WienerFilter

Introduction

Contact Person: Dayane Goncalves

Wiener filter is a estimation method proposed by Nobert Wiener during the 1940s and published in 1949. The motivation for this algorithm is to improve the energy estimation in the online processing of signals from E Tilecal cells (high occupancy channels) inside the Read-Out Drivers (RODs) modules.

Note: The efficiency of Wiener Filter to estimate energy in E TileCal cells is currently being studied and its implementation is being considered for Phase-II.


Wiener Filter algorithm description

The bases of this algorithm are described by S. Haykin in ISBN 978-0-132-67145-3 (2013) 108 - 140. The wiener filter uses knowledge of the statistical properties of both the signal and the noise to reconstruct an optimal estimate of the signal from a noisy data stream.

The idea is to minimize the expected value of the squared of the error:

CodeCogsEqn.gif

Let R denote the N-by-N correlation matrix of the inputs u(n), u(n-1), u(n-N+1):

CodeCogsEqn_(2).gif

Correspondingly, p is denoted as N-by-1 cross-correlation vector between the inputs of the filter and the desired response d(n):

CodeCogsEqn_(3).gif

The wiener equation can be rewrite in the compact matrix form:

CodeCogsEqn_(4).gif

CodeCogsEqn_(5).gif

The amplitude estimation is given by:

CodeCogsEqn_(6).gif

Inclusion of an additional weight in the optimization procedure:

In order to absorb bias (mean of the error) an element 1 is added to each input signal, as last element.

This way, the amplitude estimation is given by:

CodeCogsEqn_(7).gif

Where, CodeCogsEqn_(1).gif corresponds to the bias that is subtracted in each estimate

Documentation

Links of all talks done related to Wiener Filter applied in E TileCal Cells:

* https://indico.cern.ch/event/703027/contributions/2898945/attachments/1602005/2540143/EneReco2.pdf

* https://indico.cern.ch/event/702998/contributions/2883660/attachments/1594665/2525136/EneRecoSimulationMeeting.pdf

* https://indico.cern.ch/event/711640/contributions/2922427/attachments/1611342/2558719/EneRecoSimulationMeeting25Feb18.pdf

* https://indico.cern.ch/event/713504/contributions/2931428/attachments/1615227/2567383/EneRecoSimulationMeeting12Mar18.pdf

* https://indico.cern.ch/event/713504/contributions/2931428/attachments/1615227/2567383/EneRecoSimulationMeeting12Mar18.pdf

* https://indico.cern.ch/event/722757/contributions/2972311/attachments/1633941/2606130/EneRecoSimulationMeeting16Apr18.pdf

* https://indico.cern.ch/event/731568/contributions/3016241/attachments/1655459/2650141/EneRecoSimulationMeeting24May18.pdf

* https://indico.cern.ch/event/738604/contributions/3048268/attachments/1672581/2683714/EneRecoSimulationMeeting21Jun18.pdf

* https://indico.cern.ch/event/730841/contributions/3028107/attachments/1663897/2666695/TileWeekPerformance.pdf

* https://indico.cern.ch/event/746578/contributions/3087892/attachments/1693468/2725242/EneRecoSimulationMeeting26Jul18.pdf

* https://indico.cern.ch/event/759687/contributions/3151179/attachments/1724051/2784230/EneRecoSimulationMeeting27Sep18.pdf

* https://indico.cern.ch/event/691318/contributions/3195435/attachments/1743226/2821177/Wiener_filter_studies.pdf

* https://indico.cern.ch/event/775929/contributions/3225202/attachments/1757770/2850563/Energy_rec_22_11_2018.pdf

* https://indico.cern.ch/event/691322/contributions/3230600/attachments/1759226/2853636/E4_studies.pdf

* https://indico.cern.ch/event/786709/contributions/3303251/attachments/1792372/2920554/Tile06Feb19.pdf

Major updates:
-- DayaneOliveiraGoncalves- 3 Mar 2019

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