Full Survey of the TileCal correlated noise effect

Introduction


The goal of this page is to provide relevant information on the TileCal note: Full Survey of the TileCal correlated noise effect.

Contributors

  • D. Calvet, Lab. de Physique Corpusculaire (LPC), Inst. Nat. Phys. Nucl. et Particul. (IN2P3), Univ. Blaise Pascal Clermont-Fe. II, France
  • A. Onofre, M.C.N. Fiolhais, F.Veloso, LIP, Departamento de Fisica da Universidade de Coimbra, 3004-516 Coimbra, Portugal
  • L. Fiorini, Institut de Fisica d'Altes Energies, IFAE, Edifici Cn, Universitat Autonoma de Barcelona, ES-08193 Bellaterra (Barcelona), Spain
  • L. Gurriana, LIP, Av. Elias Garcia 14 - 1, 1000-149 Lisboa, Portugal
  • B. Martin, Michigan State University, USA
  • I. Minashvili, Joint Institute for Nuclear Research (JINR), Russia
  • A. A. Solodkov, Institute for High Energy Physics (IHEP), Federal Agency of Atom. Energy, Moscow Region, RU-142 284 Protvino, Russia
  • I. Vichou, University of Illinois, Department of Physics, 1110 West Green Street, Urbana, Illinois 61801, USA

Abstract

Following the study presented in the recently submitted ATLAS note Minimizing the TileCal correlated noise effect: a simple approach , a full survey of the TileCal correlations is shown in this note for different kinds of data runs. The analysis of special calibration runs allowed to adapt the correction algorithm based on a $\chi^2$ minimization, developed to remove the correlated component of the pedestal noise, to the presence of physics signals. The new method successfully removes the correlated noise component for every partition, in the extended and long barrels, of all 64 TileCal modules. Following the conclusions of previous work, the correlations pattern were included within the simulation code allowing to test the unfolding in top quark pair production events, not yet available as real data. Further work was also performed at the lab with TileCal testing modules to track the coherent noise sources.

The Method

Correlated noise plays an important role on the TileCal pedestal degradation. The simplest tool to evaluate the correlation effect is the calculation of the covariance between channels within the same partition using

$ cov(x_i,x_j) =  E[(x_i-\mu_i)(x_j-\mu_j)] = E[x_i x_j] - \mu_i \mu_j,$

where $ x_{i} $ and $ x_{j} $ are the noise signals for channels $ i $ and $ j $ respectively and $ \mu_{i} $ and $ \mu_{j} $ are their corresponding mean values and the operator $ E $ denotes the expectation value. The correlation between the channels is defined as

$ \rho(x_i,x_j) = \frac{cov(x_i,x_j)}{\sqrt{E[(x_i-\mu_i)^2]}\sqrt{E[(x_j-\mu_j)^2]}} = \frac{cov(x_i,x_j)}{ \sigma_i . \sigma_j} $

The correlated noise was studied using 10,000 events from the run 125204, a standalone bi-gain pedestal run taken in standard final front-end Tile electronics, final finger LVPS, in 2009-08-16 during cosmics data taking. The method used assumes the observed noise measurement ($ x_i $) in a particular channel $ i $ of the TileCal module, is assumed to be a combination of a genuine intrinsic noise component ($ x_i^{int} $) plus a contribution which depends on the response of all channels in the module as a whole and is probably dominated by the closest neighbours, i.e., the measurement in PMT channel $ i $, $x_i$, is assumed to be a linear combination between the intrinsic noise component ($x_i^{int}$) and a weighted sum of the signals of all the other PMTs ($N_{PMT}$) in the module i.e.,

$ x_i = x_i^{int} + \sum_{j\ne i}^{N_{PMT}} \alpha_{i,j}x_j $

The $\alpha_{i,j}$ unknown parameters establish the linear weight of the other PMTs, task left to the method to figure out. The matricial nature of $\alpha_{i,j}$ coefficients is perhaps more visible by turning the previous expression into:

$  {x_1} \sim   {\beta_1} + \alpha_{1,2}x_2  + ... + \alpha_{1,N_{PMT}}x_{N_{PMT}} $

$  {x_2} \sim  \alpha_{2,1}x_1 + {\beta_2,} + ... + \alpha_{2,N_{PMT}}x_{N_{PMT}}  $

$         .  $

$         .  $

$         .  $

$  {x_{N_{PMT}}} \sim  \alpha_{N_{PMT},1}x_1 + \alpha_{N_{PMT},2}x_{2} + ... + {\beta_{N_{PMT}}} $

For the pedestal runs, it is assumed that the $x_i$ distributions are centered at zero after calibration, before or after reconstruction. Nevertheless, pedestal offsets (represented by $\beta_i$) are still kept to take into account effects that deviate the intrinsic mean value of the channel from zero (like miscalibrations). For the set of 7 pulse samples which will correspond to signals, separated by 25~ns, the distributions are not centered at zero. Therefore, the mean value was artificially subtracted to the distributions in order to calculate the $\alpha$ matrix without the influence of the mean values. A short description of the method used to remove the effect of the correlated noise in the presence of physics signal is presented in what follows.

For each channel, the measured noise can be compared with the model above using a usual $\chi^2$ method,

$ \chi_{i}^2 = \sum_{Events} \frac{\left [ x_i - (\beta_i + \sum_{k\ne i}^{N_{PMT}} \alpha_{i,k}x_k )\right ] ^2}{\sigma_{i}^2}$,

which can be minimized (individually for each PMT channel) with respect to each one of the $\alpha_{i,j}$ and $\beta_i$ of the model,

$ \frac{\partial \chi_{i}^2}{\partial \alpha_{i,1}} = \frac{\partial \chi_{i}^2}{\partial \alpha_{i,2}} = ...= \frac{\partial \chi_{i}^2}{\partial \alpha_{i,N_{PMT}}} =  \frac{\partial \chi_{i}^2}{\partial \beta_i } = 0.$

Assuming the noise correlations are the same in the presence of physics signal or, in other words, that no cross talk is present apart from the noise correlations, the noise correlation pattern (which indeed is represented by the $\alpha$ matrix) is extracted from Sample 0, following the minimization procedure. This allows to extract all the $ \alpha$ matrix elements together with the offset terms $\beta_i$ for each one of the channels.

The reconstruction of the signal in channel $i$ ($x_i^{rec}$) is performed removing the offset evaluated during the minimization procedure $\beta_i$ and by applying the $\alpha$ matrix to the measured values of all the other PMTs of the module according to,

$ s_i^{rec}  = s_i - \left ( \alpha_{i,1}x_1 + \alpha_{i,2}x_2 + ... + \beta_i + ... + \alpha_{i,N_{PMT}}x_{N_{PMT}} \right ) $

The reconstructed signal $s_i^{rec}$ should describe the signal $s_i$ without the correlated noise component of channel $i$. However, in the presence of physics signal, a cut must be imposed so that the subtraction does not include physics signals but only noise. Therefore, only channels below 2 RMS value of the distribution are considered in the subtraction even though all channels are corrected.

Conclusions

The full survey of the TileCal modules showed that, in spite of the similar pattern observed between modules, the use of a unique generic matrix to treat all modules is discarded. The $\alpha$ matrices reflect the configuration of the TileCal readout hardware with clear clusters of neighbour channels determining the PMT signal responses. The offset values are also close to zero, as expected.

The full survey of the $\chi^2$ method implementation confirms the algorithm efficiency in removing correlations even though it cannot be applied using a general average matrix but either on a module-by-module basis. The method proposed to remove the correlated noise component of the TileCal has been tested and approved through this systematic survey of the TileCal modules as a powerful solution to the non-coherent noise presence in pedestal runs. This approach shall be regarded as well as an effective diagnosis tool to the general behaviour of TileCal modules.

The analysis of special calibration runs, where only one channel is fired at a time, excluded the presence of cross-talk allowing the method to be applied in the presence of physics signals. The method was applied to this calibration and no degradation of the signal was observed. TileCIS events allowed to understand how to apply the correction algorithm to physics signals. Furthermore, the results on Minimum Bias and simulated $t \bar t$ events show a clear improvement on the pedestal distributions and, again, signals are not degraded by the method.

In addition to the work developed at the analysis level, tests were performed at lab (Building 175). These tests showed that the use of linear power supplies introduce lower levels of noise correlation on the front end electronics, with respect to the fingers used. The bb66 is better than the old version even refurbished. The noise correlation level is lower when the amount of load is reduced (when half of the SD is disconnected) and the importance of the effect seems dominantly related with the HV micro board.

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