\section{Criteria for an optimal pedestal extraction}

\ldots {\it In this section, the distinction is made between: 
\begin{itemize}
\item Defining the pedestal RMS as contribution
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).
\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}


\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: