Index: /trunk/MagicSoft/TDAS-Extractor/Changelog
===================================================================
--- /trunk/MagicSoft/TDAS-Extractor/Changelog	(revision 5369)
+++ /trunk/MagicSoft/TDAS-Extractor/Changelog	(revision 5370)
@@ -19,4 +19,8 @@
 
                                                  -*-*- END OF LINE -*-*-
+
+2004/11/10: Markus Gaug 
+  * Pedestal.tex: put a copy of Wolfgangs two emails with definitions 
+                  and explications
 
 2004/10/27: Oscar Blanch Bigas
Index: /trunk/MagicSoft/TDAS-Extractor/Pedestal.tex
===================================================================
--- /trunk/MagicSoft/TDAS-Extractor/Pedestal.tex	(revision 5369)
+++ /trunk/MagicSoft/TDAS-Extractor/Pedestal.tex	(revision 5370)
@@ -15,7 +15,143 @@
 }
 
+\subsection{Copy of email by W. Wittek from 25 Oct 2004 and 10 Nov 2004}
+
+
+\subsubsection{Pedestal RMS}
+
+We all know how it is defined. It can be completely
+described by the matrix
+
+\begin{equation}
+   < (P_i - <P_i>) * (P_j - <P_j>) >
+\end{equation}
+
+where $i$ and $j$ denote the $i^{th}$ and $j^{th}$ FADC slice, 
+$P_i$ is the pedestal
+value in slice $i$ for an event and the average $<>$ is over many events.
+\par
+
+By definition, the pedestal RMS is independent of the signal extractor.
+Therefore no signal extractor is needed for the pedestals.
+
+\subsubsection{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
+
+\begin{equation}
+X = SE-ST 
+\end{equation}
+
+has the mean $B$ and the RMS $R$
+
+\begin{eqnarray}
+   B    &=& <X> \\
+   R^2  &=& <(X-B)^2>
+\end{eqnarray}
+
+One may also define
+
+\begin{equation}
+   D^2 = <(SE-ST)^2> = <(SE-ST-B + B)^2> = B^2 + R^2
+\end{equation}
+
+$B$ 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 $ST$.
+
+\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.
+\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
+
+\begin{equation}
+\frac{sig^2}{<Q>^2} = \frac{1}{<m_{pe}>} * 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 :
+
+\begin{equation}
+ sig^2 = RMS_Q^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
+Razmick plot
+
+\begin{equation}
+    \frac{sig^2}{<Q>^2} \quad \textit{vs.} \quad \frac{1}{<Q>}
+\end{equation}
+
+This should give a straight line passing through the origin. The slope of
+the line is equal to
+
+\begin{equation}
+    c * F^2
+\end{equation}
+
+where $c$ is the photon/ADC conversion factor  $<Q>/<m_{pe}>$.
+
 %%% Local Variables: 
 %%% mode: latex
 %%% TeX-master: "MAGIC_signal_reco"
 %%% TeX-master: "MAGIC_signal_reco"
+%%% TeX-master: "MAGIC_signal_reco"
 %%% End: