source: trunk/MagicSoft/TDAS-Extractor/Algorithms.tex@ 13414

Last change on this file since 13414 was 6748, checked in by gaug, 20 years ago
*** empty log message ***
File size: 49.1 KB
Line 
1\section{Signal Reconstruction Algorithms \label{sec:algorithms}}
2
3\subsection{Implementation of Signal Extractors in MARS}
4
5We performed all studies presented in this note using and developing the common MAGIC software framework
6MARS~\cite{MARS}.
7\par
8All signal extractor classes are stored in the MARS-directory {\textit{\bf msignal/}}.
9There, the base classes {\textit{\bf MExtractor}}, {\textit{\bf MExtractTime}}, {\textit{\bf MExtractTimeAndCharge}} and
10all individual extractors can be found. Figure~\ref{fig:extractorclasses} gives a sketch of the
11inheritances and tasks of each class.
12
13\begin{figure}[htp]
14\includegraphics[width=0.99\linewidth]{ExtractorClasses.eps}
15\caption{Sketch of the inheritances of three typical MARS signal extractor classes:
16MExtractFixedWindow, MExtractTimeFastSpline and MExtractTimeAndChargeDigitalFilter}
17\label{fig:extractorclasses}
18\end{figure}
19
20The following base classes for the extractor tasks are used:
21\begin{description}
22\item[MExtractor:\xspace] This class provides the basic data members, equal for all extractors, which are:
23 \begin{enumerate}
24 \item Global extraction ranges, defined by the variables
25 {\textit{\bf fHiGainFirst, fHiGainLast, fLoGainFirst, fLoGainLast}} and the function {\textit{\bf SetRange()}}.
26 The ranges always {\textit{\bf include}} the edge slices.
27 \item An internal variable {\textit{\bf fHiLoLast}} regulating the overlap of the desired high-gain
28 extraction range into the low-gain array.
29 \item The maximum possible FADC value, before the slice is declared as saturated, defined
30 by the variable {\textit{\bf fSaturationLimit}} (default:\,254).
31 \item The typical delay between high-gain and low-gain slices, expressed in FADC slices and parameterized
32 by the variable {\textit{\bf fOffsetLoGain}} (default:\,1.51)
33 \item Pointers to the storage containers {\textit{\bf MRawEvtData, MRawRunHeader, MPedestalCam}}
34 and~{\textit{\bf MExtractedSignalCam}}, defined by the variables
35 {\textit{\bf fRawEvt, fRunHeader, fPedestals}} and~{\textit{\bf fSignals}}.
36 \item Names of the storage containers to be searched for in the parameter list, parameterized
37 by the variables {\textit{\bf fNamePedestalCam}} and~{\textit{\bf fNameSignalCam}} (default: ``MPedestalCam''
38 and~''MExtractedSignalCam'').
39 \item The equivalent number of FADC samples, used for the calculation of the pedestal RMS and then the
40 number of photo-electrons with the F-Factor method (see eq.~\ref{eq:rmssubtraction} and
41 section~\ref{sec:photo-electrons}). This number is parameterized by the variables
42 {\textit{\bf fNumHiGainSamples}} and~{\textit{\bf fNumLoGainSamples}}.
43 \end{enumerate}
44
45 {\textit {\bf MExtractor}} is able to loop over all events, if the {\textit{\bf Process()}}-function is not overwritten.
46 It uses the following (virtual) functions, to be overwritten by the derived extractor class:
47
48 \begin{enumerate}
49 \item void {\textit {\bf FindSignalHiGain}}(Byte\_t* firstused, Byte\_t* logain, Float\_t\& sum, Byte\_t\& sat) const
50 \item void {\textit {\bf FindSignalLoGain}}(Byte\_t* firstused, Float\_t\& sum, Byte\_t\& sat) const
51 \end{enumerate}
52
53 where the pointers ``firstused'' point to the first used FADC slice declared by the extraction ranges,
54 the pointer ``logain'' points to the beginning of the ``low-gain'' FADC slices array (to be used for
55 pulses reaching into the low-gain array) and the variables ``sum'' and ``sat'' get filled with the
56 extracted signal and the number of saturating FADC slices, respectively.
57 \par
58 The pedestals get subtracted automatically {\textit {\bf after}} execution of these two functions.
59
60\item[MExtractTime:\xspace] This class provides - additionally to those already declared in {\textit{\bf MExtractor}} -
61 the basic data members, equal for all time extractors, which are:
62 \begin{enumerate}
63 \item Pointer to the storage container {\textit{\bf MArrivalTimeCam}}
64 parameterized by the variable
65 {\textit{\bf fArrTime}}.
66 \item The name of the ``MArrivalTimeCam''-container to be searched for in the parameter list,
67 parameterized by the variables {\textit{\bf fNameTimeCam}} (default: ``MArrivalTimeCam'' ).
68 \end{enumerate}
69
70 {\textit {\bf MExtractTime}} is able to loop over all events, if the {\textit{\bf Process()}}-function is not
71 overwritten.
72 It uses the following (virtual) functions, to be overwritten by the derived extractor class:
73
74 \begin{enumerate}
75 \item void {\textit {\bf FindTimeHiGain}}(Byte\_t* firstused, Float\_t\& time, Float\_t\& dtime, Byte\_t\& sat, const MPedestlPix \&ped) const
76 \item void {\textit {\bf FindTimeLoGain}}(Byte\_t* firstused, Float\_t\& time, Float\_t\& dtime, Byte\_t\& sat, const MPedestalPix \&ped) const
77 \end{enumerate}
78
79 where the pointers ``firstused'' point to the first used FADC slice declared by the extraction ranges,
80 and the variables ``time'', ``dtime'' and ``sat'' get filled with the
81 extracted arrival time, its error and the number of saturating FADC slices, respectively.
82 \par
83 The pedestals can be used for the arrival time extraction via the reference ``ped''.
84
85\item[MExtractTimeAndCharge:\xspace] This class provides - additionally to those already declared in
86 {\textit{\bf MExtractor}} and {\textit{\bf MExtractTime}} -
87 the basic data members, equal for all time and charge extractors, which are:
88 \begin{enumerate}
89 \item The actual extraction window sizes, parameterized by the variables
90 {\textit{\bf fWindowSizeHiGain}} and {\textit{\bf fWindowSizeLoGain}}.
91 \item The shift of the low-gain extraction range start w.r.t. to the found high-gain arrival
92 time, parameterized by the variable {\textit{\bf fLoGainStartShift}} (default: -2.8)
93 \end{enumerate}
94
95 {\textit {\bf MExtractTimeAndCharge}} is able to loop over all events, if the {\textit{\bf Process()}}-function is not
96 overwritten.
97 It uses the following (virtual) functions, to be overwritten by the derived extractor class:
98
99 \begin{enumerate}
100 \item void {\textit {\bf FindTimeAndChargeHiGain}}(Byte\_t* firstused, Byte\_t* logain, Float\_t\& sum, Float\_t\& dsum,
101 Float\_t\& time, Float\_t\& dtime, Byte\_t\& sat,
102 const MPedestlPix \&ped, const Bool\_t abflag) const
103 \item void {\textit {\bf FindTimeAndChargeLoGain}}(Byte\_t* firstused, Float\_t\& sum, Float\_t\& dsum,
104 Float\_t\& time, Float\_t\& dtime, Byte\_t\& sat,
105 const MPedestalPix \&ped, const Bool\_t abflag) const
106 \end{enumerate}
107
108 where the pointers ``firstused'' point to the first used FADC slice declared by the extraction ranges,
109 the pointer ``logain'' point to the beginning of the low-gain FADC slices array (to be used for
110 pulses reaching into the ``low-gain'' array),
111 the variables ``sum'', ``dsum'' get filled with the
112 extracted signal and its error. The variables ``time'', ``dtime'' and ``sat'' get filled with the
113 extracted arrival time, its error and the number of saturating FADC slices, respectively.
114 \par
115 The pedestals can be used for the extraction via the reference ``ped'', also the ``AB-flag'' is given
116 for clock noise correction.
117\end{description}
118
119
120\subsection{Pure Signal Extractors}
121
122The pure signal extractors have in common that they reconstruct only the
123charge, but not the arrival time. All extractors treated here derive from the MARS-base
124class {\textit{\bf MExtractor}} which provides the following facilities:
125
126\begin{itemize}
127\item The global extraction limits can be set from outside
128\item FADC saturation is kept track of
129\end{itemize}
130
131The following adjustable parameters have to be set from outside:
132\begin{description}
133\item[Global extraction limits:\xspace] Limits in between which the extractor is allowed
134to extract the signal, for high gain and low gain, respectively.
135\end{description}
136
137As the pulses jitter by about one FADC slice,
138not every pulse lies exactly within the optimal limits, especially if one chooses small
139extraction windows.
140Moreover, the readout position with respect to the trigger position has changed a couple
141of times during last year, therefore a very careful adjustment of the extraction limits
142is mandatory before using these extractors.
143
144\subsubsection{Fixed Window}
145
146This extractor is implemented in the MARS-class {\textit{\bf MExtractFixedWindow}}.
147It simply adds the FADC slice contents in the assigned ranges.
148As it does not correct for the clock-noise, only an even number of samples is allowed.
149Figure~\ref{fig:fixedwindowsketch} gives a sketch of the extraction ranges used in this
150paper and for two typical calibration pulses.
151
152\begin{figure}[htp]
153 \includegraphics[width=0.49\linewidth]{MExtractFixedWindow_5Led_UV.eps}
154 \includegraphics[width=0.49\linewidth]{MExtractFixedWindow_23Led_Blue.eps}
155\caption[Sketch extraction ranges MExtractFixedWindow]{%
156Sketch of the extraction ranges for the extractor {\textit{\bf MExtractFixedWindow}}
157for two typical calibration pulses (pedestals have been subtracted) and a typical inner pixel.
158The pulse would be shifted half a slice to the right for an outer pixel. }
159\label{fig:fixedwindowsketch}
160\end{figure}
161
162
163\subsubsection{Fixed Window with Integrated Cubic Spline}
164
165This extractor is implemented in the MARS-class {\textit{\bf MExtractFixedWindowSpline}}. It
166uses a cubic spline algorithm, adapted from \cite{NUMREC} and integrates the
167spline interpolated FADC slice values from a fixed extraction range. The edge slices are counted as half.
168As it does not correct for the clock-noise, only an odd number of samples is allowed.
169Figure~\ref{fig:fixedwindowsplinesketch} gives a sketch of the extraction ranges used in this
170paper and for typical calibration pulses.
171
172\begin{figure}[htp]
173 \includegraphics[width=0.49\linewidth]{MExtractFixedWindowSpline_5Led_UV.eps}
174 \includegraphics[width=0.49\linewidth]{MExtractFixedWindowSpline_23Led_Blue.eps}
175\caption[Sketch extraction ranges MExtractFixedWindowSpline]{%
176Sketch of the extraction ranges for the extractor {\textit{\bf MExtractFixedWindowSpline}}
177for two typical calibration pulses (pedestals have been subtracted) and a typical inner pixel.
178The pulse would be shifted half a slice to the right for an outer pixel. }
179\label{fig:fixedwindowsplinesketch}
180\end{figure}
181
182\subsubsection{Fixed Window with Global Peak Search}
183
184This extractor is implemented in the MARS-class {\textit{\bf MExtractFixedWindowPeakSearch}}.
185The basic idea of this extractor is to correct for coherent movements in arrival time for all pixels,
186as e.g. caused by the trigger jitter.
187In a first loop over the pixels, it determined a reference point slices number defined by the highest sum of
188consecutive non-saturating FADC slices in a (smaller) peak-search window.
189\par
190In a second loop over the pixels,
191it adds the contents of the FADC slices starting from the reference point over an extraction window of a pre-defined window size.
192It loops twice over all pixels in every event, because it has to find the reference point, first.
193As it does not correct for the clock-noise, only extraction windows with an even number of samples are allowed.
194For a high intensity calibration run causing high-gain saturation in the whole camera, this
195extractor apparently fails since only dead pixels are taken into account in the peak search
196 which cannot produce a saturated signal.
197For this special case, we modified {\textit{\bf MExtractFixedWindowPeakSearch}}
198such to define the peak search window as the one starting from the mean position of the first saturating slice.
199\par
200The following adjustable parameters have to be set from outside:
201\begin{description}
202\item[Peak Search Window:\xspace] Defines the ``sliding window'' size within which the peaking sum is
203searched for (default: 4 slices)
204\item[Offset from Window:\xspace] Defines the offset of the start of the extraction window w.r.t. the
205starting point of the obtained peak search window (default: 1 slice)
206\item[Low-Gain Peak shift:\xspace] Defines the shift in the low-gain with respect to the peak found
207in the high-gain (default: 1 slice)
208\end{description}
209
210Figure~\ref{fig:fixedwindowpeaksearchsketch} gives a sketch of the possible peak-search and extraction
211window positions in two typical calibration pulses.
212
213\begin{figure}[htp]
214 \includegraphics[width=0.49\linewidth]{MExtractFixedWindowPeakSearch_5Led_UV.eps}
215 \includegraphics[width=0.49\linewidth]{MExtractFixedWindowPeakSearch_23Led_Blue.eps}
216\caption[Sketch extraction ranges MExtractFixedWindowPeakSearch]{%
217Sketch of the extraction ranges for the extractor {\textit{\bf MExtractFixedWindowPeakSearch}}
218for two typical calibration pulses (pedestals have been subtracted) and a typical inner pixel.
219The pulse would be shifted half a slice to the right for an outer pixel. }
220\label{fig:fixedwindowpeaksearchsketch}
221\end{figure}
222
223\subsection{Combined Extractors}
224
225The combined extractors have in common that for a given pulse, they reconstruct
226both the arrival time and
227the charge.
228All combined extractors described here derive from the MARS-base
229class {\textit{\bf MExtractTimeAndCharge}} which itself derives from MExtractor and MExtractTime.
230It provides the following facilities:
231
232\begin{itemize}
233\item Only one loop over all pixels is performed.
234\item The individual FADC slice values get the clock-noise-corrected pedestals immediately subtracted.
235\item The low-gain extraction range is adapted dynamically, based on the arrival time computed from the high-gain samples.
236\item Arrival times extracted from the low-gain samples get corrected for the intrinsic time delay of the low-gain
237 pulse.
238\item The global extraction limits can be set from outside.
239\item FADC saturation is kept track of.
240\end{itemize}
241
242The following adjustable parameters have to be set from outside, additionally to those declared in the
243base classes MExtractor and MExtractTime:
244
245\begin{description}
246\item[Global extraction limits:\xspace] Limits in between which the extractor is allowed
247to search. They are fixed by the extractor for the high-gain, but re-adjusted for
248every event in the low-gain, depending on the arrival time found in the low-gain.
249However, the dynamically adjusted window is not allowed to pass beyond the global
250limits.
251\item[Low-gain start shift:\xspace] Global shift between the computed high-gain arrival
252time and the start of the low-gain extraction limit (corrected for the intrinsic time offset).
253This variable tells where the extractor is allowed to start searching for the low-gain signal
254if the high-gain arrival time is known. It avoids that the extractor gets confused by possible high-gain
255signals leaking into the ``low-gain'' region (default: -2.8).
256\end{description}
257
258\subsubsection{Sliding Window with Amplitude-Weighted Time}
259
260This extractor is implemented in the MARS-class {\textit{\bf MExtractTimeAndChargeSlidingWindow}}.
261It extracts the signal from a sliding window of an adjustable size, for high-gain and low-gain
262individually (default: 6 and 6). The signal is the one which maximizes the summed
263(clock-noise and pedestal-corrected) consecutive FADC slice contents.
264\par
265The amplitude-weighted arrival time is calculated from the window with
266the highest FADC slice contents integral using the following formula:
267
268\begin{equation}
269 t = \frac{\sum_{i=i_0}^{i_0+\mathrm{\it ws}-1} s_i \cdot i}{\sum_{i=i_0}^{i_0+\mathrm{\it ws}-1} i}
270\end{equation}
271where $i$ denotes the FADC slice index, starting from slice $i_0$
272and running over a window of size $\mathrm{\it ws}$. $s_i$ the clock-noise and
273pedestal-corrected FADC slice contents at slice position $i$.
274\par
275The following adjustable parameters have to be set from outside:
276\begin{description}
277\item[Window sizes:\xspace] Independently for high-gain and low-gain (default: 6,6)
278\end{description}
279
280Figure~\ref{fig:slidingwindowsketch} gives a sketch of the reconstructed arrival time of the possible extraction
281window positions in two typical calibration pulses.
282
283\begin{figure}[htp]
284 \includegraphics[width=0.49\linewidth]{MExtractTimeAndChargeSlidingWindow_5Led_UV.eps}
285 \includegraphics[width=0.49\linewidth]{MExtractTimeAndChargeSlidingWindow_23Led_Blue.eps}
286\caption[Sketch calculated arrival times MExtractTimeAndChargeSlidingWindow]{%
287Sketch of the calculated arrival times for the extractor {\textit{\bf MExtractTimeAndChargeSlidingWindow}}
288for two typical calibration pulses (pedestals have been subtracted) and a typical inner pixel.
289The extraction window sizes modify the position of the (amplitude-weighted) mean FADC-slices slightly.
290The pulse would be shifted half a slice to the right for an outer pixel. }
291\label{fig:slidingwindowsketch}
292\end{figure}
293
294\subsubsection{Cubic Spline with Sliding Window or Amplitude Extraction}
295
296This extractor is implemented in the MARS-class {\textit{\bf MExtractTimeAndChargeSpline}}.
297It interpolates the FADC contents using a cubic spline algorithm, adapted from \cite{NUMREC}.
298In a second step, it searches for the position of the spline maximum. From then on, two
299possibilities are offered:
300
301\begin{description}
302\item[Extraction Type Amplitude:\xspace] The amplitude of the spline maximum is taken as charge signal
303and the (precise) position of the maximum is returned as arrival time. This type is faster, since a spline integration is not performed.
304\item[Extraction Type Integral:\xspace] The integrated spline between maximum position minus
305rise time (default: 1.5 slices) and maximum position plus fall time (default: 4.5 slices)
306is taken as charge signal and the position of the half maximum left from the position of the maximum
307is returned as arrival time (default).
308The low-gain signal stretches the rise and fall time by a stretch factor (default: 1.5). This type
309is slower, but yields more precise results (see section~\ref{sec:performance}) .
310The charge integration resolution is set to 0.1 FADC slices.
311\end{description}
312
313The following adjustable parameters have to be set from outside:
314
315\begin{description}
316\item[Charge Extraction Type:\xspace] The amplitude of the spline maximum can be chosen while the position
317of the maximum is returned as arrival time. This type is fast. \\
318Otherwise, the integrated spline between maximum position minus rise time (default: 1.5 slices)
319and maximum position plus fall time (default: 4.5 slices) is taken as signal and the position of the
320half maximum is returned as arrival time (default).
321The low-gain signal stretches the rise and fall time by a stretch factor (default: 1.5). This type
322is slower, but more precise. The charge integration resolution is 0.1 FADC slices.
323\item[Rise Time and Fall Time:\xspace] Can be adjusted for the integration charge extraction type.
324\item[Resolution:\xspace] Defined as the maximum allowed difference between the calculated half maximum value and
325the computed spline value at the arrival time position. Can be adjusted for the half-maximum time extraction
326type.
327\item[Low Gain Stretch:\xspace] Can be adjusted to account for the larger rise and fall times in the
328low-gain as compared to the high gain pulses (default: 1.5)
329\end{description}
330
331Figure~\ref{fig:splinesketch} gives a sketch of the reconstructed arrival time for possible extraction
332window positions in two typical calibration pulses.
333
334\begin{figure}[htp]
335 \includegraphics[width=0.49\linewidth]{MExtractTimeAndChargeSpline_5Led_UV.eps}
336 \includegraphics[width=0.49\linewidth]{MExtractTimeAndChargeSpline_23Led_Blue.eps}
337\caption[Sketch calculated arrival times MExtractTimeAndChargeSpline]{%
338Sketch of the calculated arrival times for the extractor {\textit{\bf MExtractTimeAndChargeSpline}}
339for two typical calibration pulses (pedestals have been subtracted) and a typical inner pixel.
340The extraction window sizes modify the position of the (amplitude-weighted) mean FADC-slices slightly.
341The pulse would be shifted half a slice to the right for an outer pixel. }
342\label{fig:splinesketch}
343\end{figure}
344
345\subsubsection{Digital Filter}
346
347This extractor is implemented in the MARS-class {\textit{\bf MExtractTimeAndChargeDigitalFilter}}.
348
349
350The 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
351from discrete measurements of the signal. Thereby, the noise contribution to the amplitude reconstruction is minimized.
352
353For the digital filtering method, three assumptions have to be made:
354
355\begin{itemize}
356\item{The normalized signal shape has to be always constant, especially independent of the signal amplitude and in time.}
357\item{The noise properties have to be independent of the signal amplitude.}
358\item{The noise auto-correlation matrix does not change its form significantly with time and operation conditions.}
359\end{itemize}
360
361
362The pulse shape is mainly determined by the artificial pulse stretching by about 6\,ns on the receiver board.
363Thus the first assumption holds to a good approximation for all pulses with intrinsic signal widths much smaller than
364the shaping constant. Also the second assumption is fulfilled: Signal and noise are independent
365and the measured pulse is the linear superposition of the signal and noise. The validity of the third
366assumption is discussed below, especially for different night sky background conditions.
367
368Let $g(t)$ be the normalized signal shape, $E$ the signal amplitude and $\tau$ the time shift
369of the physical signal from the predicted signal shape. Then the time dependence of the signal, $y(t)$, is given by:
370
371\begin{equation}
372y(t)=E \cdot g(t-\tau) + b(t) \ ,
373\end{equation}
374
375where $b(t)$ is the time-dependent noise contribution. For small time shifts $\tau$ (usually smaller than
376one FADC slice width),
377the time dependence can be linearized:
378
379\begin{equation} \label{shape_taylor_approx}
380y(t)=E \cdot g(t) - E\tau \cdot \dot{g}(t) + b(t) \ ,
381\end{equation}
382
383where $\dot{g}(t)$ is the time derivative of the signal shape. Discrete
384measurements $y_i$ of the signal at times $t_i \ (i=1,...,n)$ have the form:
385
386\begin{equation}
387y_i=E \cdot g_i- E\tau \cdot \dot{g}_i +b_i \ .
388\end{equation}
389
390The correlation of the noise contributions at times $t_i$ and $t_j$ can be expressed in the
391noise autocorrelation matrix $\boldsymbol{B}$:
392
393\begin{equation}
394B_{ij} = \langle b_i b_j \rangle - \langle b_i \rangle \langle b_j
395\rangle \ .
396\label{eq:autocorr}
397\end{equation}
398%\equiv \langle b_i b_j \rangle with $\langle b_i \rangle = 0$.
399
400The signal amplitude $E$, and the product $E \tau$ of amplitude and time shift, can be estimated from the given set of
401measurements $\boldsymbol{y} = (y_1, ... ,y_n)$ by minimizing the deviation of the measured FADC slice contents from the
402known pulse shape with respect to the known noise auto-correlation:
403
404\begin{eqnarray}
405\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) \\
406&=& (\boldsymbol{y} - E
407\boldsymbol{g} - E\tau \dot{\boldsymbol{g}})^T \boldsymbol{B}^{-1} (\boldsymbol{y} - E \boldsymbol{g}- E\tau \dot{\boldsymbol{g}}) \ ,
408\end{eqnarray}
409
410where the last expression uses the matrix formalism.
411$\chi^2$ is a continuous function of $\tau$ and will have to be discretized itself for numerical computation applications.
412$\chi^2$ is in principle independent of the noise level if always the appropriate noise autocorrelation matrix is used.
413In our case however, we decided to use one matrix $\boldsymbol{B}$ for all levels of night-sky background.
414Changes in the noise level lead only to a multiplicative factor for all matrix elements and thus do not affect the
415position of the minimum of $\chi^2$.
416The minimum of $\chi^2$ is obtained for:
417
418\begin{equation}
419\frac{\partial \chi^2(E, E\tau)}{\partial E} = 0 \qquad \text{and} \qquad \frac{\partial \chi^2(E, E\tau)}{\partial(E\tau)} = 0 \ .
420\end{equation}
421
422
423Taking into account that $\boldsymbol{B}$ is a symmetric matrix, this leads to the following
424two equations for the estimated amplitude $\overline{E}$ and the estimation for the product of amplitude
425and time offset $\overline{E\tau}$:
426
427\begin{eqnarray}
4280&=&-\boldsymbol{g}^T\boldsymbol{B}^{-1}\boldsymbol{y}
429 +\boldsymbol{g}^T\boldsymbol{B}^{-1}\boldsymbol{g}\overline{E}
430 +\boldsymbol{g}^T\boldsymbol{B}^{-1}\dot{\boldsymbol{g}}\overline{E\tau}
431\\
4320&=&-\dot{\boldsymbol{g}}^T\boldsymbol{B}^{-1}\boldsymbol{y}
433 +\dot{\boldsymbol{g}}^T\boldsymbol{B}^{-1}\boldsymbol{g}\overline{E}
434 +\dot{\boldsymbol{g}}^T\boldsymbol{B}^{-1}\dot{\boldsymbol{g}}\overline{E\tau} \ .
435\end{eqnarray}
436
437Solving these equations one gets the following solutions:
438
439\begin{equation}
440\overline{E}(\tau) = \boldsymbol{w}_{\text{amp}}^T (\tau)\boldsymbol{y} \quad \mathrm{with} \quad
441 \boldsymbol{w}_{\text{amp}}
442 = \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}}}
443 {(\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 } \ ,
444\end{equation}
445
446\begin{equation}
447\overline{E\tau}(\tau)= \boldsymbol{w}_{\text{time}}^T(\tau) \boldsymbol{y} \quad
448 \mathrm{with} \quad \boldsymbol{w}_{\text{time}}
449 = \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}}}
450 {(\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 } \ .
451\end{equation}
452
453Thus $\overline{E}$ and $\overline{E\tau}$ are given by a weighted sum of the discrete measurements $y_i$
454with the weights for the amplitude, $w_{\text{amp}}(\tau)$, and time shift, $w_{\text{time}}(\tau)$.
455\par
456The time dependence gets discretized once again leading to a set of weight samples which themselves depend on the
457discretized time $\tau$.
458\par
459Note the remaining time dependency of the two weight samples. This follows from the dependence of $\boldsymbol{g}$ and
460$\dot{\boldsymbol{g}}$ on the relative position of the signal pulse with respect to FADC slices positions.
461\par
462Because of the truncation of the Taylor series in equation (\ref{shape_taylor_approx}) the above results are
463only valid for vanishing time offsets $\tau$. For larger time offsets, one has to iterate the problem using
464the time shifted signal shape $g(t-\tau)$.
465
466The covariance matrix $\boldsymbol{V}$ of $\overline{E}$ and $\overline{E\tau}$ is given by:
467
468\begin{equation}
469\left(\boldsymbol{V}^{-1}\right)_{ij}
470 =\frac{1}{2}\left(\frac{\partial^2 \chi^2(E, E\tau)}{\partial \alpha_i \partial \alpha_j} \right) \quad
471 \text{with} \quad \alpha_i,\alpha_j \in \{E, E\tau\} \ .
472\end{equation}
473
474The expected contribution of the noise to the estimated amplitude, $\sigma_E$, is:
475
476\begin{equation}
477\sigma_E^2=\boldsymbol{V}_{E,E}
478 =\frac{\dot{\boldsymbol{g}}^T\boldsymbol{B}^{-1}\dot{\boldsymbol{g}}}
479 {(\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\label{eq:of_noise}
481\end{equation}
482
483The expected contribution of the noise to the estimated timing, $\sigma_{\tau}$, is:
484
485\begin{equation}
486E^2 \cdot \sigma_{\tau}^2 < \sigma^2_{E \tau} =\boldsymbol{V}_{E\tau,E\tau}
487 =\frac{{\boldsymbol{g}}^T\boldsymbol{B}^{-1}{\boldsymbol{g}}}
488 {(\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} \ .
489\label{eq:of_noise_time}
490\end{equation}
491
492Both equations~\ref{eq:of_noise} and~\ref{eq:of_noise_time} are independent of the signal amplitude.
493\par
494In the MAGIC MC simulations~\cite{MC-Camera}, a night-sky background rate of 0.13 photoelectrons per ns,
495an FADC gain of 7.8 FADC counts per photo-electron and an intrinsic FADC noise of 1.3 FADC counts
496per FADC slice is implemented.
497These numbers simulate the night sky background conditions for an extragalactic source and result
498in a noise contribution of about 4 FADC counts per single FADC slice:
499$\sqrt{B_{ii}} \approx 4$~FADC counts.
500Using the digital filter with weights determined for 6~FADC slices ($i=0...5$) the errors of the
501reconstructed signal and time amount to:
502
503\begin{equation}
504\sigma_E \approx 8.3 \ \mathrm{FADC\ counts} \ (\approx 1.1\,\mathrm{phe}) \qquad
505\sigma_{\tau} \approx \frac{6.5\ \Delta T_{\mathrm{FADC}}}{(E\ /\ \mathrm{FADC\ counts})} \ (\approx \frac{2.8\,\mathrm{ns}}{E\,/\ \mathrm{N_{phe}}})\ ,
506\label{eq:of_noise_calc}
507\end{equation}
508
509where $\Delta T_{\mathrm{FADC}} = 3.33$ ns is the sampling interval of the MAGIC FADCs.
510The error in the reconstructed signal corresponds to about one photo electron.
511For signals of the size of two photo electrons, the timing error is about 1.4\,ns.
512\par
513
514An IACT has typically two types of background noise:
515On the one hand, there is the constantly present electronics noise,
516while on the other hand, the light of the night sky introduces a sizeable background
517to the measurement of the Cherenkov photons from air showers.
518
519The electronics noise is largely white, i.e. uncorrelated in time.
520The noise from the night sky background photons is the superposition of the
521detector response to single photo electrons following a Poisson distribution in time.
522Figure \ref{fig:noise_autocorr_allpixels} shows the noise
523autocorrelation matrix for an open camera. The large noise autocorrelation of the current FADC
524system is due to the pulse shaping (with the shaping constant equivalent to about two FADC slices).
525
526In general, the amplitude and time weights, $\boldsymbol{w}_{\text{amp}}$ and $\boldsymbol{w}_{\text{time}}$,
527depend on the pulse shape, the derivative of the pulse shape and the noise autocorrelation.
528In the high gain samples, the correlated night sky background noise dominates over
529the white electronics noise. As a consequence, different noise levels cause the elements of the noise autocorrelation
530matrix to change by the same factor, which cancels out in the weights calculation. Figure~\ref{fig:noise_autocorr_allpixels}
531shows the noise autocorrelation matrix for two different levels of night sky background (top) and the ratio between the
532corresponding elements of both (bottom). The central regions of $\pm$3 FADC slices around the diagonal (which is used to
533calculate the weights) deviate by less than 10\%.
534Thus, the weights are to a reasonable approximation independent of the night sky background noise level in the high gain.
535\par
536In the low gain samples the correlated noise of the LONS is in the same order of magnitude as the white electronics
537and digitization noise.
538Moreover, the noise autocorrelation for the low gain samples cannot be determined directly from the data. The low gain is only switched on
539if the pulse exceeds a preset threshold. There are no pedestals in the low gain available. Thus the noise auto-correlation determined from
540MC simulations for an extragalactic background is also used to compute the weights for cosmics and calibration pulses.
541
542%\begin{figure}[h!]
543%\begin{center}
544%\includegraphics[totalheight=7cm]{noise_autocorr_AB_36038_TDAS.eps}
545%\end{center}
546%\caption[Noise autocorrelation one pixel.]{Noise autocorrelation
547%matrix $\boldsymbol{B}$ for open camera including the noise due to night sky background fluctuations
548%for one single pixel (obtained from 1000 events).}
549%\label{fig:noise_autocorr_1pix}
550%\end{figure}
551
552\begin{figure}[htp]
553\begin{center}
554\includegraphics[totalheight=7cm]{noise_38995_smallNSB_all396.eps}
555\includegraphics[totalheight=7cm]{noise_39258_largeNSB_all396.eps}
556\includegraphics[totalheight=7cm]{noise_small_over_large.eps}
557\end{center}
558\caption[Noise autocorrelation average all pixels.]{Noise autocorrelation
559matrix $\boldsymbol{B}$ for open camera and averaged over all pixels. The top figure shows $\boldsymbol{B}$
560obtained with camera pointing off the galactic plane (and low night sky background fluctuations).
561The central figure shows $\boldsymbol{B}$ with the camera pointing into the galactic plane
562(high night sky background) and the
563bottom plot shows the ratio between both. One can see that the entries of $\boldsymbol{B}$ do not
564simply scale with the amount of night sky background.}
565\label{fig:noise_autocorr_allpixels}
566\end{figure}
567
568Using the average reconstructed pulpo pulse shapes, as shown in figure \ref{fig:pulse_shapes}, and the reconstructed noise
569autocorrelation matrices from pedestal runs with random triggers, the digital filter weights are computed.
570As the pulse shapes in the high and low gain and for cosmics, calibration and pulpo events are somewhat different,
571dedicated digital filter weights are computed for these event classes. Also filter weights optimized for MC simulations are calculated.
572High/low gain filter weights are computed for the following event classes:
573
574\begin{enumerate}
575\item{cosmics weights: for cosmics events}
576\item{calibration weights UV: for UV calibration pulses}
577\item{calibration weights blue: for blue and green calibration pulses}
578\item{MC weights: for MC simulations}
579\item{pulpo weights: for pulpo runs.}
580\end{enumerate}
581
582
583\begin{table}[h]{\normalsize\center
584\begin{tabular}{lllll}
585 \hline
586 & high gain shape & high gain noise & low gain shape & low gain noise
587\\ cosmics & 25945 (pulpo) & 38995 (extragal.) & 44461 (pulpo) & MC low
588\\ UV & 36040 (UV) & 38995 (extragal.) & 44461 (pulpo) & MC low
589\\ blue & 31762 (blue) & 38995 (extragal.) & 31742 (blue) & MC low
590\\ MC & MC & MC high & MC & MC low
591\\ pulpo & 25945 (pulpo) & 38993 (no LONS) & 44461 (pulpo) & MC low
592\\
593\hline
594\end{tabular}
595\caption{The used runs for the pulse shapes and noise auto-correlations for the digital filter weights of the different event types.}\label{table:weight_files}}
596\end{table}
597
598
599
600
601 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.
602The 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
603FADC 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.
604A binning resolution of $0.1\,T_{\text{ADC}}$ has been chosen.
605
606
607\begin{figure}[h!]
608\begin{center}
609\includegraphics[totalheight=7cm]{w_time_MC_input_TDAS.eps}
610\end{center}
611\caption[Time weights.]{Time weights $w_{\mathrm{time}}(t_0) \ldots w_{\mathrm{time}}(t_5)$ for a window size of 6 FADC slices for the pulse shape
612used 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
613FADC 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
614of $0.1\,T_{\text{ADC}}$ has been chosen.} \label{fig:w_time_MC_input_TDAS}
615\end{figure}
616
617\begin{figure}[h!]
618\begin{center}
619\includegraphics[totalheight=7cm]{w_amp_MC_input_TDAS.eps}
620\end{center}
621\caption[Amplitude weights.]{Amplitude weights $w_{\mathrm{amp}}(t_0) \ldots w_{\mathrm{amp}}(t_5)$ for a window size of 6 FADC slices for the
622pulse shape used in the MC simulations. The first weight $w_{\mathrm{amp}}(t_0)$ is plotted as a function of the relative time $t_{\text{rel}}$
623the 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.
624A binning resolution of $0.1\, T_{\text{ADC}}$ has been chosen.} \label{fig:w_amp_MC_input_TDAS}
625\end{figure}
626
627In the current implementation a two step procedure is applied to reconstruct the signal. The weight functions $w_{\mathrm{amp}}(t)$
628and $w_{\mathrm{time}}(t)$ are computed numerically with a resolution of $1/10$ of an FADC slice.
629In the first step the quantities $e_{i_0}$ and $(e\tau)_{i_0}$ are computed using a window of $n$ slices:
630
631\begin{equation}
632e_{i_0}=\sum_{i=i_0}^{i_0+n-1} w_{\mathrm{amp}}(t_i)y(t_{i+i_0}) \qquad (e\tau)_{i_0}=\sum_{i=i_0}^{i_0+n-1} w_{\mathrm{time}}(t_i)y(t_{i+i_0})
633\end{equation}
634
635for all possible signal start slices $i_0$. Let $i_0^*$ be the signal start slice yielding the largest $e_{i_0}$.
636Then in a second step the timing offset $\tau$ is calculated:
637
638\begin{equation}
639\tau=\frac{(e\tau)_{i_0^*}}{e_{i_0^*}}
640\label{eq:offsettau}
641\end{equation}
642
643Using this value of $\tau$, another iteration is performed:
644
645\begin{equation}
646E=\sum_{i=i_0^*}^{i_0^*+n-1} w_{\mathrm{amp}}(t_i - \tau)y(t_{i+i_0^*}) \qquad
647 E \theta=\sum_{i=i_0^*}^{i_0^*+n-1} w_{\mathrm{time}}(t_i - \tau)y(t_{i+i_0^*}) \ .
648\end{equation}
649
650The reconstructed signal is then taken to be $E$ and the reconstructed arrival time $t_{\text{arrival}}$ is
651
652\begin{equation}
653t_{\text{arrival}} = i_0^* + \tau + \theta \ .
654\end{equation}
655
656
657
658Figure \ref{fig:amp_sliding} shows the result of the applied amplitude and time weights to the recorded FADC time slices of one
659simulated MC pulse. The left plot displays the result of the applied amplitude weights
660$e(t_0)=\sum_{i=0}^{i=n-1} w_{\mathrm{amp}}(t_0+i \cdot T_{\text{ADC}})y(t_0+i \cdot T_{\text{ADC}})$ and
661the right plot shows the result of the applied timing weights
662$e\tau(t_0)=\sum_{i=0}^{i=n-1} w_{\mathrm{time}}(t_0+i \cdot T_{\text{ADC}})y(t_0+i \cdot T_{\text{ADC}})$ as a function of the time shift $t_0$.
663
664
665% This does not apply for MAGIC as the LONs are giving always a correlated noise (in addition to the artificial shaping)
666
667%In the case of an uncorrelated noise with zero mean the noise autocorrelation matrix is:
668
669%\begin{equation}
670%\boldsymbol{B}_{ij}= \langle b_i b_j \rangle \delta_{ij} = \sigma^2(b_i) \ ,
671%\end{equation}
672
673%where $\sigma(b_i)$ is the standard deviation of the noise of the discrete measurements. Equation (\ref{of_noise}) than becomes:
674
675
676%\begin{equation}
677%\frac{\sigma^2(b_i)}{\sigma_E^2} = \sum_{i=1}^{n}{g_i^2} - \frac{\sum_{i=1}^{n}{g_i \dot{g}_i}}{\sum_{i=1}^{n}{\dot{g}_i^2}} \ .
678%\end{equation}
679
680
681\begin{figure}[h!]
682\begin{center}
683\includegraphics[totalheight=7cm]{amp_sliding.eps}
684\includegraphics[totalheight=7cm]{time_sliding.eps}
685\end{center}
686\caption[Digital filter weights applied.]{Digital filter weights applied to the recorded FADC time slices of
687one simulated MC pulse. The left plot shows the result of the applied amplitude weights
688$e(t_0)=\sum_{i=0}^{i=n-1} w_{\mathrm{amp}}(t_0+i \cdot T_{\text{ADC}})y(t_0+i \cdot T_{\text{ADC}})$ and
689the right plot displays the result of the applied timing weights
690$e\tau(t_0)=\sum_{i=0}^{i=n-1} w_{\mathrm{time}}(t_0+i \cdot T_{\text{ADC}})y(t_0+i \cdot T_{\text{ADC}})$ as a function of the time shift $t_0$.}
691\label{fig:amp_sliding}
692\end{figure}
693
694Figure \ref{fig:shape_fit_TDAS} shows the signal pulse shape of a typical MC event together with the simulated FADC slices
695of the signal pulse plus noise. The digital filter has been applied to reconstruct the signal size and timing.
696Using this information together with the average normalized MC pulse shape the simulated signal pulse shape is reconstructed
697and shown as well.
698
699
700\begin{figure}[h!]
701\begin{center}
702\includegraphics[totalheight=7cm]{shape_fit_TDAS.eps}
703\end{center}
704\caption[Shape fit.]{Simulated signal pulse shape and FADC slices for a typical MC event. The FADC measurements are affected by noise. Using the digital filter and the average MC pulse shape the signal shape is reconstructed. The event shown is the same as in figure \ref{fig:amp_sliding}.} \label{fig:shape_fit_TDAS}
705\end{figure}
706%Full fit to the MC pulse shape with the MC input shape and a numerical fit using the digital filter
707
708
709
710The following free adjustable parameters have to be set from outside:
711
712\begin{description}
713\item[Weights File:\xspace] An ascii-file containing the weights, the binning resolution and
714the window size. Currently, the following weight files have been created:
715\begin{itemize}
716\item "cosmics\_weights.dat'' with a window size of 6 FADC slices
717\item "cosmics\_weights4.dat'' with a window size of 4 FADC slices
718\item "calibration\_weights\_blue.dat'' with a window size of 6 FADC slices
719\item "calibration\_weights4\_blue.dat'' with a window size of 4 FADC slices
720\item "calibration\_weights\_UV.dat'' with a window size of 6 FADC slices and in the low-gain the
721calibration weights obtained from blue pulses\footnote{UV-pulses saturating the high-gain are not yet
722available.}.
723\item "calibration\_weights4\_UV.dat'' with a window size of 4 FADC slices and in the low-gain the
724calibration weights obtained from blue pulses\footnote{UV-pulses saturating the high-gain are not yet
725available.}.
726\end{itemize}
727\end{description}
728
729Figure~\ref{fig:dfsketch} gives a sketch of the reconstructed arrival time of the possible extraction
730window positions in two typical calibration pulses.
731
732\begin{figure}[htp]
733 \includegraphics[width=0.49\linewidth]{MExtractTimeAndChargeDigitalFilter_5Led_UV.eps}
734 \includegraphics[width=0.49\linewidth]{MExtractTimeAndChargeDigitalFilter_23Led_Blue.eps}
735\caption[Sketch calculated arrival times MExtractTimeAndChargeDigitalFilter]{%
736Sketch of the calculated arrival times for the extractor {\textit{MExtractTimeAndChargeDigitalFilter}}
737for two typical calibration pulses (pedestals have been subtracted) and a typical inner pixel.
738The extraction window sizes modify the position of the (amplitude-weighted) mean FADC-slices slightly.
739The pulse would be shifted half a slice to the right for an outer pixels. }
740\label{fig:dfsketch}
741\end{figure}
742
743\subsubsection{Digital Filter with Global Peak Search}
744
745This extractor is implemented in the MARS-class {\textit{\bf MExtractTimeAndChargeDigitalFilterPeakSearch}}.
746
747The idea of this extractor is to combine {\textit{\bf MExtractFixedWindowPeakSearch}} (the same reference point for all pixels) and
748{\textit{\bf MExtractTimeAndChargeDigitalFilter}}, where the reference point is determined pixel by pixel, in order to correct for coherent movements in arrival time
749for all pixels and still use the digital filter fit capabilities.
750 \par
751
752In a first loop over the pixels, it fixes a reference point (slice number) defined by the highest sum of
753consecutive non-saturating FADC slices in a (smaller) peak-search window.
754\par
755In a second loop over the pixels,
756it uses the digital filter algorithm within a reduced extraction window.
757It loops twice over all pixels in every event, because it has to find the reference point, first.
758
759As in the case of {\textit{\bf MExtractFixedWindowPeakSearch}}, for a high intensity calibration run
760causing high-gain saturation in the whole camera, this
761extractor apparently fails since only dead pixels
762are taken into account in the peak search which cannot produce a saturated signal.
763
764\par
765For this special case, the extractor then defines the peak search window
766as the one starting from the mean position of the first saturating slice.
767\par
768The following adjustable parameters have to be set from outside, additionally to the ones to be
769set in {\textit{\bf MExtractTimeAndChargeDigitalFilter}}:
770\begin{description}
771\item[Peak Search Window:\xspace] Defines the ``sliding window'' size within which the peaking sum is
772searched for (default: 2 slices)
773\item[Offset left from Peak:\xspace] Defines the left offset of the start of the extraction window w.r.t. the
774starting point of the obtained peak search window (default: 3 slices)
775\item[Offset right from Peak:\xspace] Defines the right offset of the of the extraction window w.r.t. the
776starting point of the obtained peak search window (default: 3 slices)
777\item[Limit for high gain failure events:\xspace] Defines the limit of the number of events which failed
778to be in the high-gain window before the run is rejected.
779\item[Limit for low gain failure events:\xspace] Defines the limit of the number of events which failed
780to be in the low-gain window before the run is rejected.
781\end{description}
782
783In principle, the ``offsets'' can be chosen very small, because both showers and calibration pulses spread
784over a very small time interval, typically less than one FADC slice. However, the MAGIC DAQ produces
785artificial jumps of two FADC slices from time to time\footnote{in 5\% of the events per pixel in December 2004},
786so the 3 slices are made in order not to reject these pixels already with the extractor.
787
788\subsubsection{Real Fit to the Expected Pulse Shape }
789
790The digital filter is a sophisticated numerical tool to fit the read-out FADC samples with the expected wave form taking the autocorrelation of the noise into account. In order to cross-check the results a pulse shape fit has been implemented using the root TH1::Fit routine. For each event the FADC samples of each pixel are filled into a histogram and fit by the expected wave form having the time shift and the area of the fit pulse as free parameters. The results are in very good agreement with the results of the digital filter.
791
792Figure \ref{fig:probability_fit} shows the distribution of the fit probability for simulated MC pulses. Both electronics and NSB noise are simulated. The distribution is mainly flat with a slight excess in the very lowest probability bins.
793\par
794
795This extractor is not (yet) implemented as a MARS-class.
796
797
798
799\begin{figure}[h!]
800\begin{center}
801\includegraphics[totalheight=7cm]{probability_fit_0ns.eps}
802\end{center}
803\caption[Fit Probability.]{Probability of the fit with the input signal shape to the simulated FADC samples
804including electronics and NSB noise.} \label{fig:probability_fit}
805\end{figure}
806
807
808
809\subsection{Used Extractors for this Analysis}
810
811We tested in this TDAS the following parameterized extractors:
812
813\begin{description}
814\item[MExtractFixedWindow]: with the following initialization, if {\textit{maxbin}} defines the
815 mean position of the high-gain FADC slice which carries the pulse maximum \footnote{The function
816{\textit{MExtractor::SetRange(higain first, higain last, logain first, logain last)}} sets the extraction
817range with the high gain start bin {\textit{higain first}} to (including) the last bin {\textit{higain last}}.
818Analog for the low gain extraction range. Note that in MARS, the low-gain FADC samples start with
819the index 0 again, thus {\textit{maxbin+0.5}} means in reality {\textit{maxbin+15+0.5}}. }
820:
821\begin{enumerate}
822\item SetRange({\textit{maxbin}}-1,{\textit{maxbin}}+2,{\textit{maxbin}}+0.5,{\textit{maxbin}}+3.5);
823\item SetRange({\textit{maxbin}}-1,{\textit{maxbin}}+2,{\textit{maxbin}}-0.5,{\textit{maxbin}}+4.5);
824\item SetRange({\textit{maxbin}}-2,{\textit{maxbin}}+3,{\textit{maxbin}}-0.5,{\textit{maxbin}}+4.5);
825\item SetRange({\textit{maxbin}}-2,{\textit{maxbin}}+5,{\textit{maxbin}}-0.5,{\textit{maxbin}}+6.5);
826\item SetRange({\textit{maxbin}}-3,{\textit{maxbin}}+10,{\textit{maxbin}}-1.5,{\textit{maxbin}}+7.5);
827\suspend{enumerate}
828\item[MExtractFixedWindowSpline]: with the following initialization, if {\textit{maxbin}} defines the
829 mean position of the high-gain FADC slice carrying the pulse maximum \footnote{The function
830{\textit{MExtractor::SetRange(higain first, higain last, logain first, logain last)}} sets the extraction
831range with the high gain start bin {\textit{higain first}} to (including) the last bin {\textit{higain last}}.
832Analog for the low gain extraction range. Note that in MARS, the low-gain FADC samples start with
833the index 0 again, thus {\textit{maxbin+0.5}} means in reality {\textit{maxbin+15+0.5}}.}:
834\resume{enumerate}
835\item SetRange({\textit{maxbin}}-1,{\textit{maxbin}}+3,{\textit{maxbin}}+0.5,{\textit{maxbin}}+4.5);
836\item SetRange({\textit{maxbin}}-1,{\textit{maxbin}}+3,{\textit{maxbin}}-0.5,{\textit{maxbin}}+5.5);
837\item SetRange({\textit{maxbin}}-2,{\textit{maxbin}}+4,{\textit{maxbin}}-0.5,{\textit{maxbin}}+5.5);
838\item SetRange({\textit{maxbin}}-2,{\textit{maxbin}}+6,{\textit{maxbin}}-0.5,{\textit{maxbin}}+7.5);
839\item SetRange({\textit{maxbin}}-3,{\textit{maxbin}}+11,{\textit{maxbin}}-1.5,{\textit{maxbin}}+8.5);
840\suspend{enumerate}
841\item[MExtractFixedWindowPeakSearch]: with the following initialization: \\
842SetRange(0,18,2,14); and:
843\resume{enumerate}
844\item SetWindows(2,2,2); SetOffsetFromWindow(0);
845\item SetWindows(4,4,2); SetOffsetFromWindow(1);
846\item SetWindows(4,6,4); SetOffsetFromWindow(0);
847\item SetWindows(6,6,4); SetOffsetFromWindow(1);
848\item SetWindows(8,8,4); SetOffsetFromWindow(1);
849\item SetWindows(14,10,4); SetOffsetFromWindow(2);
850\suspend{enumerate}
851\item[MExtractTimeAndChargeSlidingWindow]: with the following initialization: \\
852\resume{enumerate}
853\item SetWindowSize(2,2); SetRange(5,11,7,11);
854\item SetWindowSize(4,4); SetRange(5,13,6,12);
855\item SetWindowSize(4,6); SetRange(5,13,5,13);
856\item SetWindowSize(6,6); SetRange(4,14,5,13);
857\item SetWindowSize(8,8); SetRange(4,16,4,14);
858\item SetWindowSize(14,10); SetRange(5,10,7,11);
859\suspend{enumerate}
860\item[MExtractTimeAndChargeSpline]: with the following initialization:
861\resume{enumerate}
862\item SetChargeType(MExtractTimeAndChargeSpline::kAmplitude); \\
863SetRange(5,10,7,10);
864\suspend{enumerate}
865SetChargeType(MExtractTimeAndChargeSpline::kIntegral); \\
866and:
867\resume{enumerate}
868\item SetRiseTime(0.5); SetFallTime(0.5); SetRange(5,10,7,11);
869\item SetRiseTime(0.5); SetFallTime(1.5); SetRange(5,11,7,12);
870\item SetRiseTime(1.0); SetFallTime(3.0); SetRange(4,12,5,13);
871\item SetRiseTime(1.5); SetFallTime(4.5); SetRange(4,14,3,13);
872\suspend{enumerate}
873\item[MExtractTimeAndChargeDigitalFilter]: with the following initialization:
874\resume{enumerate}
875\item SetWeightsFile(``cosmics\_weights.dat''); SetRange(4,14,5,13);
876\item SetWeightsFile(``cosmics\_weights4.dat''); SetRange(5,13,6,12);
877\item SetWeightsFile(``calibration\_weights\_UV.dat'');
878\item SetWeightsFile(``calibration\_weights4\_UV.dat'');
879\item SetWeightsFile(``calibration\_weights\_blue.dat'');
880\item SetWeightsFile(``calibration\_weights4\_blue.dat'');
881\end{enumerate}
882\end{description}
883
884References: \cite{OF77,OF94}.
885
886
887%%% Local Variables:
888%%% mode: latex
889%%% TeX-master: "MAGIC_signal_reco"
890%%% TeX-master: "MAGIC_signal_reco"
891%%% TeX-master: "Algorithms"
892%%% End:
Note: See TracBrowser for help on using the repository browser.