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trunk/MagicSoft/TDAS-Extractor/Pedestal.tex (modified) (14 diffs)
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trunk/MagicSoft/TDAS-Extractor/Pedestal.tex
r6622 r6665 6 6 can be completely described by the noise-autocorrelation matrix $\boldsymbol{B}$ 7 7 (eq.~\ref{eq:autocorr}), 8 where the square root of the diagonal elements give what is usually denoted as the `` Pedestal RMS''.8 where the square root of the diagonal elements give what is usually denoted as the ``pedestal RMS''. 9 9 \par 10 10 … … 18 18 19 19 \begin{equation} 20 \frac{Var[Q]}{<Q>^2} = \frac{1}{<n_{ phe}>} * F^220 \frac{Var[Q]}{<Q>^2} = \frac{1}{<n_{\mathrm{phe}}>} * F^2 21 21 \end{equation} 22 22 23 Here, $Q$ is the signal fluctuation due to the numberof signal photo-electrons24 (equiv. to the signal $S$) , and $Var[Q]$ the fluctuationsof the true signal $Q$23 Here, $Q$ is the signal due to a number $n_{\mathrm{phe}}$ of signal photo-electrons 24 (equiv. to the signal $S$) after subtraction of the pedestal. $Var[Q]$ is the fluctuation of the true signal $Q$ 25 25 due to the Poisson fluctuations of the number of photo-electrons. Because of: 26 26 27 27 \begin{eqnarray} 28 28 \widehat{Q} &=& Q + X \\ 29 Var (\widehat{Q}) &=& Var(Q) + Var(X)\\30 Var (Q) &=& Var(\widehat{Q}) - Var(X)29 Var[\widehat{Q}] &=& Var[Q] + Var[X] \\ 30 Var[Q] &=& Var[\widehat{Q}] - Var[X] 31 31 \end{eqnarray} 32 32 33 Here, $Var[X]$ is the fluctuation due to the signal extraction, mainly as a result of the background fluctuations and 34 the numerical precision of the extraction algorithm. 35 \par 33 36 Only in the case that the intrinsic extractor resolution $R$ at fixed background $BG$ does not depend on the signal 34 37 intensity\footnote{Theoretically, this is the case for the digital filter, eq.~\ref{eq:of_noise}.}, … … 36 39 37 40 \begin{eqnarray} 38 Var (Q) &\approx& Var(\widehat{Q}) - Var(\widehat{Q})\,\vline_{\,Q=0}41 Var[Q] &\approx& Var[\widehat{Q}] - Var[\widehat{Q}]\,\vline_{\,Q=0} 39 42 \label{eq:rmssubtraction} 40 43 \end{eqnarray} … … 57 60 %where $c$ is the photon/ADC conversion factor $<Q>/<m_{pe}>$.} 58 61 59 One can then retrieve $R$ 60 by applying the signal extractor to a {\textit{\bf fixed window}} of pedestal events, where the 62 One can determine $R$ by applying the signal extractor with a {\textit{\bf fixed window}} to pedestal events, where the 61 63 bias vanishes and measure $Var(\widehat{Q})\,\vline_{\,Q=0}$. 62 64 … … 70 72 signal amplitude $S$ and dependent only on the background $BG$ (eq.~\ref{eq:of_noise}). 71 73 %It can then be obtained from the calculation of the variance $Var[\widehat{Q}]$ 72 %by applying the extractor to a fixed window of pure background events (``pedestal events'') 73 %and get rid of the bias in that way. 74 \par 75 76 In order to calculate the bias and Mean-squared error, we proceed in the following ways: 74 %by applying the extractor with a fixed window to pure background events (``pedestal events''). 75 \par 76 77 In order to calculate the statistical parameters, we proceed in the following ways: 77 78 \begin{enumerate} 78 79 \item Determine $R$ by applying the signal extractor to a fixed window 79 80 of pedestal events. The background fluctuations can be simulated with different 80 81 levels of night sky background and the continuous light source, but no signal size 81 dependenc y can be retrieved withthis method.82 dependence can be retrieved by this method. 82 83 \item Determine $B$ and $MSE$ from MC events with added noise. 83 84 % Assuming that $MSE$ and $B$ are negligible for the events without noise, one can 84 With this method, one can get a dependenc y of both values fromthe size of the signal,85 With this method, one can get a dependence of both values on the size of the signal, 85 86 although the MC might contain systematic differences with respect to the real data. 86 87 \item Determine $MSE$ from the error retrieved from the fit results of $\widehat{S}$, which is possible for the … … 102 103 Difference in mean pedestal (per FADC slice) between extraction algorithm 103 104 applied on a fixed window of 1 FADC slice (``extractor random'') and a simple addition of 104 2 FADC slices (``fundamental''). On the left, a run with closed camera has been taken, in the center105 2 fixed FADC slices (``fundamental''). On the left, a run with closed camera has been taken, in the center 105 106 an opened camera observing an extra-galactic star field and on the right, an open camera being 106 107 illuminated by the continuous light of the calibration (level: 100). Every entry corresponds to one … … 119 120 Difference in mean pedestal (per FADC slice) between extraction algorithm 120 121 applied on a fixed window of 2 FADC slices (``extractor random'') and a simple addition of 121 2 FADC slices (``fundamental''). On the left, a run with closed camera has been taken, in the center122 2 FADC fixed slices (``fundamental''). On the left, a run with closed camera has been taken, in the center 122 123 an opened camera observing an extra-galactic star field and on the right, an open camera being 123 124 illuminated by the continuous light of the calibration (level: 100). Every entry corresponds to one … … 138 139 applied on a fixed window of 6 FADC slices and time-randomized weights (``extractor random'') 139 140 and a simple addition of 140 6 FADC slices (``fundamental''). On the left, a run with closed camera has been taken, in the center141 6 FADC fixed slices (``fundamental''). On the left, a run with closed camera has been taken, in the center 141 142 an opened camera observing an extra-galactic star field and on the right, an open camera being 142 143 illuminated by the continuous light of the calibration (level: 100). Every entry corresponds to one … … 150 151 of Pedestal Events} 151 152 152 By applying the signal extractor to a fixed window ofpedestal events, we153 By applying the signal extractor with a fixed window to pedestal events, we 153 154 determine the parameter $R$ for the case of no signal ($Q = 0$)\footnote{% 154 155 In the case of … … 264 265 of Pedestal Events} 265 266 266 By applying the signal extractor to a global extraction window ofpedestal events, allowing267 By applying the signal extractor with a global extraction window to pedestal events, allowing 267 268 it to ``slide'' and maximize the encountered signal, we 268 269 determine the bias $B$ and the mean-squared error $MSE$ for the case of no signal ($S=0$). … … 288 289 at 4 slices. The global winners is extractor~\#29 289 290 (digital filter with integration of 4 slices). All sliding window extractors -- except \#21 -- 290 have a smaller mean-square error than the resolution of the fixed window reference extractor . This means291 have a smaller mean-square error than the resolution of the fixed window reference extractor (row\ 1,\#4). This means 291 292 that the global error of the sliding window extractors is smaller than the one of the fixed window extractors 292 even if the first have a bias.293 with 8~FADC slices even if the first have a bias. 293 294 \par 294 295 The important information for the image cleaning is the number of photo-electrons above which the probability for obtaining … … 314 315 \hline 315 316 \hline 316 \multicolumn{16}{|c|}{Statistical Parameters for $S=0$ } \\317 \multicolumn{16}{|c|}{Statistical Parameters for $S=0$ units in $N_{\mathrm{phe}}$} \\ 317 318 \hline 318 319 \hline … … 344 345 \end{tabular} 345 346 \vspace{1cm} 346 \caption{The statistical parameters bias, resolution and mean error for the sliding window 347 algorithm. The first line displays the resolution of the smallest existing robust fixed--window extractor 347 \caption{The statistical parameters bias, resolution and mean error for the algorithms which can be applied to sliding 348 windows (SW) and/or fixed windows (FW) of pedestal events. 349 The first line displays the resolution of the smallest existing robust fixed--window extractor 348 350 for reference. All units in equiv. 349 351 photo-electrons, uncertainty: 0.1 phes. All extractors were allowed to move 5 FADC slices plus 350 their window size. The ``winners'' for each roware marked in red. Global winners (within the given352 their window size. The ``winners'' for each column are marked in red. Global winners (within the given 351 353 uncertainty) are the extractors Nr. \#24 (MExtractTimeAndChargeSpline with an integration window of 352 354 1 FADC slice) and Nr.\#29 … … 567 569 \end{figure} 568 570 569 Figure~\ref{fig:df:convfit} shows the obtained ``conversion factors'' and ``F-Factor'' computed as :571 Figure~\ref{fig:df:convfit} shows the obtained ``conversion factors'' and ``F-Factor'' computed as~\cite{MAGIC-calibration}: 570 572 571 573 \begin{eqnarray}
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