Index: trunk/MagicSoft/TDAS-Extractor/Algorithms.tex =================================================================== --- trunk/MagicSoft/TDAS-Extractor/Algorithms.tex (revision 5919) +++ trunk/MagicSoft/TDAS-Extractor/Algorithms.tex (revision 5925) @@ -325,5 +325,5 @@ the computed spline value at the arrival time position. Can be adjusted for the half-maximum time extraction type. -\item[LoGainStretch:\xspace] Can be adjusted to account for the bigger rise and fall time in the +\item[Low Gain Stretch:\xspace] Can be adjusted to account for the larger rise and fall times in the low-gain as compared to the high gain pulses (default: 1.5) \end{description} @@ -333,8 +333,8 @@ \includegraphics[width=0.49\linewidth]{MExtractTimeAndChargeSpline_23Led_Blue.eps} \caption[Sketch calculated arrival times MExtractTimeAndChargeSpline]{% -Sketch of the calculated arrival times for the extractor {\textit{MExtractTimeAndChargeSpline}} +Sketch of the calculated arrival times for the extractor {\textit{\bf MExtractTimeAndChargeSpline}} for two typical calibration pulses (pedestals have been subtracted) and a typical inner pixel. The extraction window sizes modify the position of the (amplitude-weighted) mean FADC-slices slightly. -The pulse would be shifted half a slice to the right for an outer pixels. } +The pulse would be shifted half a slice to the right for an outer pixel. } \label{fig:splinesketch} \end{figure} @@ -342,10 +342,11 @@ \subsubsection{Digital Filter} -This extractor is implemented in the MARS-class {\textit{MExtractTimeAndChargeDigitalFilter}}. - - -The goal of the digital filtering method \cite{OF94,OF77} is to optimally reconstruct the amplitude and time origin of a signal with a known signal shape from discrete measurements of the signal. Thereby the noise contribution to the amplitude reconstruction is minimized. - -For the digital filtering method two assumptions have to be made: +This extractor is implemented in the MARS-class {\textit{\bf MExtractTimeAndChargeDigitalFilter}}. + + +The goal of the digital filtering method \cite{OF94,OF77} is to optimally reconstruct the amplitude and time origin of a signal with a known signal shape +from discrete measurements of the signal. Thereby, the noise contribution to the amplitude reconstruction is minimized. + +For the digital filtering method, three assumptions have to be made: \begin{itemize} @@ -355,5 +356,10 @@ \end{itemize} -Let $g(t)$ be the normalized signal shape, $E$ the signal amplitude and $\tau$ the time shift of the physical signal from the predicted signal shape. Then the time dependence of the signal, $y(t)$, is given by: +\par +\ldots {\textit{\bf IS THIS TRUE FOR MAGIC???? }} \ldots +\par + +Let $g(t)$ be the normalized signal shape, $E$ the signal amplitude and $\tau$ the time shift +of the physical signal from the predicted signal shape. Then the time dependence of the signal, $y(t)$, is given by: \begin{equation} @@ -361,5 +367,5 @@ \end{equation} -where $b(t)$ is the time dependent noise distribution. For small time shifts $\tau$ (usually smaller than +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: @@ -376,8 +382,9 @@ \end{equation} -The correlation of the noise contributions at times $t_i$ and $t_j$ is expressed in the noise autocorrelation matrix $\boldsymbol{B}$: +The correlation of the noise contributions at times $t_i$ and $t_j$ can be expressed in the +noise autocorrelation matrix $\boldsymbol{B}$: \begin{equation} -\boldsymbol{B_{ij}} = \langle b_i b_j \rangle - \langle b_i \rangle \langle b_j +B_{ij} = \langle b_i b_j \rangle - \langle b_i \rangle \langle b_j \rangle \ . \label{eq:autocorr} @@ -385,14 +392,17 @@ %\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 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: \begin{eqnarray} -\chi^2(E, E\tau) &=& \sum_{i,j}(y_i-E g_i-E\tau \dot{g}_i) \boldsymbol{B}^{-1}_{ij} (y_j - E g_j-E\tau \dot{g}_j) \\ +\chi^2(E, E\tau) &=& \sum_{i,j}(y_i-E g_i-E\tau \dot{g}_i) B^{-1}_{ij} (y_j - E g_j-E\tau \dot{g}_j) \\ &=& (\boldsymbol{y} - E \boldsymbol{g} - E\tau \dot{\boldsymbol{g}})^T \boldsymbol{B}^{-1} (\boldsymbol{y} - E \boldsymbol{g}- E\tau \dot{\boldsymbol{g}}) \ , \end{eqnarray} -where the last expression is matricial. The minimum is obtained for: +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. +The minimum of $\chi^2$ is obtained for: \begin{equation} @@ -410,5 +420,5 @@ \end{eqnarray} -Solving these equations one gets the solutions: +Solving these equations one gets the following solutions: \begin{equation} @@ -422,6 +432,8 @@ 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, $w_{\text{amp},i}$, and time shift, $w_{\text{time},i}$. - +with the digital filtering weights for the amplitude, $w_{\text{amp}}(t)$, and time shift, $w_{\text{time}}(t)$. +Note the remaining time dependence of the two weights 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. +\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 @@ -449,5 +461,5 @@ \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}} \ , +\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} @@ -717,3 +729,4 @@ %%% TeX-master: "MAGIC_signal_reco" %%% TeX-master: "MAGIC_signal_reco" +%%% TeX-master: "MAGIC_signal_reco" %%% End: