\section{Pulse Shape Reconstruction}

The clock of the FADC read-out is not synchronized with the trigger. Therefore the relative position of the recorded signal samples with respect to the position of the signal shape varies event by event. The time between the trigger decision and the first read-out sample is uniformly distributed in the range
$t_{\text{rel}} \in [0,3.33[$ ns and can be determined using the reconstructed arrival time $t_{\mathrm{arrival}}$.%directly by a time to digital converter (TDC) or 

The asynchronous sampling of the pulse shape allows to determine an average pulse shape from the recorded signal samples: The recorded signal samples are 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 the signal charge reconstruction.

Figure \ref{fig:pulpo_shape_high} show the average reconstructed signal of a fast pulser in the so called ``pulpo setup''. 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 50 ns with respect to the high gain pulse.

Figures \ref{fig:pulpo_shape_low} show the average normalized reconstructed pulse shapes for pulse generator pulses (pulpo setup) 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 electric 50 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 shape 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 6.5 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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