Index: trunk/MagicSoft/TDAS-Extractor/Calibration.tex =================================================================== --- trunk/MagicSoft/TDAS-Extractor/Calibration.tex (revision 6665) +++ trunk/MagicSoft/TDAS-Extractor/Calibration.tex (revision 6666) @@ -90,5 +90,5 @@ \par Although we had looked at and tested all colour and extractor combinations resulting from these data, -we refrain ourselves to show here only exemplary behaviour and results of extractors. +we restrict ourselves to show here only exemplary behaviour and results of extractors. All plots, including those which are not displayed in this TDAS, can be retrieved from the following locations: @@ -107,9 +107,10 @@ \begin{enumerate} -\item The reconstructed mean signal is less than 2.5 times the extractor resolution $R$ from zero. +\item The reconstructed mean signal $\widehat{Q}$ is less than 2.5 times the extractor resolution $R$: $\widehat{Q}<2.5\cdot R$. (2.5 Pedestal RMS in the case of the simple fixed window extractors, see section~\ref{sec:pedestals}). This criterium essentially cuts out dead pixels. -\item The reconstructed mean signal error is smaller than its value. This criterium cuts out +\item The error of the mean reconstructed signal $\delta \widehat{Q}$ is larger than the mean reconstructed signal $\widehat{Q}$: + $\delta \widehat{Q} > \widehat{Q}$. This criterion cuts out signal distributions which fluctuate so much that their RMS is bigger than its mean value. This criterium cuts out ``ringing'' pixels or mal-functioning extractors. @@ -121,5 +122,5 @@ \item All pixels with reconstructed negative mean signal or with a mean numbers of photo-electrons smaller than one. Pixels with a negative pedestal RMS subtracted -sigma occur, especially when stars are focused onto that pixel during the pedestal taking (resulting +sigma occur, especially when stars are focused onto that pixel during the pedestal run (resulting in a large pedestal RMS), but have moved to another pixel during the calibration run. In this case, the number of photo-electrons would result artificially negative. If these @@ -129,5 +130,5 @@ Moreover, the number of events are counted which have been reconstructed outside a 5$\sigma$ region -from the mean signal. These events are called ``outliers''. Figure~\ref{fig:outlier} shows a typical +from the mean signal $<\widehat{Q}>$. These events are called ``outliers''. Figure~\ref{fig:outlier} shows a typical outlier obtained with the digital filter applied on a low-gain signal, and figure~\ref{fig:unsuited:all} shows the average number of all excluded pixels and outliers obtained from all 19 calibration configurations. @@ -229,5 +230,5 @@ \begin{equation} -Var(S) = F^2 \cdot Var(N_{phe}) \cdot \frac{^2}{^2} +Var[S] = F^2 \cdot Var[N_{phe}] \cdot \frac{^2}{^2} \label{eq:excessnoise} \end{equation} @@ -236,10 +237,10 @@ and assuming that the variance of the number of photo-electrons is equal to the mean number of photo-electrons (because of the Poisson distribution), -one obtains an expression to retrieve the mean number of photo-electrons impinging on the photo-multiplier from the +one obtains an expression to retrieve the mean number of photo-electrons released at the photo-multiplier cathode from the mean extracted signal, $\widehat{S}$, and the RMS of the extracted signal obtained from pure pedestal runs $R$ (see section~\ref{sec:ffactor}): \begin{equation} - \approx F^2 \cdot \frac{<\widehat{S}>^2}{Var(\widehat{S}) - R^2} + \approx F^2 \cdot \frac{<\widehat{S}>^2}{Var[\widehat{S}] - R^2} \label{eq:pheffactor} \end{equation} @@ -247,5 +248,5 @@ In theory, eq.~\ref{eq:pheffactor} must not depend on the extractor! Effectively, we will use it to test the quality of our extractors by requiring that a valid extractor yields the same number of photo-electrons -for all pixels of a same type and does not deviate from the number obtained with other extractors. +for all pixels individually and does not deviate from the number obtained with other extractors. As the camera is flat-fielded, but the number of photo-electrons impinging on an inner and an outer pixel is different, we also use the ratio of the mean numbers of photo-electrons from the outer pixels to the one @@ -265,5 +266,5 @@ There is a considerable difference for all shown non-standard pulses. Especially the pulses from green and blue LEDs -show a clear dependency of the number of photo-electrons on the extraction window. Only the largest +show a clear dependence of the number of photo-electrons on the extraction window. Only the largest extraction windows seem to catch the entire range of (jittering) secondary pulses and get the ratio of outer vs. inner pixels right. However, they (obviously) over-estimate the number of photo-electrons @@ -384,6 +385,5 @@ Figure~\ref{fig:linear:phevscharge4} shows the conversion factor $c_{phe}$ obtained for different light intensities and colours for three exemplary inner and three exemplary outer pixels using a fixed window on -8 FADC slices. The conversion factor seems to be linear to a good approximation, -except for two cases: +8 FADC slices. The conversion factor seems to be linear to a good approximation, with the following restrictions: \begin{itemize} \item The green pulses yield systematically low conversion factors @@ -394,5 +394,5 @@ \end{itemize} -We conclude that, apart from the two reasons above, +We conclude that, with the above restrictions, the fixed window extractor \#4 is a linear extractor for both high-gain and low-gain regions, separately. @@ -402,5 +402,5 @@ using an integrated spline and a fixed window with global peak search, respectively, over an extraction window of 8 FADC slices. The same behaviour is obtained as before. These extractors are -linear to a good approximation, except for the two cases mentionned above. +linear to a good approximation, except for the two cases mentioned above. \par @@ -447,5 +447,5 @@ \par Figure~\ref{fig:linear:phevscharge20} shows the conversion factors using a sliding window of 6 FADC slices. -The linearity is maintained like in the previous examples, except for the smallest signals the effect +The linearity is maintained like in the previous examples, except that for the smallest signals the effect of the bias is already visible. \par @@ -463,5 +463,5 @@ Figure~\ref{fig:linear:phevscharge23} shows the conversion factors using the amplitude-extracting spline (extractor \#23). -Here, the linearity is worse than in the previous sample. A very clear difference between high-gain and +Here, the linearity is worse than in the previous examples. A very clear difference between high-gain and low-gain regions can be seen as well as a bigger general spread in conversion factors. In order to investigate if there is a common, systematic effect of the extractor, we show the averaged conversion factors over all @@ -514,5 +514,5 @@ Looking at figure~\ref{fig:linear:phevscharge25}, one can see that raising the integration window -to two effective FADC slices in the high-gain and three effective FADC slices in the low-gain +by two effective FADC slices in the high-gain and three effective FADC slices in the low-gain (extractor \#25), the stability is completely resumed, except for a systematic increase of the conversion factor above 200 photo-electrons. @@ -545,5 +545,5 @@ expected one. The effect is not as problematic as it may appear here, because the actual calibration will not calculate the number of photo-electrons (with the F-Factor method) for every signal intensity. -Thus, one possible reason for the instability falls away in the cosmics analysis. However, the limits +Thus, one possible reason for the instability is not relevant in the cosmics analysis. However, the limits of this extraction are visible here and should be monitored further. @@ -590,9 +590,9 @@ The calibration LEDs -deliver a fast-rising pulses, uniform over the camera in signal size and time. -We estimate the time-uniformity to better -than 300\,ps, a limit due to the different travel times of the light between inner and outer parts of the -camera. - +deliver fast-rising pulses, uniform over the camera in signal size and time. +We estimate the time-uniformity to as good as about~30\,ps, a limit due to the different travel times of the light +from the light source to the inner and outer parts of the camera. For cosmics data, however, the staggering of the +mirrors limits the time uniformity to about 600\,ps. +\par The extractors \#17--33 are able to compute the arrival time of each pulse. Since the calibration does not permit a precise measurement of the absolute arrival time, we measure @@ -609,5 +609,5 @@ systematic delays in the signal travel time, and a sigma $\sigma(\delta t_i)$, a measure of the combined time resolutions of pixel $i$ and pixel 1. Assuming that the PMTs and readout channels are -of a same kind, we obtain an approximate time resolution of pixel $i$: +of the same kind, we obtain an approximate time resolution of pixel $i$: \begin{equation} @@ -649,5 +649,6 @@ (fig.~\ref{fig:reltimesinnerledblue2} bottom), the distribution is perfectly Gaussian and the resolution good, however a rather slight change from the blue calibration pulse weights to cosmics pulses weights (top) -adds a secondary peak of events with mis-reconstructed arrival times. +adds a secondary peak of events with mis-reconstructed arrival times. In principle, the $\chi^2$ of the digital filter +fit gives an information about whether the correct shape has been used. \begin{figure}[htp] Index: trunk/MagicSoft/TDAS-Extractor/Criteria.tex =================================================================== --- trunk/MagicSoft/TDAS-Extractor/Criteria.tex (revision 6665) +++ trunk/MagicSoft/TDAS-Extractor/Criteria.tex (revision 6666) @@ -30,5 +30,5 @@ \subsection{Bias and Mean-squared Error} -Consider a large number of same signals $S$. By applying a signal extractor +Consider a large number of identical signals $S$, corresponding to a fixed number of photo-electrons. By applying a signal extractor we obtain a distribution of estimated signals $\widehat{S}$ (for fixed $S$ and fixed background fluctuations $BG$). The distribution of the quantity