Changeset 6437 for trunk/MagicSoft/TDAS-Extractor/Algorithms.tex
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- 02/13/05 21:07:35 (20 years ago)
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trunk/MagicSoft/TDAS-Extractor/Algorithms.tex (modified) (12 diffs)
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trunk/MagicSoft/TDAS-Extractor/Algorithms.tex
r6435 r6437 16 16 \begin{figure}[htp] 17 17 \includegraphics[width=0.99\linewidth]{ExtractorClasses.eps} 18 \caption{Sketch of the inheritances of three ex amplary MARS signal extractor classes:18 \caption{Sketch of the inheritances of three exemplary MARS signal extractor classes: 19 19 MExtractFixedWindow, MExtractTimeFastSpline and MExtractTimeAndChargeDigitalFilter} 20 20 \label{fig:extractorclasses} … … 302 302 \begin{description} 303 303 \item[Extraction Type Amplitude:\xspace] The amplitude of the spline maximum is taken as charge signal 304 and the (precise e) position of the maximum is returned as arrival time. This type is faster, since it305 performs not spline inte rgraion.304 and the (precise) position of the maximum is returned as arrival time. This type is faster, since it 305 performs not spline integration. 306 306 \item[Extraction Type Integral:\xspace] The integrated spline between maximum position minus 307 307 rise time (default: 1.5 slices) and maximum position plus fall time (default: 4.5 slices) … … 359 359 360 360 361 The pulse shape is mainly determined by the artificial pulse stretching by about 6 ns on the receiver board. Thus the first assumption is hold. Also the second assumption is fullfilled: Signal and noise are independent and the measured pulse is the linear superposition of the signal and noise. The validity of the third assumption is discussed below, especially for diffent night sky background conditions. 361 The pulse shape is mainly determined by the artificial pulse stretching by about 6 ns on the receiver board. 362 Thus the first assumption holds. Also the second assumption is fulfilled: Signal and noise are independent 363 and the measured pulse is the linear superposition of the signal and noise. The validity of the third 364 assumption is discussed below, especially for different night sky background conditions. 362 365 363 366 Let $g(t)$ be the normalized signal shape, $E$ the signal amplitude and $\tau$ the time shift … … 452 455 \par 453 456 Because of the truncation of the Taylor series in equation (\ref{shape_taylor_approx}) the above results are 454 only valid for vanishing time offsets $\tau$. For non-zero time offsets one has to iterate the problem using457 only valid for vanishing time offsets $\tau$. For non-zero time offsets, one has to iterate the problem using 455 458 the time shifted signal shape $g(t-\tau)$. 456 459 … … 458 461 459 462 \begin{equation} 460 \left(\boldsymbol{V}^{-1}\right)_{i ,j}463 \left(\boldsymbol{V}^{-1}\right)_{ij} 461 464 =\frac{1}{2}\left(\frac{\partial^2 \chi^2(E, E\tau)}{\partial \alpha_i \partial \alpha_j} \right) \quad 462 465 \text{with} \quad \alpha_i,\alpha_j \in \{E, E\tau\} \ . … … 482 485 483 486 484 In the MAGIC MC simulations \cite{MC-Camera} a LONS rate of 0.13 photoelectrons per ns, an FADC gain of 7.8 FADC counts per photoelectron and an intrinsic FADC noise of 1.3 FADC counts per FADC slice is implemented. This simulates the night sky background conditions for an extragalactic source. This results in a noise of about 4 FADC counts per single FADC slice: $<b_i^2> \approx 4$~FADC counts. Using the digital filter with weights parameterized over 6 FADC slices ($i=1...5$) the error of the reconstructed signal and time is give by: 485 486 \begin{equation} 487 \sigma_E \approx 8.3 \ \mathrm{FADC\ counts} \qquad \sigma_{\tau} \approx \frac{6.5\ \Delta T_{\mathrm{FADC}}}{(E\ /\ \mathrm{FADC\ counts})} \ , 487 In the MAGIC MC simulations~\cite{MC-Camera}, an night-sky background rate of 0.13 photoelectrons per ns, 488 an FADC gain of 7.8 FADC counts per photo-electron and an intrinsic FADC noise of 1.3 FADC counts 489 per FADC slice is implemented. 490 These numbers simulate the night sky background conditions for an extragalactic source and result 491 in a noise contribution of about 4 FADC counts per single FADC slice: 492 $\sqrt{B_{ii}} \approx 4$~FADC counts. 493 Using the digital filter with weights parameterized over 6 FADC slices ($i=0...5$) the errors of the 494 reconstructed signal and time amount to: 495 496 \begin{equation} 497 \sigma_E \approx 8.3 \ \mathrm{FADC\ counts} \ (\approx 1.1\,\mathrm{phe}) \qquad 498 \sigma_{\tau} \approx \frac{6.5\ \Delta T_{\mathrm{FADC}}}{(E\ /\ \mathrm{FADC\ counts})} \ (\approx \frac{2.8\,\mathrm{ns}}{E\,/\ \mathrm{N_{phe}}})\ , 488 499 \label{eq:of_noise_calc} 489 500 \end{equation} 490 501 491 where $\Delta T_{\mathrm{FADC}} = 3.33$ ns is the sampling interval of the MAGIC FADCs. The error in the reconstructed signal correspons to about one photo electron. For signals of two photo electrons size the timing error is about 1 ns. 492 493 %For the MAGIC signals, as implemented in the MC simulations \cite{MC-Camera}, a pedestal RMS of a single FADC slice of 6 FADC counts introduces an error in the reconstructed signal and time of: 494 495 For an IACT there are two types of background noise. On the one hand, there is the constantly present 496 electronics noise, 497 on the other hand, the light of the night sky introduces a sizeable background noise to the measurement of 498 Cherenkov photons from air showers. 499 500 The electronics noise is largely white, uncorrelated in time. The noise from the night sky background photons 501 is the superposition of the 502 where $\Delta T_{\mathrm{FADC}} = 3.33$ ns is the sampling interval of the MAGIC FADCs. 503 The error in the reconstructed signal corresponds to about one photo electron. 504 For signals of the size of two photo electrons, the timing error is a bit higher than 1\,ns. 505 \par 506 507 An IACT has typically two types of background noise: 508 On the one hand, there is the constantly present electronics noise, 509 while on the other hand, the light of the night sky introduces a sizeable background 510 to the measurement of the Cherenkov photons from air showers. 511 512 The electronics noise is largely white, i.e. uncorrelated in time. 513 The noise from the night sky background photons is the superposition of the 502 514 detector response to single photo electrons following a Poisson distribution in time. 503 515 Figure \ref{fig:noise_autocorr_allpixels} shows the noise 504 autocorrelation matrix for an open camera. The large noise autocorrelation in time of the current FADC 505 system is due to the pulse shaping with a shaping constant of 6 ns. 506 507 In general, the amplitude and time weights, $\boldsymbol{w}_{\text{amp}}$ and $\boldsymbol{w}_{\text{time}}$, depend on the pulse shape, the 508 derivative of the pulse shape and the noise autocorrelation. In the high gain samples the correlated night sky background noise dominates over 509 the white electronics noise. Thus different noise levels just cause the noise autocorrelation matrix $\boldsymbol{B}$ to change by a same factor, 510 which cancels out in the weights calculation. Thus in the high gain the weights are to a very good approximation independent of the night 511 sky background noise level. 516 autocorrelation matrix for an open camera. The large noise autocorrelation of the current FADC 517 system is due to the pulse shaping (with the shaping constant equivalent to about two FADC slices). 518 519 In general, the amplitude and time weights, $\boldsymbol{w}_{\text{amp}}$ and $\boldsymbol{w}_{\text{time}}$, 520 depend on the pulse shape, the derivative of the pulse shape and the noise autocorrelation. 521 In the high gain samples, the correlated night sky background noise dominates over 522 the white electronics noise. Thus, different noise levels just cause the members of the noise autocorrelation 523 matrix to change by a same factor, 524 which cancels out in the weights calculation. 525 Thus, the weights are to a very good approximation independent from the night 526 sky background noise level in the high gain. 512 527 513 528 Contrary to that in the low gain samples ... . … … 518 533 519 534 520 \begin{figure}[h!]521 \begin{center}522 \includegraphics[totalheight=7cm]{noise_autocorr_AB_36038_TDAS.eps}523 \end{center}524 \caption[Noise autocorrelation one pixel.]{Noise autocorrelation525 matrix $\boldsymbol{B}$ for open camera including the noise due to night sky background fluctuations526 for one single pixel (obtained from 1000 events).}527 \label{fig:noise_autocorr_1pix}528 \end{figure}535 %\begin{figure}[h!] 536 %\begin{center} 537 %\includegraphics[totalheight=7cm]{noise_autocorr_AB_36038_TDAS.eps} 538 %\end{center} 539 %\caption[Noise autocorrelation one pixel.]{Noise autocorrelation 540 %matrix $\boldsymbol{B}$ for open camera including the noise due to night sky background fluctuations 541 %for one single pixel (obtained from 1000 events).} 542 %\label{fig:noise_autocorr_1pix} 543 %\end{figure} 529 544 530 545 \begin{figure}[htp] … … 641 656 $e\tau(t_0)=\sum_{i=0}^{i=n-1} w_{\mathrm{time}}(t_0+i \cdot T_{\text{ADC}})y(t_0+i \cdot T_{\text{ADC}})$ as a function of the time shift $t_0$.} 642 657 \label{fig:amp_sliding} 643 \end{figure} 658 \end{figure}in the high gain 644 659 645 660 … … 679 694 \item "calibration\_weights4\_blue.dat'' with a window size of 4 FADC slices 680 695 \item "calibration\_weights\_UV.dat'' with a window size of 6 FADC slices and in the low-gain the 681 calibration weig ths obtained from blue pulses\footnote{UV-pulses saturating the high-gain are not yet696 calibration weights obtained from blue pulses\footnote{UV-pulses saturating the high-gain are not yet 682 697 available.}. 683 698 \item "calibration\_weights4\_UV.dat'' with a window size of 4 FADC slices and in the low-gain the 684 calibration weig ths obtained from blue pulses\footnote{UV-pulses saturating the high-gain are not yet699 calibration weights obtained from blue pulses\footnote{UV-pulses saturating the high-gain are not yet 685 700 available.}. 686 701 \item "cosmics\_weights\_logaintest.dat'' with a window size of 6 FADC slices and swapped high-gain and low-gain … … 779 794 780 795 \begin{description} 781 \item[MExtractFixedWindow]: with the following in tialization, if {\textit{maxbin}} defines the796 \item[MExtractFixedWindow]: with the following initialization, if {\textit{maxbin}} defines the 782 797 mean position of the high-gain FADC slice which carries the pulse maximum \footnote{The function 783 798 {\textit{MExtractor::SetRange(higain first, higain last, logain first, logain last)}} sets the extraction 784 799 range with the high gain start bin {\textit{higain first}} to (including) the last bin {\textit{higain last}}. 785 Analogu ouslyfor the low gain extraction range. Note that in MARS, the low-gain FADC samples start with800 Analogue for the low gain extraction range. Note that in MARS, the low-gain FADC samples start with 786 801 the index 0 again, thus {\textit{maxbin+0.5}} means in reality {\textit{maxbin+15+0.5}}. } 787 802 : … … 797 812 {\textit{MExtractor::SetRange(higain first, higain last, logain first, logain last)}} sets the extraction 798 813 range with the high gain start bin {\textit{higain first}} to (including) the last bin {\textit{higain last}}. 799 Analogu ouslyfor the low gain extraction range. Note that in MARS, the low-gain FADC samples start with814 Analogue for the low gain extraction range. Note that in MARS, the low-gain FADC samples start with 800 815 the index 0 again, thus {\textit{maxbin+0.5}} means in reality {\textit{maxbin+15+0.5}}.}: 801 816 \resume{enumerate} … … 874 889 %%% mode: latex 875 890 %%% TeX-master: "MAGIC_signal_reco" 891 %%% TeX-master: "MAGIC_signal_reco" 876 892 %%% End:
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