Changeset 6559

Timestamp:

02/16/05 20:38:27 (20 years ago)

Author:

gaug

Message:

*** empty log message ***

Location:

trunk/MagicSoft/TDAS-Extractor

Files:

• Criteria.tex (modified) (1 diff)
• Pedestal.tex (modified) (1 diff)

trunk/MagicSoft/TDAS-Extractor/Criteria.tex

@@ -1 @@
-\section{Criteria for the Optimal Signal Extraction (STILL TO DO!!)}
+\section{Criteria for the Optimal Signal Extraction}
 
-The goal for the optimal signal reconstruction algorithm is to compute an unbiased estimate of the strenght and arrival time of the Cherenkov signal with the highest possible resolution for all signal intensities. The algorithm shall be stable with respect to changes in observation conditions and background levels. Also the needed computing time is of concern.
-
+The goal for the optimal signal reconstruction algorithm is to compute an unbiased estimate of the strength and arrival time of the 
+Cherenkov signal with the highest possible resolution for all signal intensities. The MAGIC telescope has been optimized to 
+lower the energy treshold of observation in any respect. Particularly the choice for an FADC system has been made with an eye on the 
+possibility to extract the smallest possible signals from air showers. It would be inconsequent not to continue the optimization procedure
+in the signal extraction algorithms and the subsequent image cleaning. 
+\par
+In the image analysis, one hake the decision whether the extracted signal of a certain pixel is considered as signal or background. 
+Those considered as signal are further used to compute the image parameters while the background ones are simply rejected. The calculation 
+of the second moments of the image ``ellipse'' usually fails when applied to un-cleaned images, therefore the decision is yes or no. Moreover,
+already low contributions of mis-estimated background can degrade the resolution of the image parameters considerably. If one wants to 
+lower the threshold for signal recognition, it is therefore mandatory to increase the efficiency with which the background is recognized as 
+such. If the background resolution is bad, the signal threshold goes up and vice versa.
+\par
+The algorithm must be stable with respect to changes 
+in observation conditions and background levels and between signals induced from gamma or hadronic showers or from muons. 
 The reconstructed signal shall be proportional to the total integrated charge in the FADCs due to the PMT pulse from the Cherenkov signal.
 
-Discussion about the signal to noise for the image cleaning a la Maxim.
+Also the needed computing time is of concern.
 
+\subsection{Bias and Mean-squared Error}
 
+Consider a large number of same signals $S$. 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
 
-\subsection{Resolution and Bias} 
-\ldots {\textit{The jitter to identical input pulses is measured, for times, amplitudes, 
-high-gain and low-gain pulses and different signal and background levels }}
+\begin{equation}
+X = \widehat{S}-S 
+\end{equation}
+
+has the mean $B$ and the Variance $MSE$ defined as:
+
+\begin{eqnarray}
+   B   \ \ \ \  = \ \ \ \ \ \ <X> \ \ \ \ \  &=& \ \ <\widehat{S}> \ -\ S\\
+   R   \ \ \ \  = \ <(X-B)^2> &=& \ Var[\widehat{S}]\\
+   MSE \      = \ \ \ \ \ <X^2> \ \ \ \  &=& \ Var[\widehat{S}] +\ B^2
+\end{eqnarray}
+
+The parameter $B$ is also called the {\textit{\bf BIAS}} of the estimator and $MSE$ 
+the {\textit{\bf MEAN-SQUARED ERROR}} which combines the variance of $\widehat{S}$ and 
+the bias. Both depend generally on the size of $S$ and the background fluctuations $BG$, 
+thus: $B = B(S,BG)$ and $MSE = MSE(S,BG)$.
+
+\par
+Usually, one measures easily the parameter $R$, but needs the $MSE$ for statistical analysis (e.g. 
+in the image cleaning). 
+However, only in case of a vanishing bias $B$, the two numbers are equal. Otherwise, 
+the bias $B$ has to be known beforehand. Note that every sliding window extractor has a 
+bias, especially at low or vanishing signals $S$.
 
 \subsection{Linearity}

trunk/MagicSoft/TDAS-Extractor/Pedestal.tex

@@ -11 +11 @@
 By definition, the $\boldsymbol{B}$ and thus the ``pedestal RMS'' 
 is independent from the signal extractor.
-
-\subsection{Bias and Mean-squared Error}
-
-Consider a large number of same signals $S$. 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
-
-\begin{equation}
-X = \widehat{S}-S 
-\end{equation}
-
-has the mean $B$ and the Variance $MSE$ defined as:
-
-\begin{eqnarray}
-   B   \ \ \ \  = \ \ \ \ \ \ <X> \ \ \ \ \  &=& \ \ <\widehat{S}> \ -\ S\\
-   R   \ \ \ \  = \ <(X-B)^2> &=& \ Var[\widehat{S}]\\
-   MSE \      = \ \ \ \ \ <X^2> \ \ \ \  &=& \ Var[\widehat{S}] +\ B^2
-\end{eqnarray}
-
-The parameter $B$ is also called the {\textit{\bf BIAS}} of the estimator and $MSE$ 
-the {\textit{\bf MEAN-SQUARED ERROR}} which combines the variance of $\widehat{S}$ and 
-the bias. Both depend generally on the size of $S$ and the background fluctuations $BG$, 
-thus: $B = B(S,BG)$ and $MSE = MSE(S,BG)$.
-
-\par
-Usually, one measures easily the parameter $R$, but needs the $MSE$ for statistical analysis (e.g. 
-in the image cleaning). 
-However, only in case of a vanishing bias $B$, the two numbers are equal. Otherwise, 
-the bias $B$ has to be known beforehand. Note that every sliding window extractor has a 
-bias, especially at low or vanishing signals $S$.
 
 \subsection{Pedestal Fluctuations as Contribution to the Signal Fluctuations \label{sec:ffactor}}