Index: trunk/MagicSoft/TDAS-Extractor/Changelog =================================================================== --- trunk/MagicSoft/TDAS-Extractor/Changelog (revision 5534) +++ trunk/MagicSoft/TDAS-Extractor/Changelog (revision 5536) @@ -19,4 +19,8 @@ -*-*- END OF LINE -*-*- + +2004/12/01: Markus Gaug + * Pedestals.tex: Modified writing a bit, added subsection about applying + extractor to pedestals 2004/11/10: Hendrik Bartko Index: trunk/MagicSoft/TDAS-Extractor/Pedestal.tex =================================================================== --- trunk/MagicSoft/TDAS-Extractor/Pedestal.tex (revision 5534) +++ trunk/MagicSoft/TDAS-Extractor/Pedestal.tex (revision 5536) @@ -4,22 +4,20 @@ \begin{itemize} \item Defining the pedestal RMS as contribution -to the extracted signal fluctuations (later used in the calibration) + to the extracted signal fluctuations (later used in the calibration) \item Defining the Pedestal Mean and RMS as the result of distributions obtained by -applying the extractor to pedestal runs (yielding biases and modified widths). + applying the extractor to pedestal runs (yielding biases and modified widths). \item Deriving the correct probability for background fluctuations based on the extracted signal height. ( including biases and modified widths). \end{itemize} -\ldots Florian + ??? -\newline -\newline } -\subsection{Copy of email by W. Wittek from 25 Oct 2004 and 10 Nov 2004} +\subsection{Pedestal RMS} -\subsubsection{Pedestal RMS} +\vspace{1cm} +\ldots {\it Modified email by W. Wittek from 25 Oct 2004 and 10 Nov 2004} +\vspace{1cm} -We all know how it is defined. It can be completely -described by the matrix +The Pedestal RMS can be completely described by the matrix \begin{equation} @@ -27,18 +25,18 @@ \end{equation} -where $i$ and $j$ denote the $i^{th}$ and $j^{th}$ FADC slice, +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. +value in slice $i$ for an event and the average $<>$ is over many events (usually 1000). \par -By definition, the pedestal RMS is independent of the signal extractor. -Therefore no signal extractor is needed for the pedestals. +By definition, the pedestal RMS is independent from the signal extractor. +Therefore, no signal extractor is needed to calculate the pedestals. -\subsubsection{Bias and Error} +\subsection{Bias and Error} Consider a large number of signals (FADC spectra), all with the same integrated charge $ST$ (true signal). By applying some signal extractor we obtain a distribution of extracted signals $SE$ (for fixed $ST$ and -fixed background fluctuations). The distribution of the quantity +fixed background fluctuations $BG$). The distribution of the quantity \begin{equation} @@ -60,77 +58,31 @@ $B$ is the bias, $R$ is the RMS of the distribution of $X$ and $D$ is something -like the (asymmetric) error of $SE$. It is the distribution of $X$ (or its -parameters $B$ and $R$) which we are eventually interested in. The distribution -of $X$, and thus the parameters $B$ and $R$, depend on $ST$ and the size of the -background fluctuations. -\par - -By applying the signal extractor to pedestal events you want to -determine these parameters, I guess. - -\par -By applying it with max. peak search you get information about the bias $B$ -for very low signals, not for high signals. By applying it to a fixed window, -without max.peak search, you may get something like $R$ for high signals (but -I am not sure). - -\par -For the normal image cleaning, knowledge of $B$ is sufficient, because the -error $R$ is not used anyway. You only want to cut off the low signals. - -\par -For the model analysis you need both, $B$ and $R$, because you want to keep small -signals. Unfortunately $B$ and $R$ depend on the size of the signal ($ST$) and on -the size of the background fluctuations (BG). However, applying the signal -extractor to pedestal events gives you only 1 number, dependent on BG but -independent of $ST$. +like the (asymmetric) error of $SE$. +The distribution of $X$, and thus the parameters $B$ and $R$, +depend on the size of $ST$ and the size of the background fluctuations $BG$. \par -Where do we get the missing information from ? I have no simple solution or -answer, but I would think -\begin{itemize} -\item that you have to determine the bias from MC -\item and you may gain information about $R$ from the fitted error of $SE$, which is - known for every pixel and event -\end{itemize} - -The question is 'How do we determine the $R$ ?'. A proposal which -has been discussed in various messages is to apply the signal extractor to -pedestal events. One can do that, however, this will give you information -about the bias and the error of the extracted signal only for signals -whose size is in the order of the pedestal fluctuations. This is certainly -useful for defining the right level for the image cleaning. +For the normal image cleaning, knowledge of $B$ is sufficient and the +error $R$ should be know in order to calculate a correct background probability. \par - -However, because the bias $B$ and the error of the extracted signal $R$ depend on -the size of the signal, applying the signal extractor to pedestal events -won't give you the right answer for larger signals, for example for the -calibration signals. - -The basic relation of the F-method is +Also for the model analysis $B$ and $R$ are needed, because you want to keep small +signals. +\par +In the case of the calibration with the F-Factor methoid, +the basic relation is: \begin{equation} -\frac{sig^2}{^2} = \frac{1}{} * F^2 +\frac{(\Delta ST)^2}{^2} = \frac{1}{} * F^2 \end{equation} -Here $sig$ is the fluctuation of the extracted signal $Q$ due to the -fluctuation of the number of photo electrons. $sig$ is obtained from the -measured fluctuations of $Q$ ($RMS_Q$) by subtracting the fluctuation of the -extracted signal ($R$) due to the fluctuation of the pedestal RMS : +Here $\Delta ST$ is the fluctuation of the true signal $ST$ due to the +fluctuation of the number of photo electrons. $ST$ is obtained from the +measured fluctuations of $SE$ ($RMS_{SE}$) by subtracting the fluctuation of the +extracted signal ($R$) due to the fluctuation of the pedestal. \begin{equation} - sig^2 = RMS_Q^2 - R^2 + (\Delta ST)^2 = RMS_{SE}^2 - R^2 \end{equation} - -$R$ is in general different from the pedestal RMS. It cannot be -obtained by applying the signal extractor to pedestal events, because -the calibration signal is usually large. - -In the case of the optimum filter, $R$ may be obtained from the -fitted error of the extracted signal ($dQ_{fitted}$), which one can calculate -for every event. Whether this statemebt is true should be checked by MC. -For large signals I would expect the bias of the extracted to be small and -negligible. A way to check whether the right RMS has been subtracted is to make the @@ -138,5 +90,5 @@ \begin{equation} - \frac{sig^2}{^2} \quad \textit{vs.} \quad \frac{1}{} + \frac{(\Delta ST)^2}{^2} \quad \textit{vs.} \quad \frac{1}{} \end{equation} @@ -148,5 +100,65 @@ \end{equation} -where $c$ is the photon/ADC conversion factor $/$. +where $c$ is the photon/ADC conversion factor $/$. + +\subsection{How to retrieve Bias $B$ and Error $R$} + +$R$ is in general different from the pedestal RMS. It cannot be +obtained by applying the signal extractor to pedestal events, especially +for large signals (e.g. calibration signals). +\par +In the case of the optimum filter, $R$ 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 $). +\par + +In order to get the missing information, we did the following investigations: +\begin{enumerate} +\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. +\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, 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} +\par + + +\vspace{1cm} +\ldots{\it More test plots can be found under: +http://magic.ifae.es/$\sim$markus/ExtractorPedestals/ } +\vspace{1cm} %%% Local Variables: @@ -155,3 +167,4 @@ %%% TeX-master: "MAGIC_signal_reco" %%% TeX-master: "MAGIC_signal_reco" +%%% TeX-master: "MAGIC_signal_reco." %%% End: