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01/29/05 15:01:10 (20 years ago)
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gaug
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  • trunk/MagicSoft/TDAS-Extractor/Algorithms.tex

    r5993 r6113  
    1616\begin{figure}[htp]
    1717\includegraphics[width=0.99\linewidth]{ExtractorClasses.eps}
    18 \caption{Sketch of the inheritances of three examplary MARS signal extractor classes: MExtractFixedWindow, MExtractTimeFastSpline and MExtractTimeAndChargeDigitalFilter}
     18\caption{Sketch of the inheritances of three examplary MARS signal extractor classes:
     19MExtractFixedWindow, MExtractTimeFastSpline and MExtractTimeAndChargeDigitalFilter}
    1920\label{fig:extractorclasses}
    2021\end{figure}
    … …  
    402403\end{eqnarray}
    403404
    404 where the last expression is matricial. $\chi^2$ is a continuous function of $\tau$ and will have to be discretized itself for a
    405 desired resolution.  $\chi^2$ is also proportional to the auto noise-correlation matrix where increases in the noise level lead to
    406 a multiplicative factor for all matrix elements and thus do not affect the position of the maximum of $\chi^2$.
     405where the last expression is matricial.
     406$\chi^2$ is a continuous function of $\tau$ and will have to be discretized itself for a
     407desired resolution. 
     408$\chi^2$ is in principle independent from the noise auto-correlation matrix if always the correct noise level is calculated there.
     409In our case however, we decided to use one same matrix $\boldsymbol{B}$ for all levels of night-sky background since increases
     410in the noise level lead only to a multiplicative factor for all matrix elements and thus do not affect the position of the minimum of $\chi^2$.
    407411The minimum of $\chi^2$ is obtained for:
    408412
    … …  
    417421
    418422\begin{eqnarray}
    419 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}
     4230&=&-\boldsymbol{g}^T\boldsymbol{B}^{-1}\boldsymbol{y}
     424        +\boldsymbol{g}^T\boldsymbol{B}^{-1}\boldsymbol{g}\overline{E}
     425        +\boldsymbol{g}^T\boldsymbol{B}^{-1}\dot{\boldsymbol{g}}\overline{E\tau}
    420426\\
    421 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} \ .
     4270&=&-\dot{\boldsymbol{g}}^T\boldsymbol{B}^{-1}\boldsymbol{y}
     428        +\dot{\boldsymbol{g}}^T\boldsymbol{B}^{-1}\boldsymbol{g}\overline{E}
     429        +\dot{\boldsymbol{g}}^T\boldsymbol{B}^{-1}\dot{\boldsymbol{g}}\overline{E\tau} \ .
    422430\end{eqnarray}
    423431
    … …  
    425433
    426434\begin{equation}
    427 \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 } \ ,
    428 \end{equation}
    429 
    430 \begin{equation}
    431 \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 } \ .
    432 \end{equation}
    433 
     435\overline{E}(\tau) = \boldsymbol{w}_{\text{amp}}^T (\tau)\boldsymbol{y} \quad \mathrm{with} \quad
     436        \boldsymbol{w}_{\text{amp}}
     437        = \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}}} 
     438        {(\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 } \ ,
     439\end{equation}
     440
     441\begin{equation}
     442\overline{E\tau}(\tau)= \boldsymbol{w}_{\text{time}}^T(\tau) \boldsymbol{y} \quad
     443        \mathrm{with} \quad \boldsymbol{w}_{\text{time}}
     444        = \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}}} 
     445        {(\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 } \ .
     446\end{equation}
    434447
    435448Thus $\overline{E}$ and $\overline{E\tau}$ are given by a weighted sum of the discrete measurements $y_i$
    436449with the digital filtering weights for the amplitude, $w_{\text{amp}}(\tau)$, and time shift, $w_{\text{time}}(\tau)$
    437 where the time dependency gets discretized once again leading to a set of weights samples depending on the
     450where the time dependency gets discretized once again leading to a set of weights samples which themselves depend on the
    438451discretized time $\tau$.
    439 
    440 \par
    441 \ldots {\textit{\bf IS THIS CORRECT LIKE THIS???}} \ldots
    442 \par
    443 
    444 Note the remaining time dependence of the two weights which follow from the dependency of $\boldsymbol{g}$ and
     452\par
     453Note the remaining time dependency of the two weights samples which follow from the dependency of $\boldsymbol{g}$ and
    445454$\dot{\boldsymbol{g}}$ on the position of the pulse with respect to the FADC bin positions.
    446455\par
    … …  
    452461
    453462\begin{equation}
    454 \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\} \ .
     463\left(\boldsymbol{V}^{-1}\right)_{i,j}
     464        =\frac{1}{2}\left(\frac{\partial^2 \chi^2(E, E\tau)}{\partial \alpha_i \partial \alpha_j} \right) \quad
     465        \text{with} \quad \alpha_i,\alpha_j \in \{E, E\tau\} \ .
    455466\end{equation}
    456467
    … …  
    458469
    459470\begin{equation}\label{of_noise}
    460 \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} \ .
     471\sigma_E^2=\boldsymbol{V}_{E,E}
     472        =\frac{\dot{\boldsymbol{g}}^T\boldsymbol{B}^{-1}\dot{\boldsymbol{g}}}
     473        {(\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} \ .
    461474\end{equation}
    462475
    … …  
    464477
    465478\begin{equation}\label{of_noise_time}
    466 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} \ .
     479E^2 \cdot \sigma_{\tau}^2=\boldsymbol{V}_{E\tau,E\tau}
     480        =\frac{{\boldsymbol{g}}^T\boldsymbol{B}^{-1}{\boldsymbol{g}}}
     481        {(\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} \ .
    467482\end{equation}
    468483
    … …  
    506521\includegraphics[totalheight=7cm]{noise_autocorr_AB_36038_TDAS.eps}
    507522\end{center}
    508 \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}
     523\caption[Noise autocorrelation.]{Noise autocorrelation matrix for open camera including the noise due to night sky background fluctuations.}
     524        \label{fig:noise_autocorr_AB_36038_TDAS}
    509525\end{figure}
    510526
    … …  
    546562\end{figure}
    547563
    548 
    549 
    550 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:
     564In the current implementation a two step procedure is applied to reconstruct the signal. The weight functions $w_{\mathrm{amp}}(t)$
     565and $w_{\mathrm{time}}(t)$ are computed numerically with a resolution of $1/10$ of an FADC slice.
     566In the first step the quantities $e_{i_0}$ and $e\tau_{i_0}$ are computed using a window of $n$ slices:
    551567
    552568\begin{equation}
    … …  
    554570\end{equation}
    555571
    556 for all possible signal start slices $i_0$. Let $i_0^*$ be the signal start slice with the largest $e_{i_0}$. Then in a second step the timing offset $\tau$ is calculated:
     572for all possible signal start slices $i_0$. Let $i_0^*$ be the signal start slice with the largest $e_{i_0}$.
     573Then in a second step the timing offset $\tau$ is calculated:
    557574
    558575\begin{equation}
    … …  
    563580
    564581\begin{equation}
    565 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^*}) \ .
     582E=\sum_{i=i_0^*}^{i_0^*+n-1} w_{\mathrm{amp}}(t_i - \tau)y(t_{i+i_0^*}) \qquad
     583        E \theta=\sum_{i=i_0^*}^{i_0^*+n-1} w_{\mathrm{time}}(t_i - \tau)y(t_{i+i_0^*}) \ .
    566584\end{equation}
    567585
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