Index: trunk/MagicSoft/TDAS-Extractor/Calibration.tex
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+\section{Calibration \label{sec:calibration}}
+
+In this section, we describe the tests performed using light pulses of different colour, 
+pulse shapes and intensities with the MAGIC LED Calibration Pulser Box \cite{hardware-manual}. 
+\par
+The LED pulser system is able to provide fast light pulses of 3--4\,ns FWHM 
+with intensities ranging from 3--4 to more than 500 photo-electrons in one inner photo-multiplier of the 
+camera. These pulses can be produced in three colours {\textit {\bf green, blue}} and 
+{\textit{\bf UV}}.
+
+\begin{table}[htp]
+\centering
+\begin{tabular}{|c|c|c|c|c|c|c|}
+\hline
+\hline
+\multicolumn{7}{|c|}{The possible pulsed light colours} \\
+\hline
+\hline
+Colour &  Wavelength & Spectral Width & Min. Nr. &  Max. Nr. & Secondary & FWHM \\
+      & [nm]         & [nm]           &  Phe's   &  Phe's    & Pulses  &  Pulse [ns]\\
+\hline
+Green &  520      & 40      & 6          &  120      & yes  & 3--4  \\
+\hline
+Blue &  460       & 30      & 6          &  500      & yes  & 3--4 \\
+\hline
+UV   &  375       & 12      & 3          &  50       & no   & 2--3 \\ 
+\hline
+\hline
+\end{tabular}
+\caption{The pulser colours available from the calibration system}
+\label{tab:pulsercolours}
+\end{table}
+
+Table~\ref{tab:pulsercolours} lists the available colours and intensities and 
+figures~\ref{fig:pulseexample1leduv} and~\ref{fig:pulseexample23ledblue} show exemplary pulses 
+as registered by the FADCs.
+Whereas the UV-pulse is very stable, the green and blue pulses show sometimes smaller secondary 
+pulses after about 10--40\,ns from the main pulse.
+One can see that the very stable UV-pulses are unfortunately only available in such intensities as to 
+not saturate the high-gain readout channel. However, the brightest combination of light pulses easily 
+saturates all channels in the camera, but does not reach a saturation of the low-gain readout.
+\par
+Our tests can be classified into three subsections:
+
+\begin{enumerate}
+\item Un-calibrated pixels and events: These tests measure the percentage of failures of the extractor 
+resulting either in a pixel declared as un-calibrated or in an event which produces a signal ouside 
+of the expected Gaussian distribution.
+\item Number of photo-electrons: These tests measure the reconstructed numbers of photo-electrons, their 
+spread over the camera and the ratio of the obtained mean values for outer and inner pixels, respectively.
+\item Linearity tests: These tests measure the linearity of the extractor with respect to pulses of 
+different intensity and colour.
+\item Time resolution: These tests show the time resolution and stability obtained with different 
+intensities and colours.
+\end{enumerate}
+
+\begin{figure}[htp]
+\centering
+\includegraphics[width=0.48\linewidth]{1LedUV_Pulse_Inner.eps}
+\includegraphics[width=0.48\linewidth]{1LedUV_Pulse_Outer.eps}
+\caption{Example of a calibration pulse from the lowest available intensity (1\,Led UV). 
+The left plot shows the signal obtained in an inner pixel, the right one the signal in an outer pixel.
+Note that the pulse height fluctuates much more than suggested from these pictures. Especially, a 
+zero-pulse is also possible.}
+\label{fig:pulseexample1leduv}
+\end{figure}
+
+\begin{figure}[htp]
+\centering
+\includegraphics[width=0.48\linewidth]{23LedsBlue_Pulse_Inner.eps}
+\includegraphics[width=0.48\linewidth]{23LedsBlue_Pulse_Outer.eps}
+\caption{Example of a calibration pulse from the highest available mono-chromatic intensity (23\,Leds Blue). 
+The left plot shows the signal obtained in an inner pixel, the right one the signal in an outer pixel.
+One the left side of both plots, the (saturated) high-gain channel is visible, 
+on the right side from FADC slice 18 on, 
+the delayed low-gain 
+pulse appears. Note that in the left plot, there is a secondary pulses visible in the tail of the 
+high-gain pulse. }
+\label{fig:pulseexample23ledblue}
+\end{figure}
+
+We used data taken on the 7$^{th}$ of June, 2004 with different pulser LED combinations, each taken with 
+16384 events. The corresponding MAGIC data run numbers range from nr. 31741 to 31772. These data was taken 
+before the latest camera repair access which resulted in a replacement of about 2\% of the pixels known to be 
+mal-functionning at that time.
+There is thus a lower limit to the number of un-calibrated pixels of about 1.5--2\% of known 
+mal-functionning photo-multipliers.
+\par
+Although we had looked at and tested all colour and extractor combinations resulting from these data, 
+we refrain ourselves to show here only exemplary behaviour and results of extractors. 
+All plots, including those which are not displayed in this TDAS, can be retrieved from the following 
+locations:
+
+\begin{verbatim}
+http://www.magic.ifae.es/~markus/pheplots/
+http://www.magic.ifae.es/~markus/timeplots/
+\end{verbatim}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\subsection{Un-Calibrated Pixels and Events}
+
+The MAGIC calibration software incorporates a series of checks to sort out mal-functionning pixels. 
+Except for the software bug searching criteria, the following exclusion criteria can apply:
+
+\begin{enumerate}
+\item The reconstructed mean signal is less than 2.5 times the extractor resolution $R$ from zero. 
+(2.5 Pedestal RMS in the case of the simple fixed window extractors, see section~\ref{sec:pedestals}). 
+This criterium essentially cuts out 
+dead pixels.
+\item The reconstructed mean signal error is smaller than its value. This criterium cuts out 
+signal distributions which fluctuate so much that their RMS is bigger than its mean value. This 
+criterium cuts out ``ringing'' pixels or mal-functionning extractors. 
+\item The reconstructed mean number of photo-electrons lies 4.5 sigma outside 
+the distribution of photo-electrons obtained with the inner or outer pixels in the camera, respectively. 
+This criterium cuts out pixels channels with apparently deviating (hardware) behaviour compared to 
+the rest of the camera readout\footnote{This criteria is not applied any more in the standard analysis, 
+although here, we kept using it}.
+\item All pixels with reconstructed negative mean signal or with a 
+mean numbers of photo-electrons smaller than one. Pixels with a negative pedestal RMS subtracted 
+sigma occur, especially when stars are focussed onto that pixel during the pedestal taking (resulting 
+in a large pedestal RMS), but have moved to another pixel during the calibration run. In this case, the 
+number of photo-electrons would result artificially negative. If these 
+channels do not show any other deviating behaviour, their number of photo-electrons gets replaced by the 
+mean number of photo-electrons in the camera, and the channel is further calibrated as normal.
+\end{enumerate}
+
+Moreover, the number of events are counted which have been reconstructed outside a 5 sigma region 
+from the mean signal. These events are called ``outliers''. Figure~\ref{fig:outlier} shows a typical 
+outlier obtained with the digital filter applied to a low-gain signal.
+
+\begin{figure}[htp]
+\centering
+\includegraphics[width=0.95\linewidth]{Outlier.eps}
+\caption{Example of an event classified as ``un-calibrated''. The histogram has been obtained 
+using the digital filter (extractor \#32) applied to a high-intensity blue pulse (run 31772). 
+The event marked as ``outlier'' clearly has been mis-reconstructed. It lies outside the 5 sigma 
+region from the fitted mean.}
+\label{fig:outlier}
+\end{figure}
+
+The following figures~\ref{fig:unsuited:5ledsuv},~\ref{fig:unsuited:1leduv},~\ref{fig:unsuited:2ledsgreen}
+and~\ref{fig:unsuited:23ledsblue} show the resulting numbers of un-calibrated pixels and events for 
+different colours and intensities. Because there is a strong anti-correlation between the number of 
+excluded channels and the number of outliers per event, we have chosen to show these numbers together. 
+
+\par
+
+\begin{figure}[htp]
+\centering
+\includegraphics[height=0.95\textheight]{UnsuitVsExtractor-5LedsUV-Colour-13.eps}
+\caption{Uncalibrated pixels and pixels outside of the Gaussian distribution for a typical calibration  
+pulse of UV-light which does not saturate the high-gain readout.}
+\label{fig:unsuited:5ledsuv}
+\end{figure}
+
+\begin{figure}[htp]
+\centering
+\includegraphics[height=0.95\textheight]{UnsuitVsExtractor-1LedUV-Colour-04.eps}
+\caption{Uncalibrated pixels and pixels outside of the Gaussian distribution for a very low 
+intensity pulse.}
+\label{fig:unsuited:1leduv}
+\end{figure}
+
+\begin{figure}[htp]
+\centering
+\includegraphics[height=0.95\textheight]{UnsuitVsExtractor-2LedsGreen-Colour-02.eps}
+\caption{Uncalibrated pixels and pixels outside of the Gaussian distribution for a typical green pulse.}
+\label{fig:unsuited:2ledsgreen}
+\end{figure}
+
+\begin{figure}[htp]
+\centering
+\includegraphics[height=0.95\textheight]{UnsuitVsExtractor-23LedsBlue-Colour-00.eps}
+\caption{Uncalibrated pixels and pixels outside of the Gaussian distribution for a high-intensity blue pulse.}
+\label{fig:unsuited:23ledsblue}
+\end{figure}
+
+One can see that in general, big extraction windows raise the 
+number of un-calibrated pixels and are thus less stable. Especially for the very low-intensity 
+\textit{\bf 1Led\,UV}-pulse, the big extraction windows summing 8 or more slices, cannot calibrate more 
+than 50\% 
+of the inner pixels (fig.~\ref{fig:unsuited:1leduv}). This is an expected behavior since big windows 
+add up more noise which in turn makes the search for the small signal more difficult.
+\par
+\ldots {\bf WHICH EXTRACTOR HAS THE LEAST NUMBER OF EXCLUDED PIXELS ???}
+\par
+In general, one can also find that all ``sliding window''-algorithms (extractors \#17-32) discard 
+less pixels than the ``fixed window''-ones (extractors \#1--16). The digital filter with 
+the correct weights (extractors \#30-33) discards the least number of pixels and is also robust against 
+slight modifications of its weights (extractors \#28--30). The robustness gets lost when the high-gain and 
+low-gain weights are inverted (extractors \#31--39, see fig.~\ref{fig:unsuited:23ledsblue}). 
+\par
+Also the ``spline'' algorithms on small  
+windows (extractors \#23--25) discard less pixels than the previous extractors.
+\par
+It seems also that the spline algorithm extracting the amplitude of the signal produces an over-proportional
+number of excluded events in the low-gain. The same, however in a less significant manner, holds for 
+the digital filter with high-low-gain inverted weights. The limit of stability with respect to 
+changes  in the pulse form seems to be reached, there.
+\par
+Concerning the numbers of outliers, one can conclude that in general, the numbers are very low never exceeding
+0.1\% except for the ampltiude-extracting spline which seems to mis-reconstruct a certain type of events.
+\par
+In conclusion, already this first test excludes all extractors with too big window sizes because 
+they are not able to extract cleanly small signals produced by about 4 photo-electrons. Moreover, 
+some extractors do not reproduce the signals as expected in the low-gain. 
+The excluded extractors are:
+\begin{itemize}
+\item: MExtractFixedWindow Nr. 3--5
+\item: MExtractFixedWindowSpline Nr. 6--11 (all)
+\item: MExtractFixedWindowPeakSearch Nr. 14--16
+\item: MExtractTimeAndChargeSlidingWindow Nr. 21--22
+\item: MExtractTimeAndChargeSpline Nr. 23 and 27
+\end{itemize}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\subsection{Number of Photo-Electrons \label{sec:photo-electrons}}
+
+Assuming that the readout chain adds only negligible noise to the one 
+introduced by the photo-multiplier itself, one can make the assumption that the variance of the 
+true (non-extracted) signal $ST$ is the amplified Poisson variance of the number of photo-electrons, 
+multiplied with the excess noise of the photo-multiplier which itself is 
+characterized by the excess-noise factor $F$.
+
+\begin{equation}
+Var(ST) = F^2 \cdot Var(N_{phe}) \cdot \frac{<ST>^2}{<N_{phe}>^2}
+\label{eq:excessnoise}
+\end{equation}
+
+After introducing the effect of the night-sky background (eq.~\ref{eq:rmssubtraction}) 
+in formula~\ref{eq:excessnoise} and assuming that the variance of the number of photo-electrons is equal 
+to the mean number of photo-electrons (because of the Poisson distribution), 
+one obtains an expression to retrieve the mean number of photo-electrons  impinging on the pixel from the 
+mean extracted signal $<SE>$, its variance $Var(SE)$ and the RMS of the extracted signal obtained from 
+pure pedestal runs $R$ (see section~\ref{sec:determiner}):
+
+\begin{equation}
+<N_{phe}> \approx F^2 \cdot \frac{<SE>^2}{Var(SE) - R^2}
+\label{eq:pheffactor}
+\end{equation}
+
+In theory, eq.~\ref{eq:pheffactor} must not depend on the extractor! Effectively, we will use it to test the 
+quality of our extractors by requiring that a valid extractor yields the same number of photo-electrons 
+for all pixels of a same type and does not deviate from the number obtained with other extractors. 
+As the camera is flat-fielded, but the number of photo-electrons impinging on an inner and an outer pixel is 
+different, we also use the ratio of the mean numbers of photo-electrons from the outer pixels to the one 
+obtained from the inner pixels as a test variable. In the ideal case, it should always yield its central 
+value of about 2.6$\pm$0.1~\cite{michele-diploma}.
+\par
+In our case, there is an additional complication due to the fact that the green and blue coloured light pulses 
+show secondary pulses which destroy the Poisson behaviour of the number of photo-electrons. We will
+have to split our sample of extractors into those being affected by the secondary pulses and those 
+being immune to this effect. 
+\par
+Figures~\ref{fig:phe:5ledsuv},~\ref{fig:phe:1leduv},~\ref{fig:phe:2ledsgreen}~and~\ref{fig:phe:23ledsblue} show 
+some of the obtained results. Although one can see a rather good stability for the standard 
+{\textit{\bf 5\,Leds\,UV}}\ pulse, except for the extractors {\textit{\bf MExtractFixedWindowPeakSearch}}, initialized 
+with an extraction window of 2 slices and  {\textit{\bf MExtractTimeAndChargeDigitalFilter}}, initialized with 
+an extraction window of 4 slices (extractor \#29).
+\par
+There is a considerable difference for all shown non-standard pulses. Especially the pulses from green 
+and blue LEDs 
+show a clear dependency  of the number of photo-electrons on the extraction window. Only the largest 
+extraction windows seem to catch the entire range of (jittering) secondary pulses and get the ratio 
+of outer vs. inner pixels right. However, they (obviously) over-estimate the number of photo-electrons 
+in the primary pulse.
+\par
+The strongest discrepancy is observed in the low-gain extraction (fig.~\ref{fig:phe:23ledsblue}) where all 
+fixed window extractors with too small extraction windows fail to reconstruct the correct numbers. 
+This has to do with the fact that 
+the fixed window extractors fail to do catch a significant part of the (larger) pulse because of the 
+1~FADC slice event-to-event jitter.
+
+
+\begin{figure}[htp]
+\centering
+\includegraphics[height=0.92\textheight]{PheVsExtractor-5LedsUV-Colour-13.eps}
+\caption{Number of photo-electrons from a typical, not saturating calibration pulse of colour UV, 
+reconstructed with each of the tested signal extractors. 
+The first plots shows the number of photo-electrons obtained for the inner pixels, the second one 
+for the outer pixels and the third shows the ratio of the mean number of photo-electrons for the 
+outer pixels divided by the mean number of photo-electrons for the inner pixels. Points 
+denote the mean of all not-excluded pixels, the error bars their RMS.}
+\label{fig:phe:5ledsuv}
+\end{figure}
+
+\begin{figure}[htp]
+\centering
+\includegraphics[height=0.92\textheight]{PheVsExtractor-1LedUV-Colour-04.eps}
+\caption{Number of photo-electrons from a typical, very low-intensity calibration pulse of colour UV, 
+reconstructed with each of the tested signal extractors. 
+The first plots shows the number of photo-electrons obtained for the inner pixels, the second one 
+for the outer pixels and the third shows the ratio of the mean number of photo-electrons for the 
+outer pixels divided by the mean number of photo-electrons for the inner pixels. Points 
+denote the mean of all not-excluded pixels, the error bars their RMS.}
+\label{fig:phe:1leduv}
+\end{figure}
+
+\begin{figure}[htp]
+\centering
+\includegraphics[height=0.92\textheight]{PheVsExtractor-2LedsGreen-Colour-02.eps}
+\caption{Number of photo-electrons from a typical, not saturating calibration pulse of colour green, 
+reconstructed with each of the tested signal extractors. 
+The first plots shows the number of photo-electrons obtained for the inner pixels, the second one 
+for the outer pixels and the third shows the ratio of the mean number of photo-electrons for the 
+outer pixels divided by the mean number of photo-electrons for the inner pixels. Points 
+denote the mean of all not-excluded pixels, the error bars their RMS.}
+\label{fig:phe:2ledsgreen}
+\end{figure}
+
+
+\begin{figure}[htp]
+\centering
+\includegraphics[height=0.92\textheight]{PheVsExtractor-23LedsBlue-Colour-00.eps}
+\caption{Number of photo-electrons from a typical, high-gain saturating calibration pulse of colour blue, 
+reconstructed with each of the tested signal extractors. 
+The first plots shows the number of photo-electrons obtained for the inner pixels, the second one 
+for the outer pixels and the third shows the ratio of the mean number of photo-electrons for the 
+outer pixels divided by the mean number of photo-electrons for the inner pixels. Points 
+denote the mean of all not-excluded pixels, the error bars their RMS.}
+\label{fig:phe:23ledsblue}
+\end{figure}
+
+One can see that all extractors using a large window belong to the class of extractors being affected 
+by the secondary pulses, except for the digital filter. The only exception to this rule is the digital filter 
+which - despite of its 6 slices extraction window - seems to filter out all the secondary pulses. 
+\par
+The extractor {\textit{\bf MExtractFixedWindowPeakSearch}} at low extraction windows apparently yields chronically low 
+numbers of photo-electrons. This is due to the fact that the decision to fix the extraction window is 
+made sometimes by an inner pixel and sometimes by an outer one since the camera is flat-fielded and the 
+pixel carrying the largest non-saturated peak-search window is more or less found by a random signal 
+fluctuation. However, inner and outer pixels have a systematic offset of about 0.5 to 1 FADC slices. 
+Thus, the extraction fluctuates artificially for one given channel which results in a systematically 
+large variance and thus in a systematically low reconstructed number of photo-electrons. This test thus 
+excludes the extractors \#11--13.
+\par
+Moreover, one can see that the extractors applying a small fixed window do not get the ratio of 
+photo-electrons correctly between outer to inner pixels for the green and blue pulses. 
+\par
+The extractor {\textit{\bf MExtractTimeAndChargeDigitalFilter}} seems to be stable against modifications in the 
+exact form of the weights in the high-gain readout channel since all applied weights yield about 
+the same number of photo-electrons and the same ratio of outer vs. inner pixels. This statement does not 
+hold any more for the low-gain, as can be seen in figure~\ref{fig:phe:23ledsblue}. There, the application 
+of high-gain weights to the low-gain signal (extractors \#34--39) produces a too low number of photo-electrons
+and also a too low ratio of outer vs. inner pixels.
+\par
+All sliding window and spline algorithms yield a stable ratio of outer vs. inner pixels in the low-gain, 
+however the effect of raising the number of photo-electrons with the extraction window is very pronounced. 
+Note that in figure~\ref{fig:phe:23ledsblue}, the number of photo-electrons rises by about a factor 1.4, 
+which is slightly higher than in the case of the high-gain channel (figure~\ref{fig:phe:2ledsgreen}). 
+\par
+Concluding, there is no fixed window extractor yielding the correct number of photo-electrons 
+for the low-gain, except for the largest extraction window of 8 and 10 low-gain slices. 
+Either the number of photo-electrons itself is wrong or the ratio of outer vs. inner pixels is 
+not correct. All sliding window algorithms seem to reproduce the correct numbers if one takes into 
+account the after-pulse behaviour of the light pulser itself. The digital filter seems to be 
+unstable against exchanging the pulse form to match the slimmer high-gain pulses, though.
+
+\par
+\ldots {\textit{\bf EXCLUDED : CW4, UV4 No stability High-gain vs. LoGain}}
+\par
+
+\subsection{Linearity Tests}
+
+In this section, we test the lineary of the extractors. As the photo-multiplier and the subsequent 
+optical transmission devices~\cite{david} is a linear device over a 
+wide dynamic range, the number of photo-electrons per charge has to remain constant over the tested 
+linearity region. We will show here only examples of extractors which were not already excluded in the 
+previous section.
+\par
+A first test concerns the stability of the conversion factor: mean number of averaged photo-electrons 
+per FADC counts over the 
+tested intensity region. A much more detailed investigation on the linearity will be shwon in a 
+separate TDAS~\cite{tdas-calibration}.
+
+
+\begin{figure}[htp]
+\centering
+\includegraphics[width=0.95\linewidth]{PheVsCharge-3.eps}
+\caption{Example of a the development of the conversion factor FADC counts to photo-electrons for two 
+exemplary inner pixels (upper plots) and two exemplary outer ones (lower plots). 
+A fixed window extractor on a window size of 6 high-gain and 6 low-gain slices has been used (extractor \#3). }
+\label{fig:linear:phevscharge3}
+\end{figure}
+
+\begin{figure}[htp]
+\centering
+\includegraphics[width=0.95\linewidth]{PheVsCharge-8.eps}
+\caption{Example of a the development of the conversion factor FADC counts to photo-electrons for two 
+exemplary inner pixels (upper plots) and two exemplary outer ones (lower plots). 
+A fixed window spline extractor on a window size of 6 high-gain and 6 low-gain slices has been used 
+(extractor \#8). }
+\label{fig:linear:phevscharge8}
+\end{figure}
+
+\begin{figure}[htp]
+\centering
+\includegraphics[width=0.95\linewidth]{PheVsCharge-14.eps}
+\caption{Example of a the development of the conversion factor FADC counts to photo-electrons for two 
+exemplary inner pixels (upper plots) and two exemplary outer ones (lower plots). 
+A fixed window peak search extractor on a window size of 6 high-gain and 6 low-gain slices has been used 
+(extractor \#14). }
+\label{fig:linear:phevscharge14}
+\end{figure}
+
+\begin{figure}[htp]
+\centering
+\includegraphics[width=0.95\linewidth]{PheVsCharge-20.eps}
+\caption{Example of a the development of the conversion factor FADC counts to photo-electrons for two 
+exemplary inner pixels (upper plots) and two exemplary outer ones (lower plots). 
+A sliding window extractor on a window size of 6 high-gain and 6 low-gain slices has been used
+ (extractor \#20). }
+\label{fig:linear:phevscharge20}
+\end{figure}
+
+\begin{figure}[htp]
+\centering
+\includegraphics[width=0.95\linewidth]{PheVsCharge-25.eps}
+\caption{Example of a the development of the conversion factor FADC counts to photo-electrons for two 
+exemplary inner pixels (upper plots) and two exemplary outer ones (lower plots). 
+An integrating spline extractor on a sliding window and a window size of 2 high-gain and 3 low-gain slices 
+has been used (extractor \#25). }
+\label{fig:linear:phevscharge25}
+\end{figure}
+
+\begin{figure}[htp]
+\centering
+\includegraphics[width=0.95\linewidth]{PheVsCharge-27.eps}
+\caption{Example of a the development of the conversion factor FADC counts to photo-electrons for two 
+exemplary inner pixels (upper plots) and two exemplary outer ones (lower plots). 
+An integrating spline extractor on a sliding window and a window size of 6 high-gain and 7 low-gain slices 
+has been used (extractor \#27). }
+\label{fig:linear:phevscharge27}
+\end{figure}
+
+\begin{figure}[htp]
+\centering
+\includegraphics[width=0.95\linewidth]{PheVsCharge-30.eps}
+\caption{Example of a the development of the conversion factor FADC counts to photo-electrons for two 
+exemplary inner pixels (upper plots) and two exemplary outer ones (lower plots). 
+A digital filter extractor on a window size of 6 high-gain and 6 low-gain slices has been used
+with UV-weights (extractor \#30). }
+\label{fig:linear:phevscharge30}
+\end{figure}
+
+\begin{figure}[htp]
+\centering
+\includegraphics[width=0.95\linewidth]{PheVsCharge-31.eps}
+\caption{Example of a the development of the conversion factor FADC counts to photo-electrons for two 
+exemplary inner pixels (upper plots) and two exemplary outer ones (lower plots). 
+A digital filter extractor on a window size of 4 high-gain and 4 low-gain slices has been used
+ (extractor \#31). }
+\label{fig:linear:phevscharge31}
+\end{figure}
+
+
+
+\subsection{Time Resolution}
+
+The extractors \#17--32 are able to extract also the arrival time of each pulse. The calibration
+delivers a fast-rising pulse, uniform over the camera in signal size and time. 
+We estimate the time-uniformity to better 
+than 300\,ps, a limit due to the different travel times of the light between inner and outer parts of the
+camera. Since the calibraion does not permit a precise measurement of the absolute arrival time, we measure 
+the relative arrival time for every channel with respect to a reference channel (usually pixel Nr.\,1):
+
+\begin{equation}
+\delta t_i = t_i - t_1
+\end{equation}
+
+where $t_i$ denotes the reconstructed arrival time of pixel number $i$ and $t_1$ the reconstructed 
+arrival time of the reference pixel nr. 1 (software numbering). For one calibration run, one can then fill 
+histograms of $\delta t_i$ for each pixel and fit them to the expected Gaussian distribution. The fits 
+yield a mean $\mu(\delta t_i)$, comparable to 
+systematic offsets in the signal delay, and a sigma $\sigma(\delta t_i)$, a measure of the 
+combined time resolutions of pixel $i$ and pixel 1. Assuming that the PMTs and readout channels are 
+of a same kind, we obtain an approximate absolute time resolution of pixel $i$ by:
+
+\begin{equation}
+t^{res}_i \approx \sigma(\delta t_i)/sqrt(2)
+\end{equation}
+
+Figures~\ref{fig:reltimesinner10leduv} and~\ref{fig:reltimesouter10leduv} show distributions of $\delta t_i$ 
+for 
+one typical inner pixel and one typical outer pixel and a non-saturating calibration pulse of UV-light, 
+obtained with three different extractors. One can see that the first two yield a Gaussian distribution 
+to a good approximation, whereas the third extractor shows a three-peak structure and cannot be fitted. 
+We discarded that particular extractor for this reason.
+
+\begin{figure}[htp]
+\centering
+\includegraphics[width=0.3\linewidth]{RelArrTime_Pixel97_10LedUV_Extractor32.eps}
+\includegraphics[width=0.32\linewidth]{RelArrTime_Pixel97_10LedUV_Extractor23.eps}
+\includegraphics[width=0.32\linewidth]{RelArrTime_Pixel97_10LedUV_Extractor17.eps}
+\caption{Example of a two distributions of relative arrival times of an inner pixel with respect to 
+the arrival time of the reference pixel Nr. 1. The left plot shows the result using the digital filter
+ (extractor \#32), the central plot shows the result obtained with the half-maximum of the spline and the 
+right plot the result of the sliding window with a window size of 2 FADC slices (extractor \#17). A 
+medium sized UV-pulse (10Leds UV) has been used which does not saturate the high-gain readout channel.}
+\label{fig:reltimesinner10leduv}
+\end{figure}
+
+\begin{figure}[htp]
+\centering
+\includegraphics[width=0.31\linewidth]{RelArrTime_Pixel400_10LedUV_Extractor32.eps}
+\includegraphics[width=0.31\linewidth]{RelArrTime_Pixel400_10LedUV_Extractor23.eps}
+\includegraphics[width=0.31\linewidth]{RelArrTime_Pixel400_10LedUV_Extractor17.eps}
+\caption{Example of a two distributions of relative arrival times of an outer pixel with respect to 
+the arrival time of the reference pixel Nr. 1. The left plot shows the result using the digital filter
+ (extractor \#32), the central plot shows the result obtained with the half-maximum of the spline and the 
+right plot the result of the sliding window with a window size of 2 FADC slices (extractor \#17). A 
+medium sized UV-pulse (10Leds UV) has been used which does not saturate the high-gain readout channel.}
+\label{fig:reltimesouter10leduv}
+\end{figure}
+
+Figures~\ref{fig:reltimesinner10ledsblue} and~\ref{fig:reltimesouter10ledsblue} show distributions of 
+$<\delta t_i>$ for 
+one typical inner and one typical outer pixel and a high-gain-saturating calibration pulse of blue-light, 
+obtained with two different extractors. One can see that the first (extractor \#23) yields a Gaussian 
+distribution to a good approximation. 
+
+\begin{figure}[htp]
+\centering
+\includegraphics[width=0.31\linewidth]{RelArrTime_Pixel97_10LedBlue_Extractor23.eps}
+\includegraphics[width=0.31\linewidth]{RelArrTime_Pixel97_10LedBlue_Extractor32.eps}
+\caption{Example of a two distributions of relative arrival times of an inner pixel with respect to 
+the arrival time of the reference pixel Nr. 1. The left plot shows the result using the half-maximum of the spline (extractor \#23), the right plot shows the result obtained with the digital filter
+(extractor \#32). A 
+medium sized Blue-pulse (10Leds Blue) has been used which saturates the high-gain readout channel.}
+\label{fig:reltimesinner10ledsblue}
+\end{figure}
+
+
+
+\begin{figure}[htp]
+\centering
+\includegraphics[width=0.31\linewidth]{RelArrTime_Pixel400_10LedBlue_Extractor23.eps}
+\includegraphics[width=0.31\linewidth]{RelArrTime_Pixel400_10LedBlue_Extractor32.eps}
+\caption{Example of a two distributions of relative arrival times of an outer pixel with respect to 
+the arrival time of the reference pixel Nr. 1. The left plot shows the result using the half-maximum of the spline (extractor \#23), the right plot shows the result obtained with the digital filter
+(extractor \#32). A 
+medium sized Blue-pulse (10Leds Blue) has been used which saturates the high-gain readout channel.}
+\label{fig:reltimesouter10ledsblue}
+\end{figure}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{figure}[htp]
+\centering
+\includegraphics[width=0.95\linewidth]{UnsuitTimeVsExtractor-5LedsUV-Colour-12.eps}
+\caption{Reconstructed arrival time resolutions from a typical, not saturating calibration pulse 
+of colour UV, reconstructed with each of the tested arrival time extractors. 
+The first plots shows the time resolutions obtained for the inner pixels, the second one 
+for the outer pixels. Points 
+denote the mean of all not-excluded pixels, the error bars their RMS.}
+\label{fig:time:5ledsuv}
+\end{figure}
+
+\begin{figure}[htp]
+\centering
+\includegraphics[width=0.95\linewidth]{UnsuitTimeVsExtractor-1LedUV-Colour-04.eps}
+\caption{Reconstructed arrival time resolutions from the lowest intensity calibration pulse 
+of colour UV (carrying a mean number of 4 photo-electrons), 
+reconstructed with each of the tested arrival time extractors. 
+The first plots shows the time resolutions obtained for the inner pixels, the second one 
+for the outer pixels. Points 
+denote the mean of all not-excluded pixels, the error bars their RMS.}
+\label{fig:time:1leduv}
+\end{figure}
+
+\begin{figure}[htp]
+\centering
+\includegraphics[width=0.95\linewidth]{UnsuitTimeVsExtractor-2LedsGreen-Colour-02.eps}
+\caption{Reconstructed arrival time resolutions from a typical, not saturating calibration pulse 
+of colour Green, reconstructed with each of the tested arrival time extractors. 
+The first plots shows the time resolutions obtained for the inner pixels, the second one 
+for the outer pixels. Points 
+denote the mean of all not-excluded pixels, the error bars their RMS.}
+\label{fig:time:2ledsgreen}
+\end{figure}
+
+\begin{figure}[htp]
+\centering
+\includegraphics[width=0.95\linewidth]{UnsuitTimeVsExtractor-23LedsBlue-Colour-00.eps}
+\caption{Reconstructed arrival time resolutions from the highest intensity calibration pulse 
+of colour blue, reconstructed with each of the tested arrival time extractors. 
+The first plots shows the time resolutions obtained for the inner pixels, the second one 
+for the outer pixels. Points 
+denote the mean of all not-excluded pixels, the error bars their RMS.}
+\label{fig:time:23ledsblue}
+\end{figure}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{figure}[htp]
+\centering
+\includegraphics[width=0.95\linewidth]{TimeResExtractor-5LedsUV-Colour-12.eps}
+\caption{Reconstructed arrival time resolutions from a typical, not saturating calibration pulse 
+of colour UV, reconstructed with each of the tested arrival time extractors. 
+The first plots shows the time resolutions obtained for the inner pixels, the second one 
+for the outer pixels. Points 
+denote the mean of all not-excluded pixels, the error bars their RMS.}
+\label{fig:time:5ledsuv}
+\end{figure}
+
+\begin{figure}[htp]
+\centering
+\includegraphics[width=0.95\linewidth]{TimeResExtractor-1LedUV-Colour-04.eps}
+\caption{Reconstructed arrival time resolutions from the lowest intensity calibration pulse 
+of colour UV (carrying a mean number of 4 photo-electrons), 
+reconstructed with each of the tested arrival time extractors. 
+The first plots shows the time resolutions obtained for the inner pixels, the second one 
+for the outer pixels. Points 
+denote the mean of all not-excluded pixels, the error bars their RMS.}
+\label{fig:time:1leduv}
+\end{figure}
+
+\begin{figure}[htp]
+\centering
+\includegraphics[width=0.95\linewidth]{TimeResExtractor-2LedsGreen-Colour-02.eps}
+\caption{Reconstructed arrival time resolutions from a typical, not saturating calibration pulse 
+of colour Green, reconstructed with each of the tested arrival time extractors. 
+The first plots shows the time resolutions obtained for the inner pixels, the second one 
+for the outer pixels. Points 
+denote the mean of all not-excluded pixels, the error bars their RMS.}
+\label{fig:time:2ledsgreen}
+\end{figure}
+
+\begin{figure}[htp]
+\centering
+\includegraphics[width=0.95\linewidth]{TimeResExtractor-23LedsBlue-Colour-00.eps}
+\caption{Reconstructed arrival time resolutions from the highest intensity calibration pulse 
+of colour blue, reconstructed with each of the tested arrival time extractors. 
+The first plots shows the time resolutions obtained for the inner pixels, the second one 
+for the outer pixels. Points 
+denote the mean of all not-excluded pixels, the error bars their RMS.}
+\label{fig:time:23ledsblue}
+\end{figure}
+
+
+\begin{figure}[htp]
+\centering
+\includegraphics[width=0.95\linewidth]{TimeResVsCharge-Area-21.eps}
+\caption{Reconstructed mean arrival time resolutions as a function of the extracted mean number of 
+photo-electrons for the weighted sliding window with a window size of 8 FADC slices (extractor \#21).
+Error bars denote the 
+spread (RMS) of the time resolutions over the investigated channels.
+The marker colours show the applied 
+pulser colour, except for the last (green) point where all three colours were used.}
+\label{fig:time:dep20}
+\end{figure}
+
+\begin{figure}[htp]
+\centering
+\includegraphics[width=0.95\linewidth]{TimeResVsCharge-Area-24.eps}
+\caption{Reconstructed mean arrival time resolutions as a function of the extracted mean number of 
+photo-electrons for the half-maximum searching spline (extractor \#23). Error bars denote the 
+spread (RMS) of the time resolutions over the investigated channels.
+The marker colours show the applied 
+pulser colour, except for the last (green) point where all three colours were used.}
+\label{fig:time:dep23}
+\end{figure}
+
+
+\begin{figure}[htp]
+\centering
+\includegraphics[width=0.95\linewidth]{TimeResVsCharge-Area-30.eps}
+\caption{Reconstructed mean arrival time resolutions as a function of the extracted signal 
+for the digital filter with UV weights and 6 slices (extractor \#30).  Error bars denote the 
+spread (RMS) of the time resolutions over the investigated channels.
+The marker colours show the applied 
+pulser colour, except for the last (green) point where all three colours were used.}
+\label{fig:time:dep30}
+\end{figure}
+
+
+\begin{figure}[htp]
+\centering
+\includegraphics[width=0.95\linewidth]{TimeResVsCharge-Area-31.eps}
+\caption{Reconstructed mean arrival time resolutions as a function of the extracted signal 
+for the digital filter with UV weights and 4 slices (extractor \#32).  Error bars denote the 
+spread (RMS) of the time resolutions over the investigated channels.
+The marker colours show the applied 
+pulser colour, except for the last (green) point where all three colours were used.}
+\label{fig:time:dep32}
+\end{figure}
+
+%%% Local Variables: 
+%%% mode: latex
+%%% TeX-master: "MAGIC_signal_reco"
+%%% End: 
Index: trunk/MagicSoft/TDAS-Extractor/Criteria.tex
===================================================================
--- trunk/MagicSoft/TDAS-Extractor/Criteria.tex	(revision 6409)
+++ trunk/MagicSoft/TDAS-Extractor/Criteria.tex	(revision 6410)
@@ -7,10 +7,4 @@
 \subsection{Stability}
 \ldots {\textit The stability of an extractor to slightly varying pulse shapes is examined. }
-\begin{itemize}
-\item Stability w.r.t. different weights files \ldots Hendrik
-\item Outliers in time and amplitude extractions \ldots ???
-\item Stability of conversion factors from the calibration \ldots Markus
-\item Stability w.r.t. the calibration secondary pulses \ldots Hendrik ???
-\end{itemize}
 
 \subsection{Linearity}
Index: trunk/MagicSoft/TDAS-Extractor/MonteCarlo.tex
===================================================================
--- trunk/MagicSoft/TDAS-Extractor/MonteCarlo.tex	(revision 6410)
+++ trunk/MagicSoft/TDAS-Extractor/MonteCarlo.tex	(revision 6410)
@@ -0,0 +1,6 @@
+\section{Monte Carlo \label{sec:mc}}
+
+%%% Local Variables: 
+%%% mode: latex
+%%% TeX-master: "MAGIC_signal_reco"
+%%% End: