Index: trunk/MagicSoft/TDAS-Extractor/Algorithms.tex
===================================================================
--- trunk/MagicSoft/TDAS-Extractor/Algorithms.tex	(revision 5375)
+++ trunk/MagicSoft/TDAS-Extractor/Algorithms.tex	(revision 5376)
@@ -232,16 +232,58 @@
 \end{equation}
 
-In the case of an uncorrelated noise with zero mean the noise autocorrelation matrix is:
-
-\begin{equation}
-\boldsymbol{B}_{ij}= \langle b_i b_j \rangle \delta_{ij} = \sigma^2(b_i) \ ,
-\end{equation}
-
-where $\sigma(b_i)$ is the standard deviation of the noise of the discrete measurements. Equation (\ref{of_noise}) than becomes:
-
-
-\begin{equation}
-\frac{\sigma^2(b_i)}{\sigma_E^2} = \sum_{i=1}^{n}{g_i^2} - \frac{\sum_{i=1}^{n}{g_i \dot{g}_i}}{\sum_{i=1}^{n}{\dot{g}_i^2}} \ .
-\end{equation}
+The expected contribution of the noise to the estimated timing ,$\sigma_{\tau}$, is:
+
+\begin{equation}\label{of_noise_time}
+E^2 \cdot \sigma_{\tau}^2=\frac{{\boldsymbol{g}}^T\boldsymbol{B}^{-1}{\boldsymbol{g}}}{(\boldsymbol{g}^T \boldsymbol{B}^{-1} \boldsymbol{g})(\dot{\boldsymbol{g}}^T\boldsymbol{B}^{-1}\dot{\boldsymbol{g}}) -(\dot{\boldsymbol{g}}^T\boldsymbol{B}^{-1}\boldsymbol{g})^2} \ .
+\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:
+
+\begin{equation}\label{of_noise}
+\sigma_E \approx 8.3 \ \mathrm{FADC\ counts} \qquad \sigma_{\tau}  \approx  \frac{6.5\ \Delta T_{\mathrm{FADC}}}{E\ /\ \mathrm{FADC\ counts}} \ ,
+\end{equation}
+
+where $\Delta T_{\mathrm{FADC}} = 3.33$ ns is the sampling intervall of the MAGIC FADCs.
+
+In the current implementation a two step procedure is applied to reconstruct the signal. The weight functions $w_{\mathrm{amp}}(t)$ and $w_{\mathrm{time}}(t)$ are computed numerically with a resolution of $1/10$ of an FADC slice. In the first step the quantities $e_{i_0}$ and $e\tau_{i_0}$ are computed:
+
+\begin{equation}
+e_{i_0}=\sum_{i=i_0}^{i=i_0+5} w_{\mathrm{amp}(t_i)}y(t_{i+i_0}) \qquad e\tau_{i_0}=\sum_{i=i_0}^{i=i_0+5} w(t_i)_{\mathrm{time}(t_i)}y(t_{i+i_0}) 
+\end{equation}
+
+for all possible signal start samples $i_0$. Let $i_0^*$ be the signal start sample with the largest $e_{i_0}$. Then in a seconst step the timing offset $\tau$ is calculated:
+
+\begin{equation}
+\tau=\frac{e\tau_{i_0^*}}{e_{i_0^*}}
+\end{equation}
+
+and the weigths iterated:
+
+\begin{equation}
+E_{i_0^*}=\sum_{i=i_0^*}^{i=i_0^*+5} w_{\mathrm{amp}(t_i - \tau)}y(t_{i+i_0^*}) \qquad E \tau_{i_0^*}=\sum_{i=i_0^*}^{i=i_0^*+5} w(t_i - \tau)_{\mathrm{time}(t_i)}y(t_{i+i_0^*}) \ .
+\end{equation}
+
+The reconstructed signal is then taken to be $E_{i_0^*}$ and the reconstructed arrival time $t_{\text{arrival}}$ is 
+
+\begin{equation}
+t_{\text{arrival}} = i_0^* + \tau_{i_0^*} + \tau \ .
+\end{equation}
+
+
+
+% This does not apply for MAGIC as the LONs are giving always a correlated noise (in addition to the artificial shaping)
+
+%In the case of an uncorrelated noise with zero mean the noise autocorrelation matrix is:
+
+%\begin{equation}
+%\boldsymbol{B}_{ij}= \langle b_i b_j \rangle \delta_{ij} = \sigma^2(b_i) \ ,
+%\end{equation}
+
+%where $\sigma(b_i)$ is the standard deviation of the noise of the discrete measurements. Equation (\ref{of_noise}) than becomes:
+
+
+%\begin{equation}
+%\frac{\sigma^2(b_i)}{\sigma_E^2} = \sum_{i=1}^{n}{g_i^2} - \frac{\sum_{i=1}^{n}{g_i \dot{g}_i}}{\sum_{i=1}^{n}{\dot{g}_i^2}} \ .
+%\end{equation}
 
 
Index: trunk/MagicSoft/TDAS-Extractor/Changelog
===================================================================
--- trunk/MagicSoft/TDAS-Extractor/Changelog	(revision 5375)
+++ trunk/MagicSoft/TDAS-Extractor/Changelog	(revision 5376)
@@ -19,4 +19,9 @@
 
                                                  -*-*- END OF LINE -*-*-
+
+2004/11/10: Hendrik Bartko
+  * Algorithms.tex: added the theoretical error on the arrival time 
+    determination, added information about the current implementation
+    of the digital filter
 
 2004/11/10: Markus Gaug