\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 equally distributed in the range
$t_{\text{rel}} \in [0,3.33[$ ns and can be determined directly by a
time to digital converter (TDC) or using the reconstructed arrival time $t_{\mathrm{arrival}}$.

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.

Figures \ref{fig:pulpo_shape_high} and \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 with respect to the high gain is due to the 50 ns on board delay line of the MAGIC receiver boards.


\begin{figure}[h!]
\begin{center}
\includegraphics[totalheight=7cm]{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_low.eps}
\end{center}
\caption[Reconstructed pulpo low gain shape.]{Average reconstructed high gain pulse shape from a pulpo run. The FWHM is about 10 ns.} \label{fig:pulpo_shape_low}
\end{figure}

Figure \ref{fig:shape_green_high} shows the average reconstructed pulse shape for green calibration LED pulses. 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_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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