Index: trunk/MagicSoft/TDAS-Extractor/Pedestal.tex =================================================================== --- trunk/MagicSoft/TDAS-Extractor/Pedestal.tex (revision 6664) +++ trunk/MagicSoft/TDAS-Extractor/Pedestal.tex (revision 6665) @@ -6,5 +6,5 @@ can be completely described by the noise-autocorrelation matrix $\boldsymbol{B}$ (eq.~\ref{eq:autocorr}), -where the square root of the diagonal elements give what is usually denoted as the ``Pedestal RMS''. +where the square root of the diagonal elements give what is usually denoted as the ``pedestal RMS''. \par @@ -18,17 +18,20 @@ \begin{equation} -\frac{Var[Q]}{^2} = \frac{1}{} * F^2 +\frac{Var[Q]}{^2} = \frac{1}{} * F^2 \end{equation} -Here, $Q$ is the signal fluctuation due to the number of signal photo-electrons -(equiv. to the signal $S$), and $Var[Q]$ the fluctuations of the true signal $Q$ +Here, $Q$ is the signal due to a number $n_{\mathrm{phe}}$ of signal photo-electrons +(equiv. to the signal $S$) after subtraction of the pedestal. $Var[Q]$ is the fluctuation of the true signal $Q$ due to the Poisson fluctuations of the number of photo-electrons. Because of: \begin{eqnarray} \widehat{Q} &=& Q + X \\ -Var(\widehat{Q}) &=& Var(Q) + Var(X) \\ -Var(Q) &=& Var(\widehat{Q}) - Var(X) +Var[\widehat{Q}] &=& Var[Q] + Var[X] \\ +Var[Q] &=& Var[\widehat{Q}] - Var[X] \end{eqnarray} +Here, $Var[X]$ is the fluctuation due to the signal extraction, mainly as a result of the background fluctuations and +the numerical precision of the extraction algorithm. +\par Only in the case that the intrinsic extractor resolution $R$ at fixed background $BG$ does not depend on the signal intensity\footnote{Theoretically, this is the case for the digital filter, eq.~\ref{eq:of_noise}.}, @@ -36,5 +39,5 @@ \begin{eqnarray} -Var(Q) &\approx& Var(\widehat{Q}) - Var(\widehat{Q})\,\vline_{\,Q=0} +Var[Q] &\approx& Var[\widehat{Q}] - Var[\widehat{Q}]\,\vline_{\,Q=0} \label{eq:rmssubtraction} \end{eqnarray} @@ -57,6 +60,5 @@ %where $c$ is the photon/ADC conversion factor $/$.} - One can then retrieve $R$ -by applying the signal extractor to a {\textit{\bf fixed window}} of pedestal events, where the + One can determine $R$ by applying the signal extractor with a {\textit{\bf fixed window}} to pedestal events, where the bias vanishes and measure $Var(\widehat{Q})\,\vline_{\,Q=0}$. @@ -70,17 +72,16 @@ signal amplitude $S$ and dependent only on the background $BG$ (eq.~\ref{eq:of_noise}). %It can then be obtained from the calculation of the variance $Var[\widehat{Q}]$ -%by applying the extractor to a fixed window of pure background events (``pedestal events'') -%and get rid of the bias in that way. -\par - -In order to calculate the bias and Mean-squared error, we proceed in the following ways: +%by applying the extractor with a fixed window to pure background events (``pedestal events''). +\par + +In order to calculate the statistical parameters, we proceed in the following ways: \begin{enumerate} \item Determine $R$ by applying the signal extractor to a fixed window of pedestal events. The background fluctuations can be simulated with different levels of night sky background and the continuous light source, but no signal size - dependency can be retrieved with this method. + dependence can be retrieved by this method. \item Determine $B$ and $MSE$ from MC events with added noise. % Assuming that $MSE$ and $B$ are negligible for the events without noise, one can - With this method, one can get a dependency of both values from the size of the signal, + With this method, one can get a dependence of both values on the size of the signal, although the MC might contain systematic differences with respect to the real data. \item Determine $MSE$ from the error retrieved from the fit results of $\widehat{S}$, which is possible for the @@ -102,5 +103,5 @@ Difference in mean pedestal (per FADC slice) between extraction algorithm applied on a fixed window of 1 FADC slice (``extractor random'') and a simple addition of -2 FADC slices (``fundamental''). On the left, a run with closed camera has been taken, in the center +2 fixed FADC slices (``fundamental''). On the left, a run with closed camera has been taken, in the center an opened camera observing an extra-galactic star field and on the right, an open camera being illuminated by the continuous light of the calibration (level: 100). Every entry corresponds to one @@ -119,5 +120,5 @@ Difference in mean pedestal (per FADC slice) between extraction algorithm applied on a fixed window of 2 FADC slices (``extractor random'') and a simple addition of -2 FADC slices (``fundamental''). On the left, a run with closed camera has been taken, in the center +2 FADC fixed slices (``fundamental''). On the left, a run with closed camera has been taken, in the center an opened camera observing an extra-galactic star field and on the right, an open camera being illuminated by the continuous light of the calibration (level: 100). Every entry corresponds to one @@ -138,5 +139,5 @@ applied on a fixed window of 6 FADC slices and time-randomized weights (``extractor random'') and a simple addition of -6 FADC slices (``fundamental''). On the left, a run with closed camera has been taken, in the center +6 FADC fixed slices (``fundamental''). On the left, a run with closed camera has been taken, in the center an opened camera observing an extra-galactic star field and on the right, an open camera being illuminated by the continuous light of the calibration (level: 100). Every entry corresponds to one @@ -150,5 +151,5 @@ of Pedestal Events} -By applying the signal extractor to a fixed window of pedestal events, we +By applying the signal extractor with a fixed window to pedestal events, we determine the parameter $R$ for the case of no signal ($Q = 0$)\footnote{% In the case of @@ -264,5 +265,5 @@ of Pedestal Events} -By applying the signal extractor to a global extraction window of pedestal events, allowing +By applying the signal extractor with a global extraction window to pedestal events, allowing it to ``slide'' and maximize the encountered signal, we determine the bias $B$ and the mean-squared error $MSE$ for the case of no signal ($S=0$). @@ -288,7 +289,7 @@ at 4 slices. The global winners is extractor~\#29 (digital filter with integration of 4 slices). All sliding window extractors -- except \#21 -- -have a smaller mean-square error than the resolution of the fixed window reference extractor. This means +have a smaller mean-square error than the resolution of the fixed window reference extractor (row\ 1,\#4). This means that the global error of the sliding window extractors is smaller than the one of the fixed window extractors -even if the first have a bias. +with 8~FADC slices even if the first have a bias. \par The important information for the image cleaning is the number of photo-electrons above which the probability for obtaining @@ -314,5 +315,5 @@ \hline \hline -\multicolumn{16}{|c|}{Statistical Parameters for $S=0$} \\ +\multicolumn{16}{|c|}{Statistical Parameters for $S=0$ units in $N_{\mathrm{phe}}$} \\ \hline \hline @@ -344,9 +345,10 @@ \end{tabular} \vspace{1cm} -\caption{The statistical parameters bias, resolution and mean error for the sliding window -algorithm. The first line displays the resolution of the smallest existing robust fixed--window extractor +\caption{The statistical parameters bias, resolution and mean error for the algorithms which can be applied to sliding +windows (SW) and/or fixed windows (FW) of pedestal events. +The first line displays the resolution of the smallest existing robust fixed--window extractor for reference. All units in equiv. photo-electrons, uncertainty: 0.1 phes. All extractors were allowed to move 5 FADC slices plus -their window size. The ``winners'' for each row are marked in red. Global winners (within the given +their window size. The ``winners'' for each column are marked in red. Global winners (within the given uncertainty) are the extractors Nr. \#24 (MExtractTimeAndChargeSpline with an integration window of 1 FADC slice) and Nr.\#29 @@ -567,5 +569,5 @@ \end{figure} -Figure~\ref{fig:df:convfit} shows the obtained ``conversion factors'' and ``F-Factor'' computed as: +Figure~\ref{fig:df:convfit} shows the obtained ``conversion factors'' and ``F-Factor'' computed as~\cite{MAGIC-calibration}: \begin{eqnarray}