Changeset 5536


Ignore:
Timestamp:
12/01/04 14:01:40 (20 years ago)
Author:
gaug
Message:
*** empty log message ***
Location:
trunk/MagicSoft/TDAS-Extractor
Files:
2 edited

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  • trunk/MagicSoft/TDAS-Extractor/Changelog

    r5376 r5536  
    1919
    2020                                                 -*-*- END OF LINE -*-*-
     21
     222004/12/01: Markus Gaug
     23  * Pedestals.tex: Modified writing a bit, added subsection about applying
     24    extractor to pedestals
    2125
    22262004/11/10: Hendrik Bartko

  • trunk/MagicSoft/TDAS-Extractor/Pedestal.tex

    r5370 r5536  
    44\begin{itemize}
    55\item Defining the pedestal RMS as contribution
    6 to the extracted signal fluctuations (later used in the calibration)
     6    to the extracted signal fluctuations (later used in the calibration)
    77\item Defining the Pedestal Mean and RMS as the result of distributions obtained by
    8 applying the extractor to pedestal runs (yielding biases and modified widths).
     8    applying the extractor to pedestal runs (yielding biases and modified widths).
    99\item Deriving the correct probability for background fluctuations based on the extracted signal height.
    1010  ( including biases and modified widths).
    1111\end{itemize}
    12 \ldots Florian + ???
    13 \newline
    14 \newline
    1512}
    1613
    17 \subsection{Copy of email by W. Wittek from 25 Oct 2004 and 10 Nov 2004}
     14\subsection{Pedestal RMS}
    1815
    1916
    20 \subsubsection{Pedestal RMS}
     17\vspace{1cm}
     18\ldots {\it  Modified email by W. Wittek from 25 Oct 2004 and 10 Nov 2004}
     19\vspace{1cm}
    2120
    22 We all know how it is defined. It can be completely
    23 described by the matrix
     21The Pedestal RMS can be completely described by the matrix
    2422
    2523\begin{equation}
    … …  
    2725\end{equation}
    2826
    29 where $i$ and $j$ denote the $i^{th}$ and $j^{th}$ FADC slice,
     27where $i$ and $j$ denote the $i^{th}$ and $j^{th}$ FADC slice and
    3028$P_i$ is the pedestal
    31 value in slice $i$ for an event and the average $<>$ is over many events.
     29value in slice $i$ for an event and the average $<>$ is over many events (usually 1000).
    3230\par
    3331
    34 By definition, the pedestal RMS is independent of the signal extractor.
    35 Therefore no signal extractor is needed for the pedestals.
     32By definition, the pedestal RMS is independent from the signal extractor.
     33Therefore, no signal extractor is needed to calculate the pedestals.
    3634
    37 \subsubsection{Bias and Error}
     35\subsection{Bias and Error}
    3836
    3937Consider a large number of signals (FADC spectra), all with the same
    4038integrated charge $ST$ (true signal). By applying some signal extractor
    4139we obtain a distribution of extracted signals $SE$ (for fixed $ST$ and
    42 fixed background fluctuations). The distribution of the quantity
     40fixed background fluctuations $BG$). The distribution of the quantity
    4341
    4442\begin{equation}
    … …  
    6058
    6159$B$ is the bias, $R$ is the RMS of the distribution of $X$ and $D$ is something
    62 like the (asymmetric) error of $SE$. It is the distribution of $X$ (or its
    63 parameters $B$ and $R$) which we are eventually interested in. The distribution
    64 of $X$, and thus the parameters $B$ and $R$, depend on $ST$ and the size of the
    65 background fluctuations.
    66 \par
    67 
    68 By applying the signal extractor to pedestal events you want to
    69 determine these parameters, I guess.
    70 
    71 \par
    72 By applying it with max. peak search you get information about the bias $B$
    73 for very low signals, not for high signals. By applying it to a fixed window,
    74 without max.peak search, you may get something like $R$ for high signals (but
    75 I am not sure).
    76 
    77 \par
    78 For the normal image cleaning, knowledge of $B$ is sufficient, because the
    79 error $R$ is not used anyway. You only want to cut off the low signals.
    80 
    81 \par
    82 For the model analysis you need both, $B$ and $R$, because you want to keep small
    83 signals. Unfortunately $B$ and $R$ depend on the size of the signal ($ST$) and on
    84 the size of the background fluctuations (BG). However, applying the signal
    85 extractor to pedestal events gives you only 1 number, dependent on BG but
    86 independent of $ST$.
     60like the (asymmetric) error of $SE$.
     61The distribution of $X$, and thus the parameters $B$ and $R$,
     62depend on the size of $ST$ and the size of the background fluctuations $BG$.
    8763
    8864\par
    8965
    90 Where do we get the missing information from ? I have no simple solution or
    91 answer, but I would think
    92 \begin{itemize}
    93 \item that you have to determine the bias from MC
    94 \item and you may gain information about $R$ from the fitted error of $SE$, which is
    95   known for every pixel and event
    96 \end{itemize}
    97 
    98 The question is 'How do we determine the $R$ ?'. A proposal which
    99 has been discussed in various messages is to apply the signal extractor to
    100 pedestal events. One can do that, however, this will give you information
    101 about the bias and the error of the extracted signal only for signals
    102 whose size is in the order of the pedestal fluctuations. This is certainly
    103 useful for defining the right level for the image cleaning.
     66For the normal image cleaning, knowledge of $B$ is sufficient and the
     67error $R$ should be know in order to calculate a correct background probability.
    10468\par
    105 
    106 However, because the bias $B$ and the error of the extracted signal $R$ depend on
    107 the size of the signal, applying the signal extractor to pedestal events
    108 won't give you the right answer for larger signals, for example for the
    109 calibration signals.
    110 
    111 The basic relation of the F-method is
     69Also for the model analysis $B$ and $R$ are needed, because you want to keep small
     70signals.
     71\par
     72In the case of the calibration with the F-Factor methoid,
     73the basic relation is:
    11274
    11375\begin{equation}
    114 \frac{sig^2}{<Q>^2} = \frac{1}{<m_{pe}>} * F^2
     76\frac{(\Delta ST)^2}{<ST>^2} = \frac{1}{<m_{pe}>} * F^2
    11577\end{equation}
    11678
    117 Here $sig$ is the fluctuation of the extracted signal $Q$ due to the
    118 fluctuation of the number of photo electrons. $sig$ is obtained from the
    119 measured fluctuations of $Q$  ($RMS_Q$) by subtracting the fluctuation of the
    120 extracted signal ($R$) due to the fluctuation of the pedestal RMS :
     79Here $\Delta ST$ is the fluctuation of the true signal $ST$ due to the
     80fluctuation of the number of photo electrons. $ST$ is obtained from the
     81measured fluctuations of $SE$  ($RMS_{SE}$) by subtracting the fluctuation of the
     82extracted signal ($R$) due to the fluctuation of the pedestal.
    12183
    12284\begin{equation}
    123  sig^2 = RMS_Q^2 - R^2
     85 (\Delta ST)^2 = RMS_{SE}^2 - R^2
    12486\end{equation}
    125 
    126 $R$ is in general different from the pedestal RMS. It cannot be
    127 obtained by applying the signal extractor to pedestal events, because
    128 the calibration signal is usually large.
    129 
    130 In the case of the optimum filter, $R$ may be obtained from the
    131 fitted error of the extracted signal ($dQ_{fitted}$), which one can calculate
    132 for every event. Whether this statemebt is true should be checked by MC.
    133 For large signals I would expect the bias of the extracted to be small and
    134 negligible.
    13587
    13688A way to check whether the right RMS has been subtracted is to make the
    … …  
    13890
    13991\begin{equation}
    140     \frac{sig^2}{<Q>^2} \quad \textit{vs.} \quad \frac{1}{<Q>}
     92    \frac{(\Delta ST)^2}{<ST>^2} \quad \textit{vs.} \quad \frac{1}{<ST>}
    14193\end{equation}
    14294
    … …  
    148100\end{equation}
    149101
    150 where $c$ is the photon/ADC conversion factor  $<Q>/<m_{pe}>$.
     102where $c$ is the photon/ADC conversion factor  $<ST>/<m_{pe}>$.
     103
     104\subsection{How to retrieve Bias $B$ and Error $R$}
     105
     106$R$ is in general different from the pedestal RMS. It cannot be
     107obtained by applying the signal extractor to pedestal events, especially
     108for large signals (e.g. calibration signals).
     109\par
     110In the case of the optimum filter, $R$ can be obtained from the
     111fitted error of the extracted signal ($\Delta(SE)_{fitted}$),
     112which one can calculate for every event.
     113
     114\vspace{1cm}
     115\ldots {\it Whether this statemebt is true should be checked by MC.}
     116\vspace{1cm}
     117
     118For large signals, one would expect the bias of the extracted signal
     119to be small and negligible (i.e. $<ST> \approx <SE>$).
     120\par
     121
     122In order to get the missing information, we did the following investigations:
     123\begin{enumerate}
     124\item Determine bias $B$ and resolution $R$ from MC events with and without added noise.
     125    Assuming that $R$ and $B$ are negligible for the events without noise, one can
     126    get a dependency of both values from the size of the signal.
     127\item Determine $R$ from the fitted error of $SE$, which is possible for the
     128    fit and the digital filter. In prinicple, all dependencies can be retrieved with this
     129    method.
     130\item Determine $R$ for low signals by applying the signal extractor to a fixed window
     131    of pedestal events. The background fluctuations can be simulated with different
     132    levels of night sky background and the continuous light, but no signal size
     133    dependency can be retrieved with the method. Its results are only valid for small
     134    signals.
     135\end{enumerate}
     136
     137\par
     138
     139\subsubsection{Determine error $R$ by applying the signal extractor to a fixed window
     140of pedestal events}
     141
     142By applying the signal extractor to pedestal events we want to
     143determine these parameters. There are the following possibilities:
     144
     145\begin{enumerate}
     146\item Applying the signal extractor allowing for a possible sliding window
     147    to get information about the bias $B$ (valid for low signals).
     148\item Applying the signal extractor to a fixed window, to get something like
     149    $R$. In the case of the digital filter, this has to be done by randomizing
     150    the time slice indices.
     151\end{enumerate}
     152
     153\vspace{1cm}
     154\ldots {\it This assumptions still have to proven, best mathematically!!! Wolfgang, Thomas???}
     155\vspace{1cm}
     156\par
     157
     158
     159\vspace{1cm}
     160\ldots{\it More test plots can be found under:
     161http://magic.ifae.es/$\sim$markus/ExtractorPedestals/ }
     162\vspace{1cm}
    151163
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