 0.1 phes/ns, respectively. These rates are about 50\% lower than those obtained
     546from the November 2004 test campaign. However, we should take into account that the method is at
    515547the limit of distinguishing single photo-electrons. It may occur often that a single photo-electron
    516548signal is too low in order to get recognized as such. We tried various pixels and found that
    … …  
    518550the same fitted rates. To conclude, one may say that there is consistency within the double-peak
    519551structure of the pedestal spectrum found by the digital filter which can be explained by the fact that
    520 single photo-electrons are found.
     552single photo-electrons are separated from the pure electronics noise.
    521553\par
    522554
    … …  
    542574\end{eqnarray}
    543575
    544 where $\mu_0$ is the mean position of the pedestal peak and $\mu_1$ the mean position of the (assumed)
     576where $\mu_0$ denotes the mean position of the pedestal peak and $\mu_1$ the mean position of the (assumed)
    545577single photo-electron peak. The obtained conversion factors are systematically lower than the ones
    546578obtained from the standard calibration and decrease with increasing window size. This is consistent
    547 with the assumption that the digital filter finds the upward fluctuating pulse out of several. Therefore,
    548 $\mu_1$ is biased against higher values. The F-Factor is also systematically low, which is also consistent
    549 with the assumption that the spacing between $\mu_1$ and $\mu_0$ is artificially high. One can also see
    550 that the error bars are too high for a ``calibration'' of the F-Factor.
    551 \par
    552 In conclusion, one can say that the digital filter is at the edge of being able to see single photo-electrons,
     579with the assumption that the digital filter finds the most upward fluctuating pulse out of several. Therefore,
     580$\mu_1$ is biased against higher values. The F-Factor is also systematically low (however with huge error bars),
     581which is also consistent
     582with the assumption that the spacing between $\mu_1$ and $\mu_0$ is artificially high.
     583Unfortunately, the error bars are too high for a ``calibration'' of the F-Factor.
     584\par
     585In conclusion, the digital filter is at the edge of being able to see single photo-electrons,
    553586however a single photo-electron calibration cannot yet be done with the current FADC system because the
    554 resolution is too poor.
     587resolution is too poor. These limitations might be overcome if a higher sampling speed is used and the artificial
     588pulse shaping removed. We expect to improve this method considerably with the new 2\,GSamples/s~FADC readout of MAGIC.
    555589
    556590\begin{figure}[htp]
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