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01/21/05 11:12:16 (20 years ago)
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gaug
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  • trunk/MagicSoft/TDAS-Extractor/Algorithms.tex

    r5919 r5925  
    325325the computed spline value at the arrival time position. Can be adjusted for the half-maximum time extraction
    326326type.
    327 \item[LoGainStretch:\xspace] Can be adjusted to account for the bigger rise and fall time in the
     327\item[Low Gain Stretch:\xspace] Can be adjusted to account for the larger rise and fall times in the
    328328low-gain as compared to the high gain pulses (default: 1.5)
    329329\end{description}
    … …  
    333333  \includegraphics[width=0.49\linewidth]{MExtractTimeAndChargeSpline_23Led_Blue.eps}
    334334\caption[Sketch calculated arrival times MExtractTimeAndChargeSpline]{%
    335 Sketch of the calculated arrival times for the extractor {\textit{MExtractTimeAndChargeSpline}}
     335Sketch of the calculated arrival times for the extractor {\textit{\bf MExtractTimeAndChargeSpline}}
    336336for two typical calibration pulses (pedestals have been subtracted) and a typical inner pixel.
    337337The extraction window sizes modify the position of the (amplitude-weighted) mean FADC-slices slightly.
    338 The pulse would be shifted half a slice to the right for an outer pixels. }
     338The pulse would be shifted half a slice to the right for an outer pixel. }
    339339\label{fig:splinesketch}
    340340\end{figure}
    … …  
    342342\subsubsection{Digital Filter}
    343343
    344 This extractor is implemented in the MARS-class {\textit{MExtractTimeAndChargeDigitalFilter}}.
    345 
    346 
    347 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.
    348 
    349 For the digital filtering method two assumptions have to be made:
     344This extractor is implemented in the MARS-class {\textit{\bf MExtractTimeAndChargeDigitalFilter}}.
     345
     346
     347The 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
     348from discrete measurements of the signal. Thereby, the noise contribution to the amplitude reconstruction is minimized.
     349
     350For the digital filtering method, three assumptions have to be made:
    350351
    351352\begin{itemize}
    … …  
    355356\end{itemize}
    356357
    357 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:
     358\par
     359\ldots {\textit{\bf IS THIS TRUE FOR MAGIC???? }} \ldots
     360\par
     361
     362Let $g(t)$ be the normalized signal shape, $E$ the signal amplitude and $\tau$ the time shift
     363of the physical signal from the predicted signal shape. Then the time dependence of the signal, $y(t)$, is given by:
    358364
    359365\begin{equation}
    … …  
    361367\end{equation}
    362368
    363 where $b(t)$ is the time dependent noise distribution. For small time shifts $\tau$ (usually smaller than
     369where $b(t)$ is the time-dependent noise contribution. For small time shifts $\tau$ (usually smaller than
    364370one FADC slice width),
    365371the time dependence can be linearized by the use of a Taylor expansion:
    … …  
    376382\end{equation}
    377383
    378 The correlation of the noise contributions at times $t_i$ and $t_j$ is expressed in the noise autocorrelation matrix $\boldsymbol{B}$:
     384The correlation of the noise contributions at times $t_i$ and $t_j$ can be expressed in the
     385noise autocorrelation matrix $\boldsymbol{B}$:
    379386
    380387\begin{equation}
    381 \boldsymbol{B_{ij}} = \langle b_i b_j \rangle - \langle b_i \rangle \langle b_j
     388B_{ij} = \langle b_i b_j \rangle - \langle b_i \rangle \langle b_j
    382389\rangle  \ .
    383390\label{eq:autocorr}
    … …  
    385392%\equiv \langle b_i b_j \rangle with $\langle b_i \rangle = 0$.
    386393
    387 The signal amplitude, $E$, and the product of amplitude and time shift, $E \tau$, can be estimated from the given set of
    388 measurements $\boldsymbol{y} = (y_1, ... ,y_n)$ by minimizing:
     394The signal amplitude $E$, and the product of amplitude and time shift $E \tau$, can be estimated from the given set of
     395measurements $\boldsymbol{y} = (y_1, ... ,y_n)$ by minimizing the excess noise contribution with respect to the known noise
     396auto-correlation:
    389397
    390398\begin{eqnarray}
    391 \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) \\
     399\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) \\
    392400&=& (\boldsymbol{y} - E
    393401\boldsymbol{g} - E\tau \dot{\boldsymbol{g}})^T \boldsymbol{B}^{-1} (\boldsymbol{y} - E \boldsymbol{g}- E\tau \dot{\boldsymbol{g}}) \ ,
    394402\end{eqnarray}
    395403
    396 where the last expression is matricial. The minimum is obtained for:
     404where the last expression is matricial. $\chi^2$ is a continuous function of $\tau$ and will have to be discretized itself for a
     405desired resolution.
     406The minimum of $\chi^2$ is obtained for:
    397407
    398408\begin{equation}
    … …  
    410420\end{eqnarray}
    411421
    412 Solving these equations one gets the solutions:
     422Solving these equations one gets the following solutions:
    413423
    414424\begin{equation}
    … …  
    422432
    423433Thus $\overline{E}$ and $\overline{E\tau}$ are given by a weighted sum of the discrete measurements $y_i$
    424 with the digital filtering weights for the amplitude, $w_{\text{amp},i}$, and time shift, $w_{\text{time},i}$.
    425 
     434with the digital filtering weights for the amplitude, $w_{\text{amp}}(t)$, and time shift, $w_{\text{time}}(t)$.
     435Note the remaining time dependence of the two weights which follow from the dependency of $\boldsymbol{g}$ and
     436$\dot{\boldsymbol{g}}$ on the position of the pulse with respect to the FADC bin positions.
     437\par
    426438Because of the truncation of the Taylor series in equation (\ref{shape_taylor_approx}) the above results are
    427439only valid for vanishing time offsets $\tau$. For non-zero time offsets one has to iterate the problem using
    … …  
    449461
    450462\begin{equation}\label{of_noise}
    451 \sigma_E \approx 8.3 \ \mathrm{FADC\ counts} \qquad \sigma_{\tau}  \approx  \frac{6.5\ \Delta T_{\mathrm{FADC}}}{E\ /\ \mathrm{FADC\ counts}} \ ,
     463\sigma_E \approx 8.3 \ \mathrm{FADC\ counts} \qquad \sigma_{\tau}  \approx  \frac{6.5\ \Delta T_{\mathrm{FADC}}}{(E\ /\ \mathrm{FADC\ counts})} \ ,
    452464\end{equation}
    453465
    … …  
    717729%%% TeX-master: "MAGIC_signal_reco"
    718730%%% TeX-master: "MAGIC_signal_reco"
     731%%% TeX-master: "MAGIC_signal_reco"
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