\section{Pulse Shape Reconstruction}

The FADC clock is not synchronized with the trigger. Therefore, the relative position of the recorded 
signal samples varies  from event to event with respect to the position of the signal shape. 
The time between the trigger decision and the first read-out sample is uniformly distributed in the range 
$t_{\text{rel}} \in [0,T_{\mathrm{FADC}}[$, where $T_{\mathrm{FADC}}=3.33$ ns is the digitization period of the MAGIC 300 MHz FADCs. 
It can be determined using the reconstructed arrival time 
$t_{\mathrm{arrival}}$.%directly by a time to digital converter (TDC) or 
\par
\ldots {\textit{MAYBE a PLOT TO DEMONSTRATE THIS?}}
\par
The asynchronous sampling of the pulse shape allows to determine an average pulse shape from the recorded 
signal samples: The recorded signal samples can be shifted in time such that the shifted arrival times 
of all events are equal. In addition, the signal samples are normalized event by event using the 
reconstructed charge of the pulse. The accuracy of the signal shape reconstruction depends on the accuracy 
of the arrival time and charge reconstruction and amounts to \ldots 

\par
{\textit{NUMBER IS MISSING !!}}
\par
\ldots

Figure~\ref{fig:pulpo_shape_high} shows the averaged and shifted reconstructed signal of a fast pulser 
in the so called pulse generator (``pulpo'') setup. 

\ldots
\par
{\textit{EXPLAIN PULPO SETUO}}
\par
\ldots


Clearly visible are the high and the low gain pulses. The low gain 
pulse is attenuated by a factor of about 10 and delayed by about 55\,ns with respect to the high gain pulse.

Figures~\ref{fig:pulpo_shape_low} shows the averaged normalized reconstructed pulse shapes for the ``pulpo'' 
pulses in the high and in the low gain, respectively. The input FWHM of the pulse generator pulses is 
about 2\,ns. The FWHM of the average reconstructed high gain pulse shape is about 6.3\,ns, while the FWHM of 
the average reconstructed low gain pulse shape is about 10\,ns. The pulse broadening of the low gain pulses 
with respect to the high gain pulses is due to the limited dynamic range of the passive 55\,ns on board 
delay line of the MAGIC receiver boards. 
%   while the FWHM of the average reconstructed low gain pulse shape is 
% Due to the electric delay line for the low gain pules on the receiver board the low gain pulse is widened with respect to the high gain. 
It has a FWHM of about 10 ns.


\begin{figure}[h!]
\begin{center}
\includegraphics[totalheight=7cm]{pulpo_shape_high_low_TDAS.eps}%{pulpo_shape_high.eps}
\end{center}
\caption[Reconstructed high gain shape.]{Average reconstructed high gain pulse shape from a pulpo run. The FWHM is about 6.2 ns.} 
\label{fig:pulpo_shape_high}
\end{figure}

\begin{figure}[h!]
\begin{center}
\includegraphics[totalheight=7cm]{pulpo_shape_high_low_MC_TDAS.eps}%{pulpo_shape_low.eps}
\end{center}
\caption[Reconstructed pulpo low gain shape.]{Average normalized reconstructed high gain and low gain pulse shapes from a pulpo run. 
The FWHM of the low gain pulse is about 10 ns. The black line corresponds to the pulse shape implemented into the MC simulations.} 
\label{fig:pulpo_shape_low}
\end{figure}

Figure \ref{fig:shape_green_high} shows the normalized average reconstructed pulse shapes for green and UV calibration LED pulses~\cite{MAGIC-calibration}
 as well as the normalized average reconstructed pulse shape for cosmics events. The pulse shape of the UV calibration pulses is quite similar to the 
reconstructed pulse shape for cosmics events, both have a FWHM of about 6.3 ns. As air showers due to hadronic cosmic rays trigger the telescope 
much more frequently than gamma showers the reconstructed pulse shape of the cosmics events corresponds mainly to hadron induced showers. 
The pulse shape due to electromagnetic air showers might be slightly different. The pulse shape for green calibration LED pulses is wider 
and has a pronounced tail.

% The pulses shape has a FWHM of about 6.5 ns and a significant tail.


\begin{figure}[h!]
\begin{center}
\includegraphics[totalheight=7cm]{shape_green_UV_data_TDAS.eps}%{shape_green_high.eps}
\end{center}
\caption[Reconstructed green calibration high gain shape.]{Average reconstructed high gain pulse shape for one green LED calibration run. 
The FWHM is about 6.5 ns.} 
\label{fig:shape_green_high}
\end{figure}



\begin{itemize}
\item{Algorithm: overlay many events}
\item{Differences cosmics / calibration}
\item{Implementation / parameterization in the MC. 
\newline
\newline
\ldots {\it MAYBE, we should create MC calibration pulses for the subsequent studies }
\newline
\newline}
\end{itemize}


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