\section{Criteria for the Optimal Signal Extraction}

The goal for the optimal signal reconstruction algorithm is to compute an unbiased estimate of the strength and arrival time of the 
Cherenkov signal with the highest possible resolution for all signal intensities. The MAGIC telescope has been optimized to 
lower the energy treshold of observation in any respect. Particularly the choice for an FADC system has been made with an eye on the 
possibility to extract the smallest possible signals from air showers. It would be inconsequent not to continue the optimization procedure
in the signal extraction algorithms and the subsequent image cleaning. 
\par
In the image analysis, one takes the decision whether the extracted signal of a certain pixel is considered as signal or background. 
Those considered as signal are further used to compute the image parameters while the background ones are simply rejected. The calculation 
of the second moments of the image ``ellipse'' usually fails when applied to un-cleaned images, therefore the decision is yes or no. Moreover,
already low contributions of mis-estimated background can degrade the resolution of the image parameters considerably. If one wants to 
lower the threshold for signal recognition, it is therefore mandatory to increase the efficiency with which the background is recognized as 
such. If the background resolution is bad, the signal threshold goes up and vice versa.
\par
The algorithm must be stable with respect to changes 
in observation conditions and background levels and between signals induced from gamma or hadronic showers or from muons. 
The reconstructed signal shall be proportional to the total integrated charge in the FADCs due to the PMT pulse from the Cherenkov signal.

Also the needed computing time is of concern.

\subsection{Bias and Mean-squared Error}

Consider a large number of same signals $S$. By applying a signal extractor
we obtain a distribution of estimated signals $\widehat{S}$ (for fixed $S$ and
fixed background fluctuations $BG$). The distribution of the quantity

\begin{equation}
X = \widehat{S}-S 
\end{equation}

has the mean $B$ and the Variance $MSE$ defined as:

\begin{eqnarray}
   B   \ \ \ \  = \ \ \ \ \ \ <X> \ \ \ \ \  &=& \ \ <\widehat{S}> \ -\ S\\
   R   \ \ \ \  = \ <(X-B)^2> &=& \ Var[\widehat{S}]\\
   MSE \      = \ \ \ \ \ <X^2> \ \ \ \  &=& \ Var[\widehat{S}] +\ B^2
\end{eqnarray}

The parameter $B$ is also called the {\textit{\bf BIAS}} of the estimator and $MSE$ 
the {\textit{\bf MEAN-SQUARED ERROR}} which combines the variance of $\widehat{S}$ and 
the bias. Both depend generally on the size of $S$ and the background fluctuations $BG$, 
thus: $B = B(S,BG)$ and $MSE = MSE(S,BG)$.

\par
Usually, one measures easily the parameter $R$, but needs the $MSE$ for statistical analysis (e.g. 
in the image cleaning). 
However, only in case of a vanishing bias $B$, the two numbers are equal. Otherwise, 
the bias $B$ has to be known beforehand. Note that every sliding window extractor has a 
bias, especially at low or vanishing signals $S$.

\subsection{Linearity}
\ldots {\textit The Nuria plots ... }

\subsection{Low Gain Extraction}
\ldots {\textit The stability of the low-gain extraction w.r.t. the high-gain extraction}


\subsection{Stability}
\ldots {\textit The stability of an extractor to slightly varying pulse shapes is examined. }


\subsection{Treatment of Calibration Pulses}



\subsection{Applicability for Different Sampling Speeds / No Pulse Shaping.}
The current read-out system of the MAGIC telescope \cite{Magic-DAQ} with 300 MSamples/s is relatively slow compared to the fast pulses of about 2 ns FWHM of Cherenkov pulses. To acquire the pulse shape an artificial pulse shaping to about 6.5 ns FWHM is used. Thereby also more LONS is integrated that acts as noise. 

For 2 ns FWHM fast pulses a 2 GSamples/s FADC provides at least 4 sampling points. This permits a reasonable reconstruction of the pulse shape. First prototype tests with fast digitization systems for MAGIC have been successfully conducted \cite{GSamlesFADC}. The signals have been reconstructed within the common MAGIC Mars software framework.


\ldots {\textit Some comments by Hendrik ...}

\subsection{CPU Requirements}
\ldots {\textit The needed CPU time for each extractor}




\subsection{Pulpo Pulses}
\subsection{Cosmics Data?}
The results of this subsection are based on the following runs taken
on the 21st of September 2004.
\begin{itemize}
\item{Run 39000}: OffCrab11 at 19.1 degrees zenith angle and 106.2
azimuth.
\item{Run 39182}: CrabNebula at 19.0 degrees zenith angle and 106.0 azimuth.
\end{itemize}


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