Index: trunk/MagicSoft/TDAS-Extractor/Pedestal.tex =================================================================== --- trunk/MagicSoft/TDAS-Extractor/Pedestal.tex (revision 6406) +++ trunk/MagicSoft/TDAS-Extractor/Pedestal.tex (revision 6407) @@ -173,5 +173,5 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsubsection{ \label{sec:determiner} Application of the Signal Extractor to a Fixed Window +\subsubsection{ \label{sec:ped:fixedwindow} Application of the Signal Extractor to a Fixed Window of Pedestal Events} @@ -283,5 +283,5 @@ -\subsubsection{ \label{sec:determiner} Application of the Signal Extractor to a Sliding Window +\subsubsection{ \label{sec:ped:slidingwindow} Application of the Signal Extractor to a Sliding Window of Pedestal Events} @@ -298,5 +298,7 @@ a sliding window. In this sample, every extractor had the freedom to move 5 slices, i.e. the global window size was fixed to five plus the extractor window size. This first line -shows the resolution of the smallest, robust fixed window algorithm in order to give a reference. +shows the resolution of the smallest existing robust fixed window algorithm in order to give the reference +value of 2.5 and 3 photo-electrons RMS. +\par One can see that the bias $B$ typically decreases with increasing window size (except for the digital filter), while the error $R$ increases with @@ -308,5 +310,7 @@ at 4 slices. The global winners are extractors \#25 (spline with integration of 1 slice) and \#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. +have a smaller mean-square error than the resolution of the fixed window reference extractor. 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. \begin{table}[htp] @@ -328,4 +332,5 @@ 4 & Fixed Win. 8 & 1.2 & -- & 0.0 & 1.2 & 2.5 & -- & 0.0 & 2.5 & 3.0 & -- & 0.0 & 3.0 \\ \hline +-- & Slid. Win. 1 & 0.4 & 0.4 & 0.4 & 0.6 & 1.2 & 1.2 & 1.3 & 1.8 & 1.4 & 1.4 & 1.5 & 2.0 \\ 17 & Slid. Win. 2 & 0.5 & 0.5 & 0.4 & 0.6 & 1.4 & 1.4 & 1.2 & 1.8 & 1.6 & 1.6 & 1.5 & 2.2 \\ 18 & Slid. Win. 4 & 0.8 & 0.8 & 0.5 & 0.9 & 1.9 & 1.9 & 1.2 & 2.2 & 2.2 & 2.3 & 1.6 & 2.8 \\ @@ -346,5 +351,5 @@ } \caption{The statistical parameters bias, resolution and mean error for the sliding window -algorithm. The first line displays the resolution of the smallest, robust fixed window extractor +algorithm. 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 @@ -356,16 +361,10 @@ \end{table} - - - -\par -Figures~\ref{fig:amp:distped} through~\ref{fig:df:distped} show the -extracted pedestal distributions for the digital filter with cosmics weights (extractor~\#28) and the -spline amplitude (extractor~\#27), respectively for one examplary channel (corresponding to pixel 200). +Figures~\ref{fig:sw:distped} through~\ref{fig:df:distped} show the +extracted pedestal distributions for some selected extractors (\#18, \#23, \#25 and \#28) + for one examplary channel (pixel 100) and two background situations: Closed camera with only electronic +noise and open camera pointing to an extra-galactic source. One can see the (asymmetric) Poisson behaviour of the -night sky background photons for the distributions with open camera and the cutoff at the lower egde -for the distribution with high-intensity continuous light due to a limited pedestal offset and the cutoff -to negative fluctuations. -\par +night sky background photons for the distributions with open camera. \begin{figure}[htp] @@ -435,6 +434,11 @@ \end{figure} -\par - +\subsection{ \label{sec:ped:singlephe} Single Photo-Electron Extraction with the Digital Filter} + +Figures~\ref{fig:df:sphespectrum} show spectra +obtained with the digital filter applied on two different global search windows. +One can clearly distinguish a pedestal peak (fitted to Gaussian with index 0) +and further, positive contributions. +\par Because the background is determined by the single photo-electrons from the night-sky background, the following possibilities can occur: @@ -444,5 +448,7 @@ finds only electronic noise. Usually, the returned signal charge is then negative. -\item The extractor finds the signal from one photo-electron +\item There is one photo-electron in the extraction window and the extractor finds it. +\item There are more than on photo-electrons in the extraction window, but separated by more than +two FADC slices whereupon the extractor finds the one with the highest charge (upward fluctuation). \item The extractor finds an overlap of two or more photo-electrons. \end{enumerate} @@ -460,5 +466,5 @@ Given a global extraction window of size $WS$ and an average rate of photo-electrons from the night-sky background $R$, we will now calculate the probability for the extractor to find zero photo-electrons in the -$WS$. The probability to find $k$ photo-electrons can be written as: +$WS$. The probability to find any number of $k$ photo-electrons can be written as: \begin{equation} @@ -472,5 +478,5 @@ \end{equation} -The probability to find more than one photo-electron is then: +The probability to find one or more photo-electrons is then: \begin{equation} @@ -478,7 +484,6 @@ \end{equation} -Figures~\ref{fig:sphe:sphespectrum} show spectra -obtained with the digital filter applied on two different global search windows. -One can clearly distinguish a pedestal peak (fitted to Gaussian with index 0), +In figures~\ref{fig:df:sphespectrum}, +one can clearly distinguish the pedestal peak (fitted to Gaussian with index 0), corresponding to the case of  $P(0)$ and further contributions of $P(1)$ and $P(2)$ (fitted to Gaussians with index 1 and 2). @@ -518,8 +523,15 @@ We estimated the effective window size $WS$ as the sum of the range in which the digital filter -amplitude weights are greater than 0.5 (1.6 FADC slices) and the global search window minus the -size of the window size of the weights (which is 6 FADC slices). Figures~\ref{fig::df:ratiofit} -show the result for two different levels of night-sky background. - +amplitude weights are greater than 0.5 (1.5 FADC slices) and the global search window minus the +size of the window size of the weights (which is 6 FADC slices). Figures~\ref{fig:df:ratiofit} +show the result for two different levels of night-sky background. The fitted rates deliver +0.08 and 0.1 phes/ns, respectively. These rates are about 50\% too low compared to the results obtained +in the November 2004 test campaign. However, we should take into account that the method is at +the limit of distinguishing single photo-electrons. It may occur often that a single photo-electron +signal is too low in order to get recognized as such. We tried various pixels and found that +some of them do not permit to apply this method at all. The ones which succeed, however, yield about +the same fitted rates. To conclude, one may say that there is consistency within the double-peak +structure of the pedestal spectrum found by the digital filter which can be explained by the fact that +single photo-electrons are found. \par @@ -534,6 +546,41 @@ In the top plot, a pedestal run with extra-galactic star background has been taken and in the bottom, a galatic star background. An exemplary pixel (Nr. 100) has been used. -Above, a rate of 0.8 phe/ns and below, a rate of 1.0 phe/ns has been obtained.} +Above, a rate of 0.08 phe/ns and below, a rate of 0.1 phe/ns has been obtained.} \label{fig:df:ratiofit} +\end{figure} + +Figure~\ref{fig:df:convfit} shows the obtained ``conversion factors'' and ``F-Factor'' computed as: + +\begin{eqnarray} +c_{phe} &=& \frac{1}{\mu_1 - \mu_0} \\ +F_{phe} &=& \sqrt{1 + \frac{\sigma_1^2 - \sigma_0^2}{(\mu_1 - \mu_0)^2} } +\end{eqnarray} + +where $\mu_0$ is the mean position of the pedestal peak and $\mu_1$ the mean position of the (assumed) +single photo-electron peak. The obtained conversion factors are systematically lower than the ones +obtained from the standard calibration and decrease with increasing window size. This is consistent +with the assumption that the digital filter finds the upward fluctuating pulse out of several. Therefore, +$\mu_1$ is biased against higher values. The F-Factor is also systematically low, which is also consistent +with the assumption that the spacing between $\mu_1$ and $\mu_0$ is artificially high. One can also see +that the error bars are too high for a ``calibration'' of the F-Factor. +\par +In conclusion, one can say that the digital filter is at the edge of being able to see single photo-electrons, +however a single photo-electron calibration cannot yet be done with the current FADC system because the +resolution is too poor. + +\begin{figure}[htp] +\centering +\includegraphics[height=0.4\textheight]{ConvFactor-28-Run38995.eps} +\vspace{\floatsep} +\includegraphics[height=0.4\textheight]{FFactor-28-Run38995.eps} +\caption{MExtractTimeAndChargeDigitalFilter: Obtained conversion factors (top) and F-Factors (bottom) +from the position and width of +the fitted Gaussian mean of the single photo-electron peak and the pedestal peak depending on +the applied global extraction window sizes. +A pedestal run with extra-galactic star background has been taken and +an exemplary pixel (Nr. 100) used. The conversion factor obtained from the +standard calibration is shown as a reference line. The obtained conversion factors are systematically +lower than the reference one.} +\label{fig:df:convfit} \end{figure}