\section{Introduction}


The MAGIC telescope aims to study the gamma ray emission from high energy phenomena and the violent physics processes in the universe 
at the lowest energy threshold possible \cite{low_energy}. 

Figure~\ref{fig:MAGIC_read-out_scheme} shows a sketch of the MAGIC read-out scheme, including the PMT camera, 
the analog-optical link, the majority trigger logic and 
FADCs. The used PMTs provide a very fast 
response to the input light signal. The response of the PMTs to sub-ns input light pulses shows a FWHM of 
1.0 - 1.2 ns and rise and fall times of 600 and 700 ps correspondingly \cite{Magic-PMT}. By modulating VCSEL 
type laser diodes in amplitude, the fast analogue signals from the PMTs are transferred via 162\,m long, 
50/125\,$\mu$m diameter optical fibers to the counting house \cite{MAGIC-analog-link-2}. After transforming the 
light back to an electrical signal, the original PMT pulse has a FWHM of about 2.2 ns and rise and fall 
times of about 1\,ns. % was 2.2 ns

%an analog optical link \ci
%te{MAGIC-analog-link-2} to the counting house.


\begin{figure}[h!]
\begin{center}
\includegraphics[width=\textwidth]{Magic_readout_scheme1.eps}
\end{center}
\caption[Current MAGIC read-out scheme.]{Current MAGIC read-out scheme: the analog PMT signals are 
transferred via an analog optical link to the counting house -- where after the trigger decision -- the signals 
are digitized by a 300 MHz FADCs system and written to the hard disk of a DAQ PC.} 
\label{fig:MAGIC_read-out_scheme}
\end{figure}

 

%After modulating VCSEL type laser diodes, after traveling through 162m of multi-mode graded index fiber of 50/125 $\mu$m diameter and.


In order to sample this pulse shape with the 300 MSamples/s FADC system, the original pulse is folded with a stretching function of 6ns leading to a FWHM  greater than 6\,ns. Because the MAGIC FADCs have a 
resolution of 8 bit only, the signals are split into two branches with gains differing by a factor 10. 
One branch is delayed by 55\,ns and then both branches are multiplexed and consecutively read-out by one FADC.
Figure~\ref{fig:pulpo_shape_high} shows a typical average of identical signals. A more detailed overview about the MAGIC read-out and DAQ system is given in \cite{Magic-DAQ}.
% The maximum sustained trigger rate could be 1 kHz. The FADCs feature a FIFO memory which allows a significantly higher short-time rate.
% Obviously by doing this, more LONS is integrated and thus the performance of the telescope on the analysis level is degraded. 


To reach the highest sensitivity and the lowest possible analysis energy threshold the recorded signals from Cherenkov light have to be accurately reconstructed. Therefore the highest possible signal to noise ratio, signal reconstruction resolution and a small bias are important.

Monte Carlo (MC) based simulations predict different time structures for gamma and hadron induced shower images as well as for images of single muons. An accurate arrival time determination may therefore improve the separation power of gamma events from the background events. Moreover, the timing information may be used in the image cleaning to discriminate between pixels which signal belongs to the shower and pixels which are affected by randomly timed background noise.


This note is structured as follows: In section 2 the average pulse shapes are reconstructed from the recorded FADC samples for calibration and cosmics pulses. These pulse shapes are compared with the pulse shape implemented in the MC. In section 3 different signal reconstruction algorithms and their implementation in the common MAGIC software framework MARS are reviewed. In section 4 criteria for an optimal signal reconstruction are developed. Thereafter the signal extraction algorithms under study are applied to pedestal, calibration and MC events in sections 5 to 7. The CPU requirements of the different algorithms are compared in section 8. Finally in section 9 the results are summarized and in section 10 a standard signal extraction algorithm for MAGIC is proposed.

\subsection{Characteristics of the current read-out system}

The following intrinsic characteristics of the current read-out system affect especially the signal reconstruction:

\begin{description}
\item[Inner and Outer pixels:\xspace] The MAGIC camera has two types of pixels which incorporate the following differences:
\begin{enumerate}
\item Size: The outer pixels have a factor four bigger area then the inner pixels~\cite{MAGIC-design}. 
Their (quantum-efficiency convoluted) effective area is about a factor 2.6 higher.
\item Gain: The camera is flat-fielded in order to yield a similiar reconstructed charge signal for the same photon illumination intensity. 
In order to achieve this, the gain of the inner pixels has been adjusted to about a factor 2.6 higher than the outer 
ones~\cite{tdas-calibration}. This results in lower effective noise charge from the night sky background for the outer pixels. 
\item Delay: The signal of the outer pixels is delayed by about 1.5\,ns with respect to the inner ones.
\end{enumerate}
\item[Clock noise:\xspace] The MAGIC 300\,MHz FADCs have an intrinsic clock noise of a few LSBs occurring with a frequency of 150\,MHz. 
This clock noise results 
in a superimposed AB-pattern for the read-out pedestals. In the standard analysis, the amplitude of this clock noise gets measured in the 
pedestal extraction algorithms and further corrected for by all signal extractors.
\item[Trigger Jitter:\xspace] The FADC clock is not synchronized with the trigger. Therefore, the relative position of the recorded 
signal samples varies uniformely by one FADC slice with respect to the position of the signal shape by one FADC slice from event to event.
\item[DAQ jumps:\xspace] Unfortunately, the position of the signal pulse with respect to the first recorded FADC sample is not constant. 
It varies randomly by an integer number of FADC slices -- typically two -- in about 1\% of the channels per event.

\end{description}

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