Changeset 6268

Timestamp:

02/04/05 19:48:06 (20 years ago)

Author:

gaug

Message:

*** empty log message ***

Location:

trunk/MagicSoft/TDAS-Extractor

Files:

• Algorithms.tex (modified) (12 diffs)
• noise_38995_smallNSB_all396.eps (added)

trunk/MagicSoft/TDAS-Extractor/Algorithms.tex

@@ -299 +299 @@
 possibilities are offered:
 
-\begin{enumerate}
+\begin{description}
 \item[Extraction Type Amplitude:\xspace] The amplitude of the spline maximum is taken as charge signal
 and the (precisee) position of the maximum is returned as arrival time. This type is faster, since it 
-performs not spline intergraion . 
+performs not spline intergraion. 
 \item[Extraction Type Integral:\xspace] The integrated spline between maximum position minus 
 rise time (default: 1.5 slices) and maximum position plus fall time (default: 4.5 slices) 
@@ -310 +310 @@
 is slower, but yields more precise results (see section~\ref{sec:performance}) . 
 The charge integration resolution is set to 0.1 FADC slices.
-\end{enumerate}
+\end{description}
 
 The following adjustable parameters have to be set from outside: 
@@ -422 +422 @@
 \begin{eqnarray}
 0&=&-\boldsymbol{g}^T\boldsymbol{B}^{-1}\boldsymbol{y}
-        +\boldsymbol{g}^T\boldsymbol{B}^{-1}\boldsymbol{g}\overline{E}
-        +\boldsymbol{g}^T\boldsymbol{B}^{-1}\dot{\boldsymbol{g}}\overline{E\tau} 
+        +\boldsymbol{g}^T\boldsymbol{B}^{-1}\boldsymbol{g}\overline{E}
+        +\boldsymbol{g}^T\boldsymbol{B}^{-1}\dot{\boldsymbol{g}}\overline{E\tau} 
 \\
 0&=&-\dot{\boldsymbol{g}}^T\boldsymbol{B}^{-1}\boldsymbol{y}
-        +\dot{\boldsymbol{g}}^T\boldsymbol{B}^{-1}\boldsymbol{g}\overline{E}
-        +\dot{\boldsymbol{g}}^T\boldsymbol{B}^{-1}\dot{\boldsymbol{g}}\overline{E\tau} \ .
+        +\dot{\boldsymbol{g}}^T\boldsymbol{B}^{-1}\boldsymbol{g}\overline{E}
+        +\dot{\boldsymbol{g}}^T\boldsymbol{B}^{-1}\dot{\boldsymbol{g}}\overline{E\tau} \ .
 \end{eqnarray}
 
@@ -434 +434 @@
 \begin{equation}
 \overline{E}(\tau) = \boldsymbol{w}_{\text{amp}}^T (\tau)\boldsymbol{y} \quad \mathrm{with} \quad 
-        \boldsymbol{w}_{\text{amp}} 
-        = \frac{ (\dot{\boldsymbol{g}}^T\boldsymbol{B}^{-1}\dot{\boldsymbol{g}}) \boldsymbol{B}^{-1} \boldsymbol{g} -(\boldsymbol{g}^T\boldsymbol{B}^{-1}\dot{\boldsymbol{g}})  \boldsymbol{B}^{-1} \dot{\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 } \ ,
+        \boldsymbol{w}_{\text{amp}} 
+        = \frac{ (\dot{\boldsymbol{g}}^T\boldsymbol{B}^{-1}\dot{\boldsymbol{g}}) \boldsymbol{B}^{-1} \boldsymbol{g} -(\boldsymbol{g}^T\boldsymbol{B}^{-1}\dot{\boldsymbol{g}})  \boldsymbol{B}^{-1} \dot{\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}
 
 \begin{equation}
 \overline{E\tau}(\tau)= \boldsymbol{w}_{\text{time}}^T(\tau) \boldsymbol{y} \quad 
-        \mathrm{with} \quad \boldsymbol{w}_{\text{time}} 
-        = \frac{ ({\boldsymbol{g}}^T\boldsymbol{B}^{-1}{\boldsymbol{g}}) \boldsymbol{B}^{-1} \dot{\boldsymbol{g}} -(\boldsymbol{g}^T\boldsymbol{B}^{-1}\dot{\boldsymbol{g}})  \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 } \ .
+        \mathrm{with} \quad \boldsymbol{w}_{\text{time}} 
+        = \frac{ ({\boldsymbol{g}}^T\boldsymbol{B}^{-1}{\boldsymbol{g}}) \boldsymbol{B}^{-1} \dot{\boldsymbol{g}} -(\boldsymbol{g}^T\boldsymbol{B}^{-1}\dot{\boldsymbol{g}})  \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}
 
@@ -462 +462 @@
 \begin{equation}
 \left(\boldsymbol{V}^{-1}\right)_{i,j}
-        =\frac{1}{2}\left(\frac{\partial^2 \chi^2(E, E\tau)}{\partial \alpha_i \partial \alpha_j} \right) \quad 
-        \text{with} \quad \alpha_i,\alpha_j \in \{E, E\tau\} \ .
+        =\frac{1}{2}\left(\frac{\partial^2 \chi^2(E, E\tau)}{\partial \alpha_i \partial \alpha_j} \right) \quad 
+        \text{with} \quad \alpha_i,\alpha_j \in \{E, E\tau\} \ .
 \end{equation}
 
@@ -470 +470 @@
 \begin{equation}\label{of_noise}
 \sigma_E^2=\boldsymbol{V}_{E,E}
-        =\frac{\dot{\boldsymbol{g}}^T\boldsymbol{B}^{-1}\dot{\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} \ .
+        =\frac{\dot{\boldsymbol{g}}^T\boldsymbol{B}^{-1}\dot{\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}
 
@@ -478 +478 @@
 \begin{equation}\label{of_noise_time}
 E^2 \cdot \sigma_{\tau}^2=\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} \ .
+        =\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}
 
@@ -496 +496 @@
 
 
-For an IACT there are two types of background noise. On the one hand, there is the constantly present electronics noise, 
-on the other hand, the light of the night sky introduces a sizeable background noise to the measurement of Cherenkov photons from air showers.
-
-The electronics noise is largely white, uncorrelated in time. The noise from the night sky background photons is the superposition of the 
-detector response to single photo electrons following a Poisson distribution in time. Figure \ref{fig:noise_autocorr_AB_36038_TDAS} shows the noise 
-autocorrelation matrix for an open camera. The large noise autocorrelation in time of the current FADC system is due to the pulse shaping with a 
-shaping constant of 6 ns. 
+For an IACT there are two types of background noise. On the one hand, there is the constantly present 
+electronics noise, 
+on the other hand, the light of the night sky introduces a sizeable background noise to the measurement of 
+Cherenkov photons from air showers.
+
+The electronics noise is largely white, uncorrelated in time. The noise from the night sky background photons 
+is the superposition of the 
+detector response to single photo electrons following a Poisson distribution in time. 
+Figure \ref{fig:noise_autocorr_allpixels} shows the noise 
+autocorrelation matrix for an open camera. The large noise autocorrelation in time of the current FADC 
+system is due to the pulse shaping with a shaping constant of 6 ns. 
 
 In general, the amplitude and time weights, $\boldsymbol{w}_{\text{amp}}$ and $\boldsymbol{w}_{\text{time}}$, depend on the pulse shape, the 
@@ -521 +525 @@
 \includegraphics[totalheight=7cm]{noise_autocorr_AB_36038_TDAS.eps}
 \end{center}
-\caption[Noise autocorrelation.]{Noise autocorrelation matrix for open camera including the noise due to night sky background fluctuations.} 
-        \label{fig:noise_autocorr_AB_36038_TDAS}
-\end{figure}
-
-
+\caption[Noise autocorrelation one pixel.]{Noise autocorrelation 
+matrix for open camera including the noise due to night sky background fluctuations for one single 
+pixel.} 
+\label{fig:noise_autocorr_1pix}
+\end{figure}
+
+\begin{figure}[htp]
+\begin{center}
+\includegraphics[totalheight=7cm]{noise_38995_smallNSB_all396.eps}
+\includegraphics[totalheight=7cm]{noise_39258_largeNSB_all396.eps}
+\includegraphics[totalheight=7cm]{noise_small_over_large.eps}
+\end{center}
+\caption[Noise autocorrelation average all pixels.]{Noise autocorrelation 
+matrix $\boldsymbol{B}$ for open camera and averaged over all pixels. The top figure shows $\boldsymbol{B}$ 
+obtained with camera pointing off the galactic plane (and low night sky background fluctuations). 
+The central figure shows $\boldsymbol{B}$ with the camera pointing into the galactic plane 
+(high night sky background) and the 
+bottom plot shows the ratio between both. One can see that the entries of $\boldsymbol{B}$ do not 
+simply scale with the amount of night sky background.} 
+\label{fig:noise_autocorr_allpixels}
+\end{figure}
 
 Using the average reconstructed pulpo pulse shape, as shown in figure \ref{fig:pulpo_shape_low}, and the 
@@ -549 +569 @@
 used in the MC simulations. The first weight $w_{\mathrm{time}}(t_0)$ 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.} \label{fig:w_time_MC_input_TDAS}
+of $0.1\,T_{\text{ADC}}$ has been chosen.} \label{fig:w_time_MC_input_TDAS}
 \end{figure}
 
@@ -581 +601 @@
 \begin{equation}
 E=\sum_{i=i_0^*}^{i_0^*+n-1} w_{\mathrm{amp}}(t_i - \tau)y(t_{i+i_0^*}) \qquad 
-        E \theta=\sum_{i=i_0^*}^{i_0^*+n-1} w_{\mathrm{time}}(t_i - \tau)y(t_{i+i_0^*}) \ .
+        E \theta=\sum_{i=i_0^*}^{i_0^*+n-1} w_{\mathrm{time}}(t_i - \tau)y(t_{i+i_0^*}) \ .
 \end{equation}
 
@@ -808 +828 @@
 %%% TeX-master: "MAGIC_signal_reco"
 %%% TeX-master: "MAGIC_signal_reco"
+%%% TeX-master: "MAGIC_signal_reco.te"
 %%% End: