Index: trunk/MagicSoft/TDAS-Extractor/Algorithms.tex
===================================================================
--- trunk/MagicSoft/TDAS-Extractor/Algorithms.tex	(revision 6660)
+++ trunk/MagicSoft/TDAS-Extractor/Algorithms.tex	(revision 6661)
@@ -369,5 +369,5 @@
 where $b(t)$ is the time-dependent noise contribution. For small time shifts $\tau$ (usually smaller than 
 one FADC slice width), 
-the time dependence can be linearized by the use of a Taylor expansion:
+the time dependence can be linearized:
 
 \begin{equation} \label{shape_taylor_approx}
@@ -392,7 +392,7 @@
 %\equiv \langle b_i b_j \rangle with $\langle b_i \rangle = 0$.
 
-The signal amplitude $E$, and the product of amplitude and time shift $E \tau$, can be estimated from the given set of
-measurements $\boldsymbol{y} = (y_1, ... ,y_n)$ by minimizing the excess noise contribution with respect to the known noise 
-auto-correlation:
+The signal amplitude $E$, and the product $E \tau$ of amplitude and time shift, can be estimated from the given set of
+measurements $\boldsymbol{y} = (y_1, ... ,y_n)$ by minimizing the deviation of the measured FADC slice contents from the 
+known pulse shape with respect to the known noise auto-correlation:
 
 \begin{eqnarray}
@@ -402,8 +402,10 @@
 \end{eqnarray}
 
-where the last expression is matricial. 
-$\chi^2$ is a continuous function of $\tau$ and will have to be discretized itself for a 
-desired resolution.  
-$\chi^2$ is in principle independent 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 $\chi^2$.
+where the last expression uses the matrix formalism. 
+$\chi^2$ is a continuous function of $\tau$ and will have to be discretized itself for numerical computation applications.
+$\chi^2$ is in principle independent of the noise level if always 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:
 
@@ -444,12 +446,14 @@
 
 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, $w_{\text{time}}(\tau)$. The time dependency gets discretized once again leading to a set of weights samples which themselves depend on the 
+with the weights for the amplitude, $w_{\text{amp}}(\tau)$, and time shift, $w_{\text{time}}(\tau)$. 
+\par
+The time dependence gets discretized once again leading to a set of weight samples which themselves depend on the 
 discretized time $\tau$.
 \par
-Note the remaining time dependency of the two weights samples. This follows from the dependency of $\boldsymbol{g}$ and 
+Note the remaining time dependency of the two weight samples. This follows from the dependence 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 
-only valid for vanishing time offsets $\tau$. For non-zero time offsets, one has to iterate the problem using 
+only valid for vanishing time offsets $\tau$. For larger time offsets, one has to iterate the problem using 
 the time shifted signal shape $g(t-\tau)$.
 
@@ -474,5 +478,5 @@
 
 \begin{equation}
-E^2 \cdot \sigma_{\tau}^2=\boldsymbol{V}_{E\tau,E\tau}
+E^2 \cdot \sigma_{\tau}^2 < \sigma^2_{E \tau} =\boldsymbol{V}_{E\tau,E\tau}
         =\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} \ .
@@ -480,6 +484,7 @@
 \end{equation}
 
-
-In the MAGIC MC simulations~\cite{MC-Camera}, an night-sky background rate of 0.13 photoelectrons per ns, 
+Both equations~\ref{eq:of_noise} and~\ref{eq:of_noise_time} are independent of the signal amplitude.
+\par
+In the MAGIC MC simulations~\cite{MC-Camera}, a night-sky background rate of 0.13 photoelectrons per ns, 
 an FADC gain of 7.8 FADC counts per photo-electron and an intrinsic FADC noise of 1.3 FADC counts 
 per FADC slice is implemented. 
@@ -487,5 +492,5 @@
 in a noise contribution of about 4 FADC counts per single FADC slice: 
 $\sqrt{B_{ii}} \approx 4$~FADC counts. 
-Using the digital filter with weights parameterized over 6 FADC slices ($i=0...5$) the errors of the 
+Using the digital filter with weights determined for 6~FADC slices ($i=0...5$) the errors of the 
 reconstructed signal and time amount to:
 
@@ -516,6 +521,6 @@
 depend on the pulse shape, the derivative of the pulse shape and the noise autocorrelation. 
 In the high gain samples, the correlated night sky background noise dominates over 
-the white electronics noise. Thus, different noise levels cause the elements of the noise autocorrelation 
-matrix to change by a same factor, which cancels out in the weights calculation. Figure~\ref{fig:noise_autocorr_allpixels} 
+the white electronics noise. As a consequence, different noise levels cause the elements of the noise autocorrelation 
+matrix to change by the same factor, which cancels out in the weights calculation. Figure~\ref{fig:noise_autocorr_allpixels} 
 shows the noise autocorrelation matrix for two different levels of night sky background (top) and the ratio between the 
 corresponding elements of both (bottom). The central regions of $\pm$3 FADC slices around the diagonal (which is used to 
@@ -523,5 +528,6 @@
 Thus, the weights are to a reasonable approximation independent of the night sky background noise level in the high gain.
 \par
-In the low gain samples the correlated noise from the LONS is in the same order of magnitude as the white electronics and digitization noise. 
+In the low gain samples the correlated noise of the LONS is in the same order of magnitude as the white electronics 
+and digitization noise. 
 Moreover, the noise autocorrelation for the low gain samples cannot be determined directly from the data. The low gain is only switched on 
 if the pulse exceeds a preset threshold. There are no pedestals in the low gain available. Thus the noise auto-correlation determined from 
@@ -554,5 +560,9 @@
 \end{figure}
 
-Using the average reconstructed pulpo pulse shapes, as shown in figure \ref{fig:pulse_shapes}, and the reconstructed noise autocorrelation matrices from a pedestal runs with random triggers, the digital filter weights are computed. As the pulse shapes in the high and low gain and for cosmics, calibration and pulpo events are somewhat different, dedicated digital filter weights are computed for these event classes. Also filter weights optimized for MC simulations are calculated. High/low gain filter weights are computed for the following event classes:
+Using the average reconstructed pulpo pulse shapes, as shown in figure \ref{fig:pulse_shapes}, and the reconstructed noise 
+autocorrelation matrices from pedestal runs with random triggers, the digital filter weights are computed. 
+As the pulse shapes in the high and low gain and for cosmics, calibration and pulpo events are somewhat different, 
+dedicated digital filter weights are computed for these event classes. Also filter weights optimized for MC simulations are calculated. 
+High/low gain filter weights are computed for the following event classes:
 
 \begin{enumerate}
@@ -583,7 +593,8 @@
 
 
- Figures \ref{fig:w_time_MC_input_TDAS} and \ref{fig:w_amp_MC_input_TDAS} show the amplitude and timing weights for the MC pulse shape. The first weight $w_{\mathrm{amp/time}}(t_0)$ is plotted as a function of the relative time $t_{\text{rel}}$ the trigger and the 
-FADC clock in the range $[-0.5,0.5[ \ T_{\text{ADC}}$, the second weight in the range $[0.5,1.5[ \ T_{\text{ADC}}$ and so on. A binning resolution 
-of $0.1\,T_{\text{ADC}}$ has been chosen. 
+ Figures \ref{fig:w_time_MC_input_TDAS} and \ref{fig:w_amp_MC_input_TDAS} show the amplitude and timing weights for the MC pulse shape. 
+The first weight $w_{\mathrm{amp/time}}(t_0)$ is plotted as a function of the relative time $t_{\text{rel}}$ between the trigger and the 
+FADC clock in the range $[-0.5,0.5[ \ T_{\text{ADC}}$, the second weight in the range $[0.5,1.5[ \ T_{\text{ADC}}$ and so on. 
+A binning resolution of $0.1\,T_{\text{ADC}}$ has been chosen. 
 
 
@@ -610,5 +621,5 @@
 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 using a window of $n$ slices:
+In the first step the quantities $e_{i_0}$ and $(e\tau)_{i_0}$ are computed using a window of $n$ slices:
 
 \begin{equation}
@@ -616,5 +627,5 @@
 \end{equation}
 
-for all possible signal start slices $i_0$. Let $i_0^*$ be the signal start slice with the largest $e_{i_0}$. 
+for all possible signal start slices $i_0$. Let $i_0^*$ be the signal start slice yielding the largest $e_{i_0}$. 
 Then in a second step the timing offset $\tau$ is calculated:
 
@@ -624,5 +635,5 @@
 \end{equation}
 
-and the weights iterated:
+Using this value of $\tau$, another iteration is performed:
 
 \begin{equation}
@@ -639,5 +650,6 @@
 
 
-Figure \ref{fig:amp_sliding} shows the result of the applied amplitude and time weights to the recorded FADC time slices of one simulated MC pulse. The left plot shows the result of the applied amplitude weights 
+Figure \ref{fig:amp_sliding} shows the result of the applied amplitude and time weights to the recorded FADC time slices of one 
+simulated MC pulse. The left plot displayes the result of the applied amplitude weights 
 $e(t_0)=\sum_{i=0}^{i=n-1} w_{\mathrm{amp}}(t_0+i \cdot T_{\text{ADC}})y(t_0+i \cdot T_{\text{ADC}})$ and 
 the right plot shows the result of the applied timing weights 
@@ -669,13 +681,13 @@
 one simulated MC pulse. The left plot shows the result of the applied amplitude weights 
 $e(t_0)=\sum_{i=0}^{i=n-1} w_{\mathrm{amp}}(t_0+i \cdot T_{\text{ADC}})y(t_0+i \cdot T_{\text{ADC}})$ and 
-the right plot shows the result of the applied timing weights 
+the right plot displayes the result of the applied timing weights 
 $e\tau(t_0)=\sum_{i=0}^{i=n-1} w_{\mathrm{time}}(t_0+i \cdot T_{\text{ADC}})y(t_0+i \cdot T_{\text{ADC}})$ as a function of the time shift $t_0$.} 
 \label{fig:amp_sliding}
 \end{figure}
 
-Figure \ref{fig:shape_fit_TDAS} shows the simulated signal pulse shape of a typical MC event together with the simulated FADC slices of the signal pulse plus noise. The digital filter has been applied to reconstruct the signal size and timing. Using this information together with the average normalized MC pulse shape the simulated signal pulse shape is reconstructed and shown as well.
-
-
-
+Figure \ref{fig:shape_fit_TDAS} shows the signal pulse shape of a typical MC event together with the simulated FADC slices 
+of the signal pulse plus noise. The digital filter has been applied to reconstruct the signal size and timing. 
+Using this information together with the average normalized MC pulse shape the simulated signal pulse shape is reconstructed 
+and shown as well.