| 1 | \section{Criteria for the Optimal Signal Extraction}
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| 2 |
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| 3 | The goal for the optimal signal reconstruction algorithm is to compute an unbiased estimate of the strength and arrival time of the
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| 4 | Cherenkov signal with the highest possible resolution for all signal intensities. The MAGIC telescope has been optimized to
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| 5 | lower the energy treshold of observation in any respect. Particularly the choice for an FADC system has been made with an eye on the
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| 6 | possibility to extract the smallest possible signals from air showers. It would be inconsequent not to continue the optimization procedure
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| 7 | in the signal extraction algorithms and the subsequent image cleaning.
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| 8 | \par
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| 9 | In the image analysis, one hake the decision whether the extracted signal of a certain pixel is considered as signal or background.
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| 10 | Those considered as signal are further used to compute the image parameters while the background ones are simply rejected. The calculation
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| 11 | of the second moments of the image ``ellipse'' usually fails when applied to un-cleaned images, therefore the decision is yes or no. Moreover,
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| 12 | already low contributions of mis-estimated background can degrade the resolution of the image parameters considerably. If one wants to
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| 13 | lower the threshold for signal recognition, it is therefore mandatory to increase the efficiency with which the background is recognized as
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| 14 | such. If the background resolution is bad, the signal threshold goes up and vice versa.
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| 15 | \par
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| 16 | The algorithm must be stable with respect to changes
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| 17 | in observation conditions and background levels and between signals induced from gamma or hadronic showers or from muons.
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| 18 | The reconstructed signal shall be proportional to the total integrated charge in the FADCs due to the PMT pulse from the Cherenkov signal.
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| 19 |
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| 20 | Also the needed computing time is of concern.
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| 21 |
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| 22 | \subsection{Bias and Mean-squared Error}
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| 23 |
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| 24 | Consider a large number of same signals $S$. By applying a signal extractor
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| 25 | we obtain a distribution of estimated signals $\widehat{S}$ (for fixed $S$ and
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| 26 | fixed background fluctuations $BG$). The distribution of the quantity
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| 27 |
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| 28 | \begin{equation}
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| 29 | X = \widehat{S}-S
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| 30 | \end{equation}
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| 31 |
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| 32 | has the mean $B$ and the Variance $MSE$ defined as:
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| 33 |
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| 34 | \begin{eqnarray}
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| 35 | B \ \ \ \ = \ \ \ \ \ \ <X> \ \ \ \ \ &=& \ \ <\widehat{S}> \ -\ S\\
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| 36 | R \ \ \ \ = \ <(X-B)^2> &=& \ Var[\widehat{S}]\\
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| 37 | MSE \ = \ \ \ \ \ <X^2> \ \ \ \ &=& \ Var[\widehat{S}] +\ B^2
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| 38 | \end{eqnarray}
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| 39 |
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| 40 | The parameter $B$ is also called the {\textit{\bf BIAS}} of the estimator and $MSE$
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| 41 | the {\textit{\bf MEAN-SQUARED ERROR}} which combines the variance of $\widehat{S}$ and
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| 42 | the bias. Both depend generally on the size of $S$ and the background fluctuations $BG$,
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| 43 | thus: $B = B(S,BG)$ and $MSE = MSE(S,BG)$.
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| 44 |
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| 45 | \par
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| 46 | Usually, one measures easily the parameter $R$, but needs the $MSE$ for statistical analysis (e.g.
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| 47 | in the image cleaning).
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| 48 | However, only in case of a vanishing bias $B$, the two numbers are equal. Otherwise,
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| 49 | the bias $B$ has to be known beforehand. Note that every sliding window extractor has a
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| 50 | bias, especially at low or vanishing signals $S$.
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| 51 |
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| 52 | \subsection{Linearity}
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| 53 | \ldots {\textit The Nuria plots ... }
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| 54 |
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| 55 | \subsection{Low Gain Extraction}
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| 56 | \ldots {\textit The stability of the low-gain extraction w.r.t. the high-gain extraction}
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| 57 |
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| 58 |
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| 59 | \subsection{Stability}
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| 60 | \ldots {\textit The stability of an extractor to slightly varying pulse shapes is examined. }
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| 61 |
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| 62 |
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| 63 | \subsection{Treatment of Calibration Pulses}
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| 64 |
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| 65 |
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| 66 |
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| 67 | \subsection{Applicability for Different Sampling Speeds / No Pulse Shaping.}
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| 68 | 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.
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| 69 |
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| 70 | 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.
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| 71 |
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| 72 |
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| 73 | \ldots {\textit Some comments by Hendrik ...}
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| 74 |
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| 75 | \subsection{CPU Requirements}
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| 76 | \ldots {\textit The needed CPU time for each extractor}
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| 77 |
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| 78 |
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| 79 |
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| 80 |
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| 81 | \subsection{Pulpo Pulses}
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| 82 | \subsection{Cosmics Data?}
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| 83 | The results of this subsection are based on the following runs taken
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| 84 | on the 21st of September 2004.
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| 85 | \begin{itemize}
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| 86 | \item{Run 39000}: OffCrab11 at 19.1 degrees zenith angle and 106.2
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| 87 | azimuth.
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| 88 | \item{Run 39182}: CrabNebula at 19.0 degrees zenith angle and 106.0 azimuth.
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| 89 | \end{itemize}
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| 90 |
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| 91 |
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| 92 | %%% Local Variables:
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| 93 | %%% mode: latex
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| 94 | %%% TeX-master: "MAGIC_signal_reco"
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| 95 | %%% End:
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