Index: /trunk/MagicSoft/TDAS-Extractor/Algorithms.tex =================================================================== --- /trunk/MagicSoft/TDAS-Extractor/Algorithms.tex (revision 5567) +++ /trunk/MagicSoft/TDAS-Extractor/Algorithms.tex (revision 5568) @@ -1,3 +1,3 @@ -\section{Signal Reconstruction Algorithms} +\section{Signal Reconstruction Algorithms \label{sec:algorithms}} \ldots {\it In this section, the extractors are described, especially w.r.t. which free parameters are left to play, @@ -176,7 +176,8 @@ The correlation of the noise contributions at times $t_i$ and $t_j$ is expressed in the noise autocorrelation matrix $\boldsymbol{B}$: -\begin{equation} +\begin{equation} \boldsymbol{B_{ij}} = \langle b_i b_j \rangle - \langle b_i \rangle \langle b_j \rangle \ . +\label{eq:autocorr} \end{equation} %\equiv \langle b_i b_j \rangle with $\langle b_i \rangle = 0$. @@ -393,3 +394,4 @@ %%% TeX-master: "MAGIC_signal_reco" %%% TeX-master: "MAGIC_signal_reco" +%%% TeX-master: "MAGIC_signal_reco" %%% End: Index: /trunk/MagicSoft/TDAS-Extractor/Changelog =================================================================== --- /trunk/MagicSoft/TDAS-Extractor/Changelog (revision 5567) +++ /trunk/MagicSoft/TDAS-Extractor/Changelog (revision 5568) @@ -19,5 +19,11 @@ -*-*- END OF LINE -*-*- -2004/11/10: Hendrik Bartko +2004/12/07: Markus Gaug + * bibfile.bib: Modified slightly citation of NUMREC + * Pedestal.tex: Modified writing a bit, added references to dig.filter + formulas. + * Algorithms.tex: Added some reference labels + +2004/12/06: Hendrik Bartko * Reconstruction.tex: Added some paragraphs how we reconstruct the average pulse shape from the recorded signal samples. Added some Index: /trunk/MagicSoft/TDAS-Extractor/Pedestal.tex =================================================================== --- /trunk/MagicSoft/TDAS-Extractor/Pedestal.tex (revision 5567) +++ /trunk/MagicSoft/TDAS-Extractor/Pedestal.tex (revision 5568) @@ -19,17 +19,15 @@ \vspace{1cm} -The Pedestal RMS can be completely described by the matrix - -\begin{equation} - < (P_i - ) * (P_j - ) > -\end{equation} - -where $i$ and $j$ denote the $i^{th}$ and $j^{th}$ FADC slice and -$P_i$ is the pedestal -value in slice $i$ for an event and the average $<>$ is over many events (usually 1000). -\par - -By definition, the pedestal RMS is independent from the signal extractor. -Therefore, no signal extractor is needed to calculate the pedestals. +The background $BG$ (Pedestal) +can be completely described by the noise-autocorrelation matrix $\boldsymbol{B}$ +(eq.~\ref{eq:autocorr}), +where the diagonal elements give what is usually denoted as the ``Pedestal RMS''. Note that +in the MAGIC readout, the diagonal elements do not scale exactly with the square root of +the number of slices as would be expected from pure stochasitic noise. + +\par + +By definition, the noise autocorrelation matrix $B$ and thus the ``pedestal RMS'' +is independent from the signal extractor. \subsection{Bias and Error} @@ -108,50 +106,44 @@ for large signals (e.g. calibration signals). \par -In the case of the optimum filter, $R$ can be obtained from the +In the case of the optimum filter, $R$ is in theory independent from the +signal amplitude $ST$ and depends only on the background $BG$, see eq.~\ref{of_noise}. +It can be obtained from the fitted error of the extracted signal ($\Delta(SE)_{fitted}$), -which one can calculate for every event. - -\vspace{1cm} -\ldots {\it Whether this statemebt is true should be checked by MC.} -\vspace{1cm} - -For large signals, one would expect the bias of the extracted signal -to be small and negligible (i.e. $ \approx $). +which one can calculate for every event or by applying the extractor to a fixed window +of pure background events (``pedestal events''). + \par In order to get the missing information, we did the following investigations: \begin{enumerate} +\item Determine $R$ by applying the signal extractor to a fixed window + of pedestal events. The background fluctuations can be simulated with different + levels of night sky background and the continuous light, but no signal size + dependency can be retrieved with the method. \item Determine bias $B$ and resolution $R$ from MC events with and without added noise. Assuming that $R$ and $B$ are negligible for the events without noise, one can get a dependency of both values from the size of the signal. \item Determine $R$ from the fitted error of $SE$, which is possible for the - fit and the digital filter. In prinicple, all dependencies can be retrieved with this - method. -\item Determine $R$ for low signals by applying the signal extractor to a fixed window - of pedestal events. The background fluctuations can be simulated with different - levels of night sky background and the continuous light, but no signal size - dependency can be retrieved with the method. Its results are only valid for small - signals. + fit and the digital filter (eq.~\ref{of_noise}). + In prinicple, all dependencies can be retrieved with this method. \end{enumerate} - -\par \subsubsection{Determine error $R$ by applying the signal extractor to a fixed window of pedestal events} -By applying the signal extractor to pedestal events we want to -determine these parameters. There are the following possibilities: - -\begin{enumerate} -\item Applying the signal extractor allowing for a possible sliding window - to get information about the bias $B$ (valid for low signals). -\item Applying the signal extractor to a fixed window, to get something like - $R$. In the case of the digital filter and the spline, this has to be done - by randomizing the time slice indices. -\end{enumerate} - -\vspace{1cm} -\ldots {\it This assumptions still have to proven, best mathematically!!! Wolfgang, Thomas???} -\vspace{1cm} +By applying the signal extractor to a fixed window of pedestal events, we +determined the parameter $R$ for the case of no signal ($ST = 0$). In the case of +all extractors using a fixed window from the beginning (extractors nr. \#1 to \#22 +in section~\ref{sec:algorithms}), the results were exactly the same as calculating +the mean and the RMS of a same (fixed) number of FADC slices (the conventional ``Pedestal +Calculation''). + +\par +In the case of the amplitude extracting spline (extractor nr. \#27), we took the +spline value at a random place within the digitizing binning resolution (0.02 FADC slices) of +one central FADC slice. +In the case of the digital filter (extractor nr. \#28), the time shift was +randomized for each event within one central FADC slice. + \par @@ -331,3 +323,4 @@ %%% TeX-master: "MAGIC_signal_reco." %%% TeX-master: "MAGIC_signal_reco" +%%% TeX-master: "Pedestal" %%% End: Index: /trunk/MagicSoft/TDAS-Extractor/bibfile.bib =================================================================== --- /trunk/MagicSoft/TDAS-Extractor/bibfile.bib (revision 5567) +++ /trunk/MagicSoft/TDAS-Extractor/bibfile.bib (revision 5568) @@ -16,7 +16,8 @@ } -@Book{NumRec, - author = "W.H.Press and S.A.Teukolsky and W.T.Vetterling and B.P.Flannery", - title = "Numerical Recipes in C++, 2nd edition", +@Book{NUMREC, + author = "W.H.Press and S.A.Teukolsky and W.T.Vetterling and B.P.Flannery", + title = "Numerical Recipes in C++", + edition = "Second", publisher = "Cambridge University Press", year = "2002"