Changeset 6661 for trunk/MagicSoft/TDAS-Extractor
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- 02/23/05 14:31:21 (20 years ago)
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trunk/MagicSoft/TDAS-Extractor/Algorithms.tex (modified) (16 diffs)
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trunk/MagicSoft/TDAS-Extractor/Algorithms.tex
r6653 r6661 369 369 where $b(t)$ is the time-dependent noise contribution. For small time shifts $\tau$ (usually smaller than 370 370 one FADC slice width), 371 the time dependence can be linearized by the use of a Taylor expansion:371 the time dependence can be linearized: 372 372 373 373 \begin{equation} \label{shape_taylor_approx} … … 392 392 %\equiv \langle b_i b_j \rangle with $\langle b_i \rangle = 0$. 393 393 394 The signal amplitude $E$, and the product of amplitude and time shift $E \tau$, can be estimated from the given set of395 measurements $\boldsymbol{y} = (y_1, ... ,y_n)$ by minimizing the excess noise contribution with respect to the known noise396 auto-correlation:394 The signal amplitude $E$, and the product $E \tau$ of amplitude and time shift, can be estimated from the given set of 395 measurements $\boldsymbol{y} = (y_1, ... ,y_n)$ by minimizing the deviation of the measured FADC slice contents from the 396 known pulse shape with respect to the known noise auto-correlation: 397 397 398 398 \begin{eqnarray} … … 402 402 \end{eqnarray} 403 403 404 where the last expression is matricial. 405 $\chi^2$ is a continuous function of $\tau$ and will have to be discretized itself for a 406 desired resolution. 407 $\chi^2$ is in principle independent of the noise level if alway the appropriate noise autocorrelation matrix is used. In our case however, we decided to use one matrix $\boldsymbol{B}$ for all levels of night-sky background. Changes in the noise level lead only to a multiplicative factor for all matrix elements and thus do not affect the position of the minimum of $\chi^2$. 404 where the last expression uses the matrix formalism. 405 $\chi^2$ is a continuous function of $\tau$ and will have to be discretized itself for numerical computation applications. 406 $\chi^2$ is in principle independent of the noise level if always the appropriate noise autocorrelation matrix is used. 407 In our case however, we decided to use one matrix $\boldsymbol{B}$ for all levels of night-sky background. 408 Changes in the noise level lead only to a multiplicative factor for all matrix elements and thus do not affect the 409 position of the minimum of $\chi^2$. 408 410 The minimum of $\chi^2$ is obtained for: 409 411 … … 444 446 445 447 Thus $\overline{E}$ and $\overline{E\tau}$ are given by a weighted sum of the discrete measurements $y_i$ 446 with the digital filtering weights for the amplitude, $w_{\text{amp}}(\tau)$, and time shift, $w_{\text{time}}(\tau)$. The time dependency gets discretized once again leading to a set of weights samples which themselves depend on the 448 with the weights for the amplitude, $w_{\text{amp}}(\tau)$, and time shift, $w_{\text{time}}(\tau)$. 449 \par 450 The time dependence gets discretized once again leading to a set of weight samples which themselves depend on the 447 451 discretized time $\tau$. 448 452 \par 449 Note the remaining time dependency of the two weight s samples. This follows from the dependencyof $\boldsymbol{g}$ and453 Note the remaining time dependency of the two weight samples. This follows from the dependence of $\boldsymbol{g}$ and 450 454 $\dot{\boldsymbol{g}}$ on the relative position of the signal pulse with respect to FADC slices positions. 451 455 \par 452 456 Because of the truncation of the Taylor series in equation (\ref{shape_taylor_approx}) the above results are 453 only valid for vanishing time offsets $\tau$. For non-zerotime offsets, one has to iterate the problem using457 only valid for vanishing time offsets $\tau$. For larger time offsets, one has to iterate the problem using 454 458 the time shifted signal shape $g(t-\tau)$. 455 459 … … 474 478 475 479 \begin{equation} 476 E^2 \cdot \sigma_{\tau}^2 =\boldsymbol{V}_{E\tau,E\tau}480 E^2 \cdot \sigma_{\tau}^2 < \sigma^2_{E \tau} =\boldsymbol{V}_{E\tau,E\tau} 477 481 =\frac{{\boldsymbol{g}}^T\boldsymbol{B}^{-1}{\boldsymbol{g}}} 478 482 {(\boldsymbol{g}^T \boldsymbol{B}^{-1} \boldsymbol{g})(\dot{\boldsymbol{g}}^T\boldsymbol{B}^{-1}\dot{\boldsymbol{g}}) -(\dot{\boldsymbol{g}}^T\boldsymbol{B}^{-1}\boldsymbol{g})^2} \ . … … 480 484 \end{equation} 481 485 482 483 In the MAGIC MC simulations~\cite{MC-Camera}, an night-sky background rate of 0.13 photoelectrons per ns, 486 Both equations~\ref{eq:of_noise} and~\ref{eq:of_noise_time} are independent of the signal amplitude. 487 \par 488 In the MAGIC MC simulations~\cite{MC-Camera}, a night-sky background rate of 0.13 photoelectrons per ns, 484 489 an FADC gain of 7.8 FADC counts per photo-electron and an intrinsic FADC noise of 1.3 FADC counts 485 490 per FADC slice is implemented. … … 487 492 in a noise contribution of about 4 FADC counts per single FADC slice: 488 493 $\sqrt{B_{ii}} \approx 4$~FADC counts. 489 Using the digital filter with weights parameterized over 6FADC slices ($i=0...5$) the errors of the494 Using the digital filter with weights determined for 6~FADC slices ($i=0...5$) the errors of the 490 495 reconstructed signal and time amount to: 491 496 … … 516 521 depend on the pulse shape, the derivative of the pulse shape and the noise autocorrelation. 517 522 In the high gain samples, the correlated night sky background noise dominates over 518 the white electronics noise. Thus, different noise levels cause the elements of the noise autocorrelation519 matrix to change by asame factor, which cancels out in the weights calculation. Figure~\ref{fig:noise_autocorr_allpixels}523 the white electronics noise. As a consequence, different noise levels cause the elements of the noise autocorrelation 524 matrix to change by the same factor, which cancels out in the weights calculation. Figure~\ref{fig:noise_autocorr_allpixels} 520 525 shows the noise autocorrelation matrix for two different levels of night sky background (top) and the ratio between the 521 526 corresponding elements of both (bottom). The central regions of $\pm$3 FADC slices around the diagonal (which is used to … … 523 528 Thus, the weights are to a reasonable approximation independent of the night sky background noise level in the high gain. 524 529 \par 525 In the low gain samples the correlated noise from the LONS is in the same order of magnitude as the white electronics and digitization noise. 530 In the low gain samples the correlated noise of the LONS is in the same order of magnitude as the white electronics 531 and digitization noise. 526 532 Moreover, the noise autocorrelation for the low gain samples cannot be determined directly from the data. The low gain is only switched on 527 533 if the pulse exceeds a preset threshold. There are no pedestals in the low gain available. Thus the noise auto-correlation determined from … … 554 560 \end{figure} 555 561 556 Using the average reconstructed pulpo pulse shapes, as shown in figure \ref{fig:pulse_shapes}, and the reconstructed noise autocorrelation matrices from a pedestal runs with random triggers, the digital filter weights are computed. As the pulse shapes in the high and low gain and for cosmics, calibration and pulpo events are somewhat different, dedicated digital filter weights are computed for these event classes. Also filter weights optimized for MC simulations are calculated. High/low gain filter weights are computed for the following event classes: 562 Using the average reconstructed pulpo pulse shapes, as shown in figure \ref{fig:pulse_shapes}, and the reconstructed noise 563 autocorrelation matrices from pedestal runs with random triggers, the digital filter weights are computed. 564 As the pulse shapes in the high and low gain and for cosmics, calibration and pulpo events are somewhat different, 565 dedicated digital filter weights are computed for these event classes. Also filter weights optimized for MC simulations are calculated. 566 High/low gain filter weights are computed for the following event classes: 557 567 558 568 \begin{enumerate} … … 583 593 584 594 585 Figures \ref{fig:w_time_MC_input_TDAS} and \ref{fig:w_amp_MC_input_TDAS} show the amplitude and timing weights for the MC pulse shape. The first weight $w_{\mathrm{amp/time}}(t_0)$ is plotted as a function of the relative time $t_{\text{rel}}$ the trigger and the 586 FADC clock in the range $[-0.5,0.5[ \ T_{\text{ADC}}$, the second weight in the range $[0.5,1.5[ \ T_{\text{ADC}}$ and so on. A binning resolution 587 of $0.1\,T_{\text{ADC}}$ has been chosen. 595 Figures \ref{fig:w_time_MC_input_TDAS} and \ref{fig:w_amp_MC_input_TDAS} show the amplitude and timing weights for the MC pulse shape. 596 The first weight $w_{\mathrm{amp/time}}(t_0)$ is plotted as a function of the relative time $t_{\text{rel}}$ between the trigger and the 597 FADC clock in the range $[-0.5,0.5[ \ T_{\text{ADC}}$, the second weight in the range $[0.5,1.5[ \ T_{\text{ADC}}$ and so on. 598 A binning resolution of $0.1\,T_{\text{ADC}}$ has been chosen. 588 599 589 600 … … 610 621 In the current implementation a two step procedure is applied to reconstruct the signal. The weight functions $w_{\mathrm{amp}}(t)$ 611 622 and $w_{\mathrm{time}}(t)$ are computed numerically with a resolution of $1/10$ of an FADC slice. 612 In the first step the quantities $e_{i_0}$ and $ e\tau_{i_0}$ are computed using a window of $n$ slices:623 In the first step the quantities $e_{i_0}$ and $(e\tau)_{i_0}$ are computed using a window of $n$ slices: 613 624 614 625 \begin{equation} … … 616 627 \end{equation} 617 628 618 for all possible signal start slices $i_0$. Let $i_0^*$ be the signal start slice withthe largest $e_{i_0}$.629 for all possible signal start slices $i_0$. Let $i_0^*$ be the signal start slice yielding the largest $e_{i_0}$. 619 630 Then in a second step the timing offset $\tau$ is calculated: 620 631 … … 624 635 \end{equation} 625 636 626 and the weights iterated:637 Using this value of $\tau$, another iteration is performed: 627 638 628 639 \begin{equation} … … 639 650 640 651 641 Figure \ref{fig:amp_sliding} shows the result of the applied amplitude and time weights to the recorded FADC time slices of one simulated MC pulse. The left plot shows the result of the applied amplitude weights 652 Figure \ref{fig:amp_sliding} shows the result of the applied amplitude and time weights to the recorded FADC time slices of one 653 simulated MC pulse. The left plot displayes the result of the applied amplitude weights 642 654 $e(t_0)=\sum_{i=0}^{i=n-1} w_{\mathrm{amp}}(t_0+i \cdot T_{\text{ADC}})y(t_0+i \cdot T_{\text{ADC}})$ and 643 655 the right plot shows the result of the applied timing weights … … 669 681 one simulated MC pulse. The left plot shows the result of the applied amplitude weights 670 682 $e(t_0)=\sum_{i=0}^{i=n-1} w_{\mathrm{amp}}(t_0+i \cdot T_{\text{ADC}})y(t_0+i \cdot T_{\text{ADC}})$ and 671 the right plot shows the result of the applied timing weights683 the right plot displayes the result of the applied timing weights 672 684 $e\tau(t_0)=\sum_{i=0}^{i=n-1} w_{\mathrm{time}}(t_0+i \cdot T_{\text{ADC}})y(t_0+i \cdot T_{\text{ADC}})$ as a function of the time shift $t_0$.} 673 685 \label{fig:amp_sliding} 674 686 \end{figure} 675 687 676 Figure \ref{fig:shape_fit_TDAS} shows the si mulated signal pulse shape of a typical MC event together with the simulated FADC slices of the signal pulse plus noise. The digital filter has been applied to reconstruct the signal size and timing. Using this information together with the average normalized MC pulse shape the simulated signal pulse shape is reconstructed and shown as well.677 678 679 688 Figure \ref{fig:shape_fit_TDAS} shows the signal pulse shape of a typical MC event together with the simulated FADC slices 689 of the signal pulse plus noise. The digital filter has been applied to reconstruct the signal size and timing. 690 Using this information together with the average normalized MC pulse shape the simulated signal pulse shape is reconstructed 691 and shown as well. 680 692 681 693
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