Index: /trunk/MagicSoft/TDAS-Extractor/Calibration.tex =================================================================== --- /trunk/MagicSoft/TDAS-Extractor/Calibration.tex (revision 6639) +++ /trunk/MagicSoft/TDAS-Extractor/Calibration.tex (revision 6640) @@ -821,10 +821,17 @@ \Delta t \approx \frac{\delta t_{\mathrm{TTS}}}{\sqrt{Q/{\mathrm{phe}}}} \end{equation} -\item The reconstruction error due to the background noise: This contribution is proportional to the -signal to square root of background light intensities. +\item The reconstruction error due to the background noise and limited extractor resolution: +This contribution is inversely proportional to the signal to square root of background light intensities. +\begin{equation} +\Delta t \approx \frac{\delta t_{\mathrm{rec}} \cdot R/\mathrm{phe}}{Q/{\mathrm{phe}}} +\end{equation} +where $R$ is the resolution defined in equation~\ref{eq:def:r}. +\item A constant offset due to the residual FADC clock jitter~\cite{florian} +\begin{equation} +\Delta t \approx \delta t_0 +\end{equation} \end{enumerate} -Additionally to these intrinsic and irreducible contributions to the timing resolutions, the limited precision of the - extractors adds an additional time spread. In the following, we show measurements of the time resolutions at different +In the following, we show measurements of the time resolutions at different signal intensities in real conditions for the calibration pulses. These set upper limits to the time resolution for cosmics since their intrinsic arrival time spread is smaller. @@ -884,17 +891,76 @@ \clearpage - -\begin{figure}[htp] -\centering -\includegraphics[width=0.47\linewidth]{TimeResVsCharge-Area-21.eps} -\includegraphics[width=0.47\linewidth]{TimeResVsCharge-Area-24.eps} -\vspace{\floatsep} -\includegraphics[width=0.47\linewidth]{TimeResVsCharge-Area-30.eps} -\includegraphics[width=0.47\linewidth]{TimeResVsCharge-Area-31.eps} +The following figure~\ref{fig:time:dep} shows the time resolution for various calibration runs taken with different colours +and light intensities as a funcion of the mean number of photo-electrons -- +reconstructed with the F-Factor method -- for four different time extractors. The dependencies have been fit to the following +empirical relation: + +\begin{equation} +\Delta T = \sqrt{\frac{A^2}{/{\mathrm{phe}}} + \frac{B^2}{^2/{\mathrm{phe^2}}} + C^2} . +\label{eq:time:fit} +\end{equation} + +The fit results are summarized in table~\ref{tab:time:fitresults}. + +\begin{table}[htp] +\scriptsize{% +\centering +\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|} +\hline +\hline +\multicolumn{10}{|c|}{\large Time Fit Results} \rule{0mm}{6mm} \rule[-2mm]{0mm}{6mm} \hspace{-3mm}\\ +\hline +\hline +\multicolumn{2}{|c|}{} & \multicolumn{4}{|c|}{\normalsize Inner Pixels} & \multicolumn{4}{|c|}{\normalsize Outer Pixels} \rule{0mm}{6mm} \rule[-2mm]{0mm}{4mm} \hspace{-3mm}\\ +\hline +{\normalsize Nr.} & {\normalsize Name } & {\normalsize A} & {\normalsize B } & {\normalsize C }& {\normalsize $\chi^2$/NDF } +& {\normalsize A } &{\normalsize B} & {\normalsize C} &{\normalsize $\chi^2$/NDF} \rule{0mm}{6mm} \rule[-2mm]{0mm}{4mm} \hspace{-3mm} \\ +\hline +21 & Sliding Window (8,8) & 3.5$\pm$0.4 & 29$\pm$1 & 0.24$\pm$0.05 & 10.2 &6.0$\pm$0.7 & 52$\pm$4 & 0.23$\pm$0.04 & 4.3 \\ +25 & Spline Half Max. & 1.9$\pm$0.2 & 3.8$\pm$1.0 & 0.15$\pm$0.02 & 1.6 &2.6$\pm$0.2 &8.3$\pm$1.9 & 0.15$\pm$0.01 & 2.3 \\ +32 & Digital Filter (6 sl.) & 1.7$\pm$0.2 & 5.7$\pm$0.8 & 0.21$\pm$0.02 & 5.0 &2.3$\pm$0.3 &13 $\pm$2 & 0.20$\pm$0.01 & 4.0 \\ +33 & Digital Filter (4 sl.) & 1.7$\pm$0.1 & 4.6$\pm$0.7 & 0.21$\pm$0.02 & 6.2 &2.3$\pm$0.2 &11 $\pm$2 & 0.20$\pm$0.01 & 5.3 \\ +\hline +\hline +\end{tabular} +\caption{The fit results obtained from the fit of equation~\ref{eq:time:fit} to the time resolutions obtained for various +intensities and colours. The fit probabilities are very small mainly because of the different intrinsic arrival time spreads of +the photon pulses from different colours. } +\label{tab:time:fitresults}. +} +\end{table} + +The low fit probabilities are partly due to the systematic differences in the pulse forms in intrinsic arrival time spreads between +pulses of different LED colours. Nevertheless, we had to include all colours in the fit to cover the full dynamic range. In general, +one can see that the time resolutions for the UV pulses are systematically better than for the other colours which we attribute to the fact +the these pulses have a smaller intrinsic pulse width -- which is very close to pulses from cosmics. Moreover, there are clear differences +visible between different time extractors, especially the sliding window extractor yields poor resolutions. The other three extractors are +compatible within the errors, with the half-maximum of the spline being slightly better. + +\par + +To summarize, we find that we can obtain a time resolution of better than 1\,ns for all pulses above a threshold of 5\ photo-electrons. +This corresponds roughly to the image cleaning threshold in case of using the best signal extractor. At the largest signals, we can +reach a time resolution of as good as 200\,ps. +\par +The expected time resolution for inner pixels and cosmics pulses can thus be conservatively estimated to be: + +\begin{equation} +\Delta T_{\mathrm{cosmics}} \approx \sqrt{\frac{4\,\mathrm{ns}^2}{/{\mathrm{phe}}} + \frac{20\,\mathrm{ns}^2}{^2/{\mathrm{phe^2}}} + 0.04\,\mathrm{ns}^2} . +\label{eq:time:fitprediction} +\end{equation} + +\begin{landscape} +\begin{figure}[htp] +\centering +\includegraphics[width=0.24\linewidth]{TimeResFitted-21.eps} +\includegraphics[width=0.24\linewidth]{TimeResFitted-25.eps} +\includegraphics[width=0.24\linewidth]{TimeResFitted-32.eps} +\includegraphics[width=0.24\linewidth]{TimeResFitted-33.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 slices (extractor \#21, top left), -the half-maximum searching spline (extractor \#24, top right), -the digital filter with UV calibration-pulse weights over 6 slices (extractor \#30, bottom left) -and the digital filter with UV calibration-pulse weights over 4 slices (extractor \#31, bottom rigth). +the half-maximum searching spline (extractor~\#25, top right), +the digital filter with correct pulse weights over 6 slices (extractor~\#30 and~\#32, bottom left) +and the digital filter with UV calibration-pulse weights over 4 slices (extractor~\#31 and~\#33, bottom rigth). Error bars denote the spread (RMS) of time resolutions of the investigated channels. The marker colours show the applied @@ -902,23 +968,23 @@ \label{fig:time:dep} \end{figure} - -\subsubsection{An Upper Limit for the Average Intrinsic Time Spread of the Photo-multipliers} - - - -\begin{figure}[htp] -\centering -\includegraphics[width=0.32\linewidth]{TimeResVsSqrtPhe-Area-24.eps} -\includegraphics[width=0.32\linewidth]{TimeResVsSqrtPhe-Area-30.eps} -\includegraphics[width=0.32\linewidth]{TimeResVsSqrtPhe-Area-31.eps} -\caption{Reconstructed arrival time resolutions as a function of the square root of the -extimated number of photo-electrons for the half-maximum searching spline (extractor \#24, left) a -and the digital filter with the calibration pulse weigths fitted to UV pulses over 6 FADC slices (extractor \#30, center) -and the digital filter with the calibration pulse weigths fitted to UV pulses over 4 FADC slices (extractor \#31, right). -The time resolutions have been fitted from -The marker colours show the applied -pulser colour, except for the last (green) point where all three colours were used.} -\label{fig:time:fit2430} -\end{figure} +\end{landscape} + +The above resolution seems to be already limited by the intrinsic resolution of the photo-multipliers and the staggering of the +mirrors in case of the MAGIC-I telescope. + +%\begin{figure}[htp] +%\centering +%\includegraphics[width=0.32\linewidth]{TimeResVsSqrtPhe-Area-24.eps} +%\includegraphics[width=0.32\linewidth]{TimeResVsSqrtPhe-Area-30.eps} +%\includegraphics[width=0.32\linewidth]{TimeResVsSqrtPhe-Area-31.eps} +%\caption{Reconstructed arrival time resolutions as a function of the square root of the +%extimated number of photo-electrons for the half-maximum searching spline (extractor \#24, left) a +%and the digital filter with the calibration pulse weigths fitted to UV pulses over 6 FADC slices (extractor \#30, center) +%and the digital filter with the calibration pulse weigths fitted to UV pulses over 4 FADC slices (extractor \#31, right). +%The time resolutions have been fitted from +%The marker colours show the applied +%pulser colour, except for the last (green) point where all three colours were used.} +%\label{fig:time:fit2430} +%\end{figure} Index: /trunk/MagicSoft/TDAS-Extractor/Criteria.tex =================================================================== --- /trunk/MagicSoft/TDAS-Extractor/Criteria.tex (revision 6639) +++ /trunk/MagicSoft/TDAS-Extractor/Criteria.tex (revision 6640) @@ -38,9 +38,9 @@ \end{equation} -has the mean $B$ and the Variance $R$ defined as: +has the mean $B$ and the resolution $R$ defined as: \begin{eqnarray} B \ \ \ \ = \ \ \ \ \ \ \ \ \ \ \ &=& \ \ <\widehat{S}> \ -\ S\\ - R^2 \ \ \ \ = \ <(X-B)^2> &=& \ Var[\widehat{S}]\\ + R^2 \ \ \ \ = \ <(X-B)^2> &=& \ Var[\widehat{S}] \label{eq:def:r}\\ MSE \ = \ \ \ \ \ \ \ \ \ &=& \ Var[\widehat{S}] +\ B^2 \end{eqnarray} Index: /trunk/MagicSoft/TDAS-Extractor/MAGIC_signal_reco.bbl =================================================================== --- /trunk/MagicSoft/TDAS-Extractor/MAGIC_signal_reco.bbl (revision 6639) +++ /trunk/MagicSoft/TDAS-Extractor/MAGIC_signal_reco.bbl (revision 6640) @@ -84,3 +84,7 @@ \newblock http://wwwmagic.mppmu.mpg.de/publications/theses/David\_thesis.ps.gz. +\bibitem{florian} +F.~Goebel, +\newblock private communication. + \end{thebibliography}