Changeset 6432 for trunk/MagicSoft/TDAS-Extractor/Algorithms.tex

Timestamp:

02/13/05 16:44:56 (20 years ago)

Author:

hbartko

Message:

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File:

• trunk/MagicSoft/TDAS-Extractor/Algorithms.tex (modified) (5 diffs)

trunk/MagicSoft/TDAS-Extractor/Algorithms.tex

@@ -355 +355 @@
 \item{The normalized signal shape has to be independent of the signal amplitude.}
 \item{The noise properties have to be independent of the signal amplitude.}
-\item{The noise auto-correlation matrix does not change its form significantly with time.}
+\item{The noise auto-correlation matrix does not change its form significantly with time and operation conditions.}
 \end{itemize}
 
-\par
-\ldots {\textit{\bf IS THIS TRUE FOR MAGIC???? }} \ldots
-\par
+
+The pulse shape is mainly determined by the artificial pulse stretching by about 6 ns on the receiver board. Thus the first assumption is hold. Also the second assumption is fullfilled: Signal and noise are independent and the measured pulse is the linear superposition of the signal and noise. The validity of the third assumption is discussed below, especially for diffent night sky background conditions.
 
 Let $g(t)$ be the normalized signal shape, $E$ the signal amplitude and $\tau$ the time shift 
@@ -407 +406 @@
 $\chi^2$ is a continuous function of $\tau$ and will have to be discretized itself for a 
 desired resolution.  
-$\chi^2$ is in principle independent from the noise auto-correlation matrix if always the correct noise level is calculated there.
-In our case however, we decided to use one same matrix $\boldsymbol{B}$ for all levels of night-sky background since increases 
-in the noise level lead only to a multiplicative factor for all matrix elements and thus do not affect the position of the minimum of $\chi^2$.
+$\chi^2$ is in principle independent of the noise level if alway the appropriate noise autocorrelation matrix is used. In our case however, we decided to use one matrix $\boldsymbol{B}$ for all levels of night-sky background. Changes in the noise level lead only to a multiplicative factor for all matrix elements and thus do not affect the position of the minimum of $\chi^2$.
 The minimum of $\chi^2$ is obtained for:
 
@@ -448 +445 @@
 
 Thus $\overline{E}$ and $\overline{E\tau}$ are given by a weighted sum of the discrete measurements $y_i$ 
-with the digital filtering weights for the amplitude, $w_{\text{amp}}(\tau)$, and time shift, $w_{\text{time}}(\tau)$
-where the time dependency gets discretized once again leading to a set of weights samples which themselves depend on the 
+with the digital filtering weights for the amplitude, $w_{\text{amp}}(\tau)$, and time shift, $w_{\text{time}}(\tau)$. The time dependency gets discretized once again leading to a set of weights samples which themselves depend on the 
 discretized time $\tau$.
 \par
-Note the remaining time dependency of the two weights samples which follow from the dependency of $\boldsymbol{g}$ and 
-$\dot{\boldsymbol{g}}$ on the position of the pulse with respect to the FADC bin positions.
+Note the remaining time dependency of the two weights samples. This follows from the dependency of $\boldsymbol{g}$ and 
+$\dot{\boldsymbol{g}}$ on the relative position of the signal pulse with respect to FADC slices positions.
 \par
 Because of the truncation of the Taylor series in equation (\ref{shape_taylor_approx}) the above results are 
@@ -485 +481 @@
 \end{equation}
 
-For the MAGIC signals, as implemented in the MC simulations, a pedestal RMS of a single FADC slice of 4 FADC counts introduces an error in the 
-reconstructed signal and time of:
+
+In the MAGIC MC simulations \cite{MC-Camera} a LONS rate of 0.13 photoelectrons per ns, an FADC gain of 7.8 FADC counts per photoelectron and an intrinsic FADC noise of 1.3 FADC counts per FADC slice is implemented. This simulates the night sky background conditions for an extragalactic source. This results in a noise of about 4 FADC counts per single FADC slice: $<b_i^2> \approx 4$~FADC counts. Using the digital filter with weights parameterized over 6 FADC slices ($i=1...5$) the error of the reconstructed signal and time is give by:
 
 \begin{equation}
@@ -493 +489 @@
 \end{equation}
 
-\par
-\ldots {\textit{\bf CALCULATE THESE NUMBERS FOR 6 SLICES! }} \ldots
-\par
-
-where $\Delta T_{\mathrm{FADC}} = 3.33$ ns is the sampling interval of the MAGIC FADCs.
-
+where $\Delta T_{\mathrm{FADC}} = 3.33$ ns is the sampling interval of the MAGIC FADCs. The error in the reconstructed signal correspons to about one photo electron. For signals of two photo electrons size the timing error is about 1 ns.
+
+%For the MAGIC signals, as implemented in the MC simulations \cite{MC-Camera}, a pedestal RMS of a single FADC slice of 6 FADC counts introduces an error in the reconstructed signal and time of:
 
 For an IACT there are two types of background noise. On the one hand, there is the constantly present