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trunk/MagicSoft/TDAS-Extractor/Pedestal.tex
r6559 r6622 6 6 can be completely described by the noise-autocorrelation matrix $\boldsymbol{B}$ 7 7 (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 fromthe signal extractor.8 where the square root of the diagonal elements give what is usually denoted as the ``Pedestal RMS''. 9 \par 10 11 By definition, $\boldsymbol{B}$ and thus the ``pedestal RMS'' 12 is independent of the signal extractor. 13 13 14 14 \subsection{Pedestal Fluctuations as Contribution to the Signal Fluctuations \label{sec:ffactor}} … … 31 31 \end{eqnarray} 32 32 33 Only in the case that the intrinsic extractor resolution $R$ at fixed background $BG$ does not depend on the signal 34 intensity\footnote{Theoretically, this is the case for the digital filter, eq.~\ref{eq:of_noise}.}, 33 35 $Var[Q]$ can be obtained from: 34 36 35 37 \begin{eqnarray} 36 Var(Q) &\approx& Var(\widehat{Q}) - Var(\widehat{Q} =0)38 Var(Q) &\approx& Var(\widehat{Q}) - Var(\widehat{Q})\,\vline_{\,Q=0} 37 39 \label{eq:rmssubtraction} 38 40 \end{eqnarray} 39 41 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 $R$ 59 60 by applying the signal extractor to a {\textit{\bf fixed window}} of pedestal events, where the 60 bias vanishes and measure $Var [\widehat{Q}=0]$.61 bias vanishes and measure $Var(\widehat{Q})\,\vline_{\,Q=0}$. 61 62 62 63 \subsection{Methods to Retrieve Bias and Mean-Squared Error} 63 64 64 65 In general, the extracted signal variance $R$ is different from the pedestal RMS. 65 It can not be obtained by applying the signal extractor to pedestal events, because of the66 (unknown) bias. 67 \par 68 In the case of the digital filter, $R$ is expected to be independent fromthe69 signal amplitude $S$ and depend sonly 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:66 It can be obtained by applying the signal extractor to pedestal events yielding the bias and 67 the resolution $R$. 68 \par 69 In the case of the digital filter, $R$ is expected to be independent of the 70 signal 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 76 In order to calculate the bias and Mean-squared error, we proceed in the following ways: 76 77 \begin{enumerate} 77 78 \item Determine $R$ by applying the signal extractor to a fixed window … … 79 80 levels of night sky background and the continuous light source, but no signal size 80 81 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 85 87 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 90 precision issues. All these systematic effects add an additional contribution to the true resolution proportional to the signal strength. 87 91 \end{enumerate} 88 92 … … 147 151 148 152 By 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 153 determine the parameter $R$ for the case of no signal ($Q = 0$)\footnote{% 154 In the case of 150 155 extractors using a fixed window (extractors nr. \#1 to \#22 151 156 in section~\ref{sec:algorithms}), the results are the same by construction 152 as calculating the pedestal RMS. 157 as calculating the RMS of the sum of a fixed number of FADC slice, traditionally 158 named ``pedestal RMS'' in MARS.}. 153 159 \par 154 160 In MARS, this functionality is implemented with a function-call to: \\ 155 161 156 {\textit{\bf MJPedestal::SetExtractionWithExtractorRndm()}} and/or\\162 {\textit{\bf MJPedestal::SetExtractionWithExtractorRndm()}} including \\ 157 163 {\textit{\bf MExtractPedestal::SetRandomCalculation()}}\\ 158 164 159 165 Besides fixing the global extraction window, additionally the following steps are undertaken 160 in order to assure that the bias vanishes:166 in order to assure an un-biased resolution. 161 167 162 168 \begin{description} 163 169 \item[\textit{MExtractTimeAndChargeSpline}:\xspace] The spline 164 maximum position -- which determines the exact extraction window -- is placed arbitrarily170 maximum position -- which determines the exact extraction window -- is placed 165 171 at a random place within the digitizing binning resolution of one central FADC slice. 166 172 \item[\textit{MExtractTimeAndChargeDigitalFilter}:\xspace] The second step timing 167 offset $\tau$ (eq.~\ref{eq:offsettau}) gets randomizedfor each event.173 offset $\tau$ (eq.~\ref{eq:offsettau}) is chosen randomly for each event. 168 174 \end{description} 169 175 … … 181 187 Figures~\ref{fig:amp:relmean} through~\ref{fig:df:relmean} 182 188 show 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.189 and for the different levels of (night-sky) background applied to 1000 pedestal events. 190 One can see that the bias vanishes to an accuracy of better than 2\% of a photo-electron 191 makefor the extractors which are used in this TDAS. 186 192 187 193 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%1 … … 218 224 Every entry corresponds to one 219 225 pixel.} 220 \label{fig: amp:relrms}226 \label{fig:int:relrms} 221 227 \end{figure} 222 228 … … 242 248 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 243 249 244 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. 250 Figures~\ref{fig:amp:relrms} through~\ref{fig:df:relrms} show the 251 differences in $R$ between the RMS of simply summing up the FADC slices over the extraction window 252 (in MARS called: ``Fundamental Pedestal RMS'') and 253 the 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 255 pixel of the camera. 248 256 The distributions have a negative mean in the case of the digital filter showing the 249 257 ``filter'' capacity of that algorithm. It ``filters out'' between 0.12 photo-electrons night sky … … 265 273 266 274 \par 267 Table~\ref{tab:bias} shows bias, resolution andmean-square error for all extractors using275 Table~\ref{tab:bias} shows the bias, the resolution and the mean-square error for all extractors using 268 276 a sliding window. In this sample, every extractor had the freedom to move 5 slices, 269 277 i.e. the global window size was fixed to five plus the extractor window size. This first line 270 278 shows 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 .279 value of 2.5 and 3 photo-electrons RMS for an extra-galactic and a galactic star-field, respectively. 272 280 \par 273 281 One can see that the bias $B$ typically decreases 274 with increasing window size (except for the digital filter), while the error $R$ increases with275 increasing window size . There is also a small difference between the obtained error on a fixed window276 extraction and the one obtained from a sliding window extraction in the case of the spline and digital282 with increasing window size, while the error $R$ increases with 283 increasing window size, except for the digital filter. There is also a small difference between the obtained error 284 on a fixed window extraction and the one obtained from a sliding window extraction in the case of the spline and digital 277 285 filter algorithms. 278 The mean-squared error has an optimum somewhere between: In the case of the286 The mean-squared error has an optimum somewhere in between: In the case of the 279 287 sliding 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\#29288 at 4 slices. The global winners is extractor~\#29 281 289 (digital filter with integration of 4 slices). All sliding window extractors -- except \#21 -- 282 290 have a smaller mean-square error than the resolution of the fixed window reference extractor. This means 283 291 that the global error of the sliding window extractors is smaller than the one of the fixed window extractors 284 292 even if the first have a bias. 285 293 \par 294 The important information for the image cleaning is the number of photo-electrons above which the probability for obtaining 295 a noise fluctuation is smaller than 0.3\% (3$\sigma$). We approximated that number with the formula: 296 297 \begin{equation} 298 N_{\mathrm{phe}}^{\mathrm{thres.}} \approx B + 3\cdot R 299 \end{equation} 300 301 Table~\ref{tab:bias} shows that most of the sliding window algorithms yield a smaller signal threshold than the fixed window ones, 302 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 303 for the galactic star-field is obtained by the digital filter fitting 4 FADC slices (extractor~\%29). 304 This is almost a factor 2 lower than the fixed window results. Also the spline integrating 1 FADC slice (extractor~\%24) yields almost 305 comparable results. 306 307 \begin{landscape} 308 %\rotatebox{90}{% 286 309 \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 322 Nr. & 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.) \\ 300 324 \hline 301 325 \hline 302 4 & Fixed Win. 8 & 1.2 & -- & 0.0 & 1.2 & 2.5 & -- & 0.0 & 2.5 & 3.0 & -- & 0.0 & 3.0 \\326 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 \\ 303 327 \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 \\ 329 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 \\ 330 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 \\ 331 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 \\ 332 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 \\ 309 333 \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\\334 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 \\ 335 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 \\ 336 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 \\ 337 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 \\ 338 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 \\ 315 339 \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}\\340 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 \\ 341 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 }\\ 318 342 \hline 319 343 \hline 320 344 \end{tabular} 321 }345 \vspace{1cm} 322 346 \caption{The statistical parameters bias, resolution and mean error for the sliding window 323 347 algorithm. The first line displays the resolution of the smallest existing robust fixed--window extractor … … 329 353 (MExtractTimeAndChargeDigitalFilter with an integration window size of 4 slices)} 330 354 \label{tab:bias} 355 } 331 356 \end{table} 357 %} 358 \end{landscape} 359 360 \clearpage 332 361 333 362 Figures~\ref{fig:sw:distped} through~\ref{fig:df4:distped} show the … … 419 448 \end{figure} 420 449 450 \clearpage 451 421 452 \subsection{ \label{sec:ped:singlephe} Single Photo-Electron Extraction with the Digital Filter} 422 453 423 Figure s~\ref{fig:df:sphespectrum} showspectra424 obtained with the digital filter applied on t wodifferent global search windows.454 Figure~\ref{fig:df:sphespectrum} shows spectra 455 obtained with the digital filter applied on three different global search windows. 425 456 One can clearly distinguish a pedestal peak (fitted to Gaussian with index 0) 426 457 and further, positive contributions. … … 434 465 Usually, the returned signal charge is then negative. 435 466 \item There is one photo-electron in the extraction window and the extractor finds it. 436 \item There are more than on photo-electronsin the extraction window, but separated by more than437 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 468 two FADC slices whereupon the extractor finds the one with the highest charge (upward fluctuation) of both. 438 469 \item The extractor finds an overlap of two or more photo-electrons. 439 470 \end{enumerate} … … 449 480 \par 450 481 451 Given a global extraction window of size $ WS$ and an average rate of photo-electrons from the night-sky482 Given a global extraction window of size $\mathrm{\it WS}$ and an average rate of photo-electrons from the night-sky 452 483 background $R$, 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: 454 485 455 486 \begin{equation} 456 P(k) = \frac{e^{-R\cdot WS} (R \cdot WS)^k}{k!}487 P(k) = \frac{e^{-R\cdot \mathrm{\it WS}} (R \cdot \mathrm{\it WS})^k}{k!} 457 488 \end{equation} 458 489 … … 460 491 461 492 \begin{equation} 462 P(0) = e^{-R\cdot WS}493 P(0) = e^{-R\cdot \mathrm{\it WS}} 463 494 \end{equation} 464 495 … … 466 497 467 498 \begin{equation} 468 P(>0) = 1 - e^{-R\cdot WS}499 P(>0) = 1 - e^{-R\cdot \mathrm{\it WS}} 469 500 \end{equation} 470 501 … … 499 530 \end{equation} 500 531 501 We tested this relation assuming that the fitted area underneath the pedestal peak Area$_0$ is532 We tested this relation assuming that the fitted area underneath the pedestal peak $\mathrm{\it Area}_0$ is 502 533 proportional 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 535 that the probability for a triple photo-electron to occur is negligible. Thus, one expects: 504 536 505 537 \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}} 507 539 \end{equation} 508 540 509 We estimated the effective window size $ WS$ as the sum of the range in which the digital filter541 We estimated the effective window size $\mathrm{\it WS}$ as the sum of the range in which the digital filter 510 542 amplitude 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). Figure s~\ref{fig:df:ratiofit}512 show the result for two different levels of night-sky background. The fitted rates deliver513 0.08 and 0.1 phes/ns, respectively. These rates are about 50\% too low compared to the resultsobtained514 inthe November 2004 test campaign. However, we should take into account that the method is at543 size of the window size of the weights (which is 6 FADC slices). Figure~\ref{fig:df:ratiofit} 544 shows the result for two different levels of night-sky background. The fitted rates deliver 545 0.08 and 0.1 phes/ns, respectively. These rates are about 50\% lower than those obtained 546 from the November 2004 test campaign. However, we should take into account that the method is at 515 547 the limit of distinguishing single photo-electrons. It may occur often that a single photo-electron 516 548 signal is too low in order to get recognized as such. We tried various pixels and found that … … 518 550 the same fitted rates. To conclude, one may say that there is consistency within the double-peak 519 551 structure of the pedestal spectrum found by the digital filter which can be explained by the fact that 520 single photo-electrons are found.552 single photo-electrons are separated from the pure electronics noise. 521 553 \par 522 554 … … 542 574 \end{eqnarray} 543 575 544 where $\mu_0$ is the mean position of the pedestal peak and $\mu_1$ the mean position of the (assumed)576 where $\mu_0$ denotes the mean position of the pedestal peak and $\mu_1$ the mean position of the (assumed) 545 577 single photo-electron peak. The obtained conversion factors are systematically lower than the ones 546 578 obtained 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, 579 with 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), 581 which is also consistent 582 with the assumption that the spacing between $\mu_1$ and $\mu_0$ is artificially high. 583 Unfortunately, the error bars are too high for a ``calibration'' of the F-Factor. 584 \par 585 In conclusion, the digital filter is at the edge of being able to see single photo-electrons, 553 586 however a single photo-electron calibration cannot yet be done with the current FADC system because the 554 resolution is too poor. 587 resolution is too poor. These limitations might be overcome if a higher sampling speed is used and the artificial 588 pulse shaping removed. We expect to improve this method considerably with the new 2\,GSamples/s~FADC readout of MAGIC. 555 589 556 590 \begin{figure}[htp]
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