Index: /trunk/MagicSoft/TDAS-Extractor/Calibration.tex =================================================================== --- /trunk/MagicSoft/TDAS-Extractor/Calibration.tex (revision 6628) +++ /trunk/MagicSoft/TDAS-Extractor/Calibration.tex (revision 6629) @@ -370,5 +370,5 @@ As the photo-multiplier and the subsequent -optical transmission devices~\cite{david} is a linear device over a +optical transmission devices~\cite{david} is a relatively linear device over a wide dynamic range, the number of photo-electrons per charge has to remain constant over the tested linearity region. @@ -427,16 +427,21 @@ \begin{figure}[h!] \centering -\includegraphics[width=0.99\linewidth]{PheVsCharge-11.eps} +\includegraphics[width=0.99\linewidth]{PheVsCharge-14.eps} \caption{Example of a the development of the conversion factor FADC counts to photo-electrons for three exemplary inner pixels (upper plots) and three exemplary outer ones (lower plots) obtained with the extractor {\textit{MExtractFixedWindowPeakSearch}} -on a window size of 2 high-gain and 2 low-gain slices (extractor \#11). } +on a window size of 6 high-gain and 6 low-gain slices (extractor \#11). } \label{fig:linear:phevscharge11} \end{figure} Figure~\ref{fig:linear:phevscharge11} shows the conversion factors using a fixed window with global peak search -integrating a window of 2 FADC slices. One can see that the linearity is completely lost! Especially in the low-gain, -the reconstructed number of photo-electrons is much too low and the conversion factors bend down. A similiar behaviour can -be found for all extractors with window sizes smaller than 6 FADC slices, especially in the low-gain region. (This behaviour +integrating a window of 6 FADC slices. One can see that the linearity is completely lost above 300 photo-electrons in the +outer pixels. Especially in the low-gain, +the reconstructed mean charge is too low and the conversion factors bend down. We show this extractor especially because it has +been used in the analysis and to derive a Crab spectrum with the consequence that the spectrum bends down at high energies. We +suppose that the loss of linearity due to usage of this extractor is responsible for the encountered problems. +A similiar behaviour can be found for all extractors with window sizes smaller than 6 FADC slices, especially in the low-gain region. +This is understandable since the low-gain pulse covers at least 6 FADC slices. +(This behaviour was already visible in the investigations on the number of photo-electrons in the previous section~\ref{sec:photo-electrons}). \par @@ -458,9 +463,9 @@ Figure~\ref{fig:linear:phevscharge23} shows the conversion factors using the amplitude-extracting spline (extractor \#23). -Here, the linearity worse than in the previous sample. A very clear difference between high-gain and +Here, the linearity is worse than in the previous sample. A very clear difference between high-gain and low-gain regions can be seen as well as a bigger general spread in conversion factors. In order to investigate if there is a common, systematic effect of the extractor, we show the averaged conversion factors over all inner and outer pixels in figure~\ref{fig:linear:phevschargearea23}. Both characteristics are maintained -there. Although the differences between high-gain and low-gain can be easily corrected for, we conclude +there. Although the differences between high-gain and low-gain could be easily corrected for, we conclude that extractor \#23 is still unstable against the linearity tests. \par @@ -482,5 +487,5 @@ Figure~\ref{fig:linear:phevscharge24} shows the conversion factors using a spline integrating over -one effective FADC slice in the high-gain and 1.5 effective FADC slices in the low-gain (extractor \#24). +one effective FADC slice in the high-gain and 1.5 effective FADC slices in the low-gain region (extractor \#24). The same problems are found as with extractor \#23, however to a much lower extent. The difference between high-gain and low-gain regions is less pronounced and the spread @@ -488,5 +493,5 @@ Figure~\ref{fig:linear:phevschargearea24} shows already rather good stability except for the two lowest intensity pulses in green and blue. We conclude that extractor \#24 is still un-stable, but -preferable to amplitude extractor. +preferable to the amplitude extractor. \par @@ -511,7 +516,6 @@ to two effective FADC slices in the high-gain and three effective FADC slices in the low-gain (extractor \#25), the stability is completely resumed, except for -a small systematic increase of the conversion factor in the low-gain range. This effect -is not very significant, however it can be seen in five out of the -six tested channels. We conclude that extractor \#25 is almost as stable as the fixed window extractors. +a systematic increase of the conversion factor above 200 photo-electrons. +We conclude that extractor \#25 is almost as stable as the fixed window extractors. \par @@ -534,5 +538,6 @@ Figure~\ref{fig:linear:phevscharge30} and~\ref{fig:linear:phevscharge31} show the conversion factors using a digital filter, -applied on 6 FADC slices and respectively 4 FADC slices with weights calculated from the UV-calibration pulse. +applied on 6 FADC slices and respectively 4 FADC slices with weights calculated from the UV-calibration pulse in the +high-gain region and from the blue calibration pulse in the low-gain region. One can see that one or two blue calibration pulses at low and intermediate intensity fall out of the linear region, moreover there is a small systematic offset between the high-gain and low-gain region. @@ -541,5 +546,5 @@ will not calculate the number of photo-electrons (with the F-Factor method) for every signal intensity. Thus, one possible reason for the instability falls away in the cosmics analysis. However, the limits -of this extraction are clearly visible here and have to be monitored further. +of this extraction are visible here and should be monitored further. \par @@ -584,10 +589,13 @@ \subsection{Relative Arrival Time Calibration} -The extractors \#17--33 are able to compute the arrival time of each pulse. The calibration LEDs +The calibration LEDs deliver a fast-rising pulses, 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 calibration 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): +camera. + +The extractors \#17--33 are able to compute the arrival time of each pulse. +Since the calibration 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 no.\,1): \begin{equation} @@ -596,5 +604,5 @@ 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). In one calibration run, one can then fill +arrival time of the reference pixel no. 1 (software numbering). In one calibration run, one can then fill histograms of $\delta t_i$ and fit them to the expected Gaussian distribution. The fits yield a mean $\mu(\delta t_i)$, comparable to @@ -604,5 +612,5 @@ \begin{equation} -t^{res}_i \approx \sigma(\delta t_i)/\sqrt(2) +t^{res}_i \approx \sigma(\delta t_i)/\sqrt{2} \end{equation} @@ -622,5 +630,5 @@ \includegraphics[width=0.45\linewidth]{RelTime_100_Extractor30.eps} \includegraphics[width=0.45\linewidth]{RelTime_100_Extractor31.eps} -\caption{Examples of a distributions of relative arrival times $\delta t_i$ of an inner pixel (Nr. 100) \protect\\ +\caption{Examples of a distributions of relative arrival times $\delta t_i$ of an inner pixel (no. 100) \protect\\ Top: {\textit{\bf MExtractTimeAndChargeSlidingWindow}} over 2 slices (\#17) and 4 slices (\#18) \protect\\ Center: {\textit{\bf MExtractTimeAndChargeSpline}} with maximum (\#23) and half-maximum pos. (\#24) \protect\\ @@ -632,5 +640,5 @@ Figures~\ref{fig:reltimesinnerledblue1} and~\ref{fig:reltimesinnerledblue2} show the distributions of $\delta t_i$ for a typical inner pixel and an intense, high-gain-saturating calibration -pulse of blue light. +pulse of blue light, obtained from the low-gain readout channel. One can see that the sliding window extractors yield double Gaussian structures, except for the largest window sizes of 8 and 10 FADC slices. Even then, the distributions are not exactly Gaussian. @@ -651,5 +659,5 @@ \includegraphics[width=0.45\linewidth]{RelTime_100_Extractor23_logain.eps} \includegraphics[width=0.45\linewidth]{RelTime_100_Extractor24_logain.eps} -\caption{Examples of a distributions of relative arrival times $\delta t_i$ of an inner pixel (Nr. 100) \protect\\ +\caption{Examples of a distributions of relative arrival times $\delta t_i$ of an inner pixel (no. 100) \protect\\ Top: {\textit{\bf MExtractTimeAndChargeSlidingWindow}} over 4 slices (\#18) and 6 slices (\#19) \protect\\ Center: {\textit{\bf MExtractTimeAndChargeSlidingWindow}} over 8 slices (\#20) and 10 slices (\#21)\protect\\ @@ -665,5 +673,5 @@ \includegraphics[width=0.45\linewidth]{RelTime_100_Extractor32_logain.eps} \includegraphics[width=0.45\linewidth]{RelTime_100_Extractor33_logain.eps} -\caption{Examples of a distributions of relative arrival times $\delta t_i$ of an inner pixel (Nr. 100) \protect\\ +\caption{Examples of a distributions of relative arrival times $\delta t_i$ of an inner pixel (no. 100) \protect\\ Top: {\textit{\bf MExtractTimeAndChargeDigitalFilter}} fitted to cosmics pulses over 6 slices (\#30) and 4 slices (\#31) \protect\\ @@ -679,5 +687,5 @@ %\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 +%the arrival time of the reference pixel no. 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 slices (extractor \#17). A @@ -691,5 +699,5 @@ %\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 +%the arrival time of the reference pixel no. 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.} @@ -704,5 +712,5 @@ %\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 +%the arrival time of the reference pixel no. 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.} @@ -710,4 +718,6 @@ %\end{figure} +\clearpage + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -716,8 +726,30 @@ As in section~\ref{sec:uncalibrated}, we tested the number of outliers from the Gaussian distribution in order to count how many times the extractor has failed to reconstruct the correct arrival time. - -\begin{figure}[htp] -\centering -\includegraphics[width=0.95\linewidth]{UnsuitTimeVsExtractor-5LedsUV-Colour-12.eps} +\par +Figure~\ref{fig:time:5ledsuv} shows the number of outliers for the different time extractors, obtained with +a UV pulse of about 20 photo-electrons. One can see that all time extractors yield an acceptable mis-reconstruction +rate of about 0.5\%, except for the maximum searching spline yields three times more mis-reconstructions. +\par +If one goes to very low-intensity pulses, as shown in figure~\ref{fig:time:1leduv}, obtained with on average 4 photo-electrons, +the number of mis-reconstructions increases considerably up to 20\% for some extractors. We interpret this high mis-reconstruction +rate to the increase possibility to mis-reconstruct a pulse from the night sky background noise instead of the signal pulse from the +calibration LEDs. One can see that the digital filter using weights on 4 FADC slices is clear inferior to the one using 6 FADC slices +in that respect. +\par +The same conclusion seems to hold for the green pulse of about 20 photo-electrons (figure~\ref{fig:time:2ledsgreen}) +where the digital filter over 6 FADC slices seems to +yield more stable results than the one over 4 FADC slices. The half-maximum searching spline seems to be superior to the maximum-searching +one. +\par +In figure~\ref{fig:time:23ledsblue}, one can see the number of outliers from an intense calibration pulse of blue light yielding about +600 photo-electrons per inner pixel. All extractors seem to be stable, except for the digital filter with weigths over 4 FADC slices. This +is expected, since the low-gain pulse is wider than 4 FADC slices. +\par +In all previous plots, the sliding window yielded the most stable results, however later we will see that this stability is only due to +an increased time spread. + +\begin{figure}[htp] +\centering +\includegraphics[height=0.35\textheight]{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. @@ -730,5 +762,5 @@ \begin{figure}[htp] \centering -\includegraphics[width=0.95\linewidth]{UnsuitTimeVsExtractor-1LedUV-Colour-04.eps} +\includegraphics[height=0.35\textheight]{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), @@ -742,5 +774,5 @@ \begin{figure}[htp] \centering -\includegraphics[width=0.95\linewidth]{UnsuitTimeVsExtractor-2LedsGreen-Colour-02.eps} +\includegraphics[height=0.35\textheight]{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. @@ -753,5 +785,5 @@ \begin{figure}[htp] \centering -\includegraphics[width=0.95\linewidth]{UnsuitTimeVsExtractor-23LedsBlue-Colour-00.eps} +\includegraphics[height=0.35\textheight]{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. @@ -762,7 +794,39 @@ \end{figure} +\clearpage + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Time Resolution} + +There are three intrinsic contributions to the timing accuracy of the signal: + +\begin{enumerate} +\item Intrinsic transit time spread TTS of the PMT. It can be in the order +of a few hundreds of ps per single photo electron. When we reconstruct +the mean pulse arrival time the error of the mean is given by the time +spread per single photo electron dividid by the square root of number of +photo electrons. +\item Intrinsic arrival time spread of the photons on the PMT. For our +calibration LEDs this can be up to about 2 ns, for muons it is about a +few hundreds of ps and for hadrons a few ns. The error of the mean +arrival time of the total pulse is again the arrival time spread of the +photons divided by the number of photo electrons. +\item reconstruction error due to noise and error of the numeric fit in +case of the digital filter. In case of the digital filter the error for +the standard noise level in the MC is about 2.7 ns divided by the signal +in photo electrons. +\end{enumerate} + +All this seems to quite agree with the results obtained with the MC +TestPulses. As 1) and 2) are proportional to one over the square root of +the signal and 3 is proportional to one over the signal, for small +pulses the noise determins the resolution, but for larger pulses the +intrinsic fluctuations limit the timing resolution. + +When we get a time resolution of about 300 ps for calibration LED pulses +we can not distinquish 1) and 2). Thus we have to wait for the exact +measurement. + \begin{figure}[htp]