**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.

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:

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

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

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

The amplitude estimation is given by:

**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:

Where, corresponds to the bias that is subtracted in each estimation

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

- Energy Reconstruction studies for HL-LHC (E4 Cells case):

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

- Wiener Filter and OF2 performancy studies for HL-LHC (E4 Cells case):

- Detailed description of Wiener filter method

- Wiener Filter performance studies for HL-LHC considering the LHC bunch train scheme

- Description of Wiener filter method and first results of its performance using real data

- Comparison of the OF2, optimised Wiener filter, a generalised version of the Wiener filter performance

- General description of the studies and a compilation of results acquired until september 2018

- Wiener filter performance studies using real data with different <μ> values

- Studies of the most appropriate energy distribution model to be used in the wiener filter method

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

- Introduce the Wiener Filter tool in the athena framework

**Major updates**:

-- DayaneOliveiraGoncalves- 3 Mar 2019

This topic: Sandbox > WebPreferences > WienerFilter

Topic revision: r2 - 2019-03-04 - DayaneOliveiraGoncalves

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