Digital Signal Processing applied to Physical Signals - CERNcds.cern.ch/record/1341287/files/CERN-THESIS-2011-008.pdf · Digital Signal Processing applied to Physical Signals Diego

POLITECNICO DI TORINO

SCUOLA DI DOTTORATO

Dottorato inIngegneria Elettronica e delle Comunicazioni

Tesi di Dottorato

Digital Signal Processing appliedto Physical Signals

Diego ALBERTO

Tutore Coordinatore del corso di dottoratoProf. Roberto Garello Prof. Ivo Montrosset

Marzo 2011

If you can talk with crowds and keep your virtue,Or walk with kings - nor lose the common touch;If neither foes nor loving friends can hurt you;If all men count with you, but none too much;If you can fill the unforgiving minuteWith sixty seconds’ worth of distance run -Yours is the Earth and everything that’s in it,And - which is more - you’ll be a Man my son!

Rudyard Kipling - If

Summary

It is well known that many of the scientific and technological discoveries of theXXI century will depend on the capability of processing and understanding a hugequantity of data. With the advent of the digital era, a fully digital and automatedtreatment can be designed and performed. From data mining to data compression,from signal elaboration to noise reduction, a processing is essential to manage andenhance features of interest after every data acquisition (DAQ) session.

In the near future, science will go towards interdisciplinary research. In this workthere will be given an example of the application of signal processing to differentfields of Physics from nuclear particle detectors to biomedical examinations.

In Chapter 1 a brief description of the collaborations that allowed this thesisis given, together with a list of the publications co-produced by the author in thesethree years. The most important notations, definitions and acronyms used in thework are also provided.

In Chapter 2, the last results on the filter designs to be implemented in thetrigger-less DAQ of the PANDA experiment are presented. Results obtained fromsimulation are used as basis for some FPGA-oriented filter projects able to processin real time data from nuclear particle detectors. For all studied filter structures,particular attention has been paid to the board inner-components consumption andto the maximum working frequency, since our aim is an on-line treatment.

In Chapter 3, from a collaboration with the INRiM institute of research, theresults of signal processing of data coming from TES single photon detectors arereported. Because of their high sensitivity and fast response, the application ofthese detectors is mandatory to all fields that deal with weak sources. It ranges fromastrophysics to structure of matter, from X-rays to infra-red wavelengths. However,the electronic DAQ system and the environmental conditions of every acquisitionrequire a digital treatment to extract the most important features of interest asthe energy resolution or the capability to really count single photons of every DAQsession.

The author had the possibility to spend a three months period as a PhD visitingstudent at CERN, Geneva. During this experience he was involved in the ALICEexperiment of the LHC project. The goal was the development of an algorithm able

to process on-line data from all detectors to be implemented on FPGA. The resultsof this study are reported in Chapter 4, subdivided in a simulation part, used tounderstand and write the algorithm, and in a real data analysis.

The last application of signal processing presented, comes from a collaborationwith the Ottica e Optometria course of studies of the University of Turin, it isintroduced in Chapter 5. The analyses conduced on human corneas aiming atdistinguishing all corneal sub-layers and estimating their thicknesses are reported.Every acquired image represents a signal to be processed. The impact of this devel-opment on medical applications is very high since, for the first time, all these clinicaltests can be made in-vivo with no adverse effect for patients and with a precisionnever reached before.

The work is concluded, in Chapter 6, with some considerations on the useful-ness, and surely the necessity, of signal processing in science experiments.

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Acknowledgments

... to (and notwithstanding) Andrea, Chiara and Arianna

I would like to thank:

• Prof. M.P.Bussa, Dr. A.Grasso, the Torino Panda-mu Group for their sci-entific support during this three years at Dipartimento di Fisica Generale ofUniversita di Torino and, in particular, Dr. M.Greco, my Physics supervisor.

• Prof. R.Garello my PhD supervisor at Dipartimento di Elettronica of Politec-nico di Torino and a dear friend.

• Dr. E.Falletti and Dr. F.Molinari for their advices in the DSP subject.

• Prof. G.Masera, Dr. A.Dassatti and Dr. L.Toscano for the useful suggestionson FPGA implementations of filter structures.

• Dr. M.Frisani for having provided the OCT corneal images and the time spentin many discussions on OCT acquisitions.

• Dr. M.Rajteri, Dr. L.Lolli and the TES group at INRiM for the collaborationon TES pulse analyses.

• Dr. L.Musa and Dr. P.G.Innocenti for the PhD visiting student period atCERN and for the possibility to cope with real data from ALICE experiment.

• Dr. M.Destefanis for the discussions regarding Physics and LATEXthemes.

• PhD students Alessandro Re, Fabrizio Sordello, Isacco Scanavino, Ivan Gnesi,Marco Musich and Thanushan Kugathasan for the pleasant lunchtime discus-sions on whatever known (or unknown) subject and for their friendship.

Finally, a special thank to all my family for the precious help given to babysittingour little children and, in particular, to my wife Gabriella, for the way in which shetrusts me.

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Ringraziamenti (Italian Version)

... a (e nonostante) Andrea, Chiara e Arianna

Vorrei ringraziare:

• la Prof. M.P.Bussa, l’Ing. A.Grasso, il gruppo Panda-mu Torino per il lorosupporto scientifico durante questi tre anni al Dipartimento di Fisica Generaledell’Universita di Torino e, in particolare, la Dr. M.Greco, mia responsabilea Fisica.

• Il Prof. R.Garello mio tutore di dottorato al Dipartimento di Elettronica delPolitecnico di Torino e caro amico.

• L’Ing. E.Falletti e l’Ing. F.Molinari per i loro consigli in materia di analisi deisegnali.

• Il Prof. G.Masera, l’Ing. A.Dassatti e l’Ing. L.Toscano per gli utili suggeri-menti sull’implementazione FPGA delle strutture di filtraggio.

• Il Dr. M.Frisani per aver fornito le immagini delle cornee analizzate e per iltempo speso in molte discussioni sulle acquisizioni OCT.

• L’Ing. M.Rajteri, l’Ing. L.Lolli ed il gruppo TES all’INRiM per la collabo-razione all’analisi degli impulsi dai rivelatori TES.

• Il Dr. L.Musa e l’Ing. P.G.Innocenti per il periodo di visita al CERN inqualita di studente di dottorato e per la possibilita di confrontarmi con i datireali dell’esperimento ALICE.

• Il Dr. M.Destefanis per le discussioni riguardo tematiche come fisica e LATEX.

• I dottorandi Alessandro Re, Fabrizio Sordello, Isacco Scanavino, Ivan Gnesi,Marco Musich e Thanushan Kugathasan per le simpatiche discussioni su qual-siasi materia nota (o ignota) all’ora di pranzo e per la loro amicizia.

Infine uno speciale ringraziamento va a tutta la mia famiglia per il preziosoaiuto fornitoci nella gestione dei bimbi e, in particolare, a mia moglie Gabriella, peril modo in cui confida in me.

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Table of contents

Summary vi

Acknowledgments vi

1 Introduction 11.1 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Scientific publication . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Notations, Definitions and Acronyms . . . . . . . . . . . . . . . . . . 5

2 Nuclear particle signals analysis for PANDA experiment 92.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 System model and simulation scheme . . . . . . . . . . . . . . . . . . 142.3 Digital conversion and filtering algorithms . . . . . . . . . . . . . . . 21

2.3.1 Butterworth LP digital filter . . . . . . . . . . . . . . . . . . . 252.3.2 MMSE noise canceller . . . . . . . . . . . . . . . . . . . . . . 27

2.4 Behaviour comparison of simulated results . . . . . . . . . . . . . . . 302.5 FPGA-oriented implementations . . . . . . . . . . . . . . . . . . . . . 35

2.5.1 Butterworth LP digital filter . . . . . . . . . . . . . . . . . . . 352.5.2 MMSE noise canceller . . . . . . . . . . . . . . . . . . . . . . 38

2.6 Discussion of FPGA-oriented results . . . . . . . . . . . . . . . . . . 402.6.1 Filter performance . . . . . . . . . . . . . . . . . . . . . . . . 402.6.2 Power consumption . . . . . . . . . . . . . . . . . . . . . . . . 452.6.3 Maximum working frequency . . . . . . . . . . . . . . . . . . 45

2.7 A statistical approach . . . . . . . . . . . . . . . . . . . . . . . . . . 462.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3 Transition-Edge Sensor single photon pulse analysis 503.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.3 Signal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.3.1 Energy Resolution . . . . . . . . . . . . . . . . . . . . . . . . 57

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3.3.2 Signal-to-Noise Ratio . . . . . . . . . . . . . . . . . . . . . . . 583.3.3 Time Jitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.3.4 Savitzky-Golay filter . . . . . . . . . . . . . . . . . . . . . . . 623.3.5 Wiener filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 713.5 Other examples of analysed datasets . . . . . . . . . . . . . . . . . . 753.6 Energy evaluation and TES amplitude response . . . . . . . . . . . . 773.7 Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . 80

4 Amplitude estimation of real signals for the ALICE experiment 814.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.2 ALICE experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.3 Self-adaptive piecewise linear filter . . . . . . . . . . . . . . . . . . . 89

4.3.1 Simulation results: ideal case . . . . . . . . . . . . . . . . . . 904.3.2 Simulation results: ideal case + noise . . . . . . . . . . . . . . 97

4.4 TPC real-data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5 Digital processing of OCT corneal images 1155.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205.3 Our purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1215.4 Our approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1215.5 OCT images and SNR . . . . . . . . . . . . . . . . . . . . . . . . . . 1265.6 Treatment of corneal marginal regions . . . . . . . . . . . . . . . . . 1305.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

6 Final conclusions 133

A The sign-LMS adaptive filter 135

B The uncertainty on TES Energy Resolution 137

C The averaging procedure and SNR improvement 140

D Wiener filter and SNR 143

E The Buzuloiu’s algorithm for 2D signals 148

F Simulation of a 2D corneal layer reconstruction 150

Bibliography 151

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Chapter 1

Introduction

1.1 Thesis outline

This PhD thesis is the fruit of a collaboration between: Dipartimento di Elettronicaof Politecnico di Torino and Universita degli Studi di Torino, Dipartimento di FisicaGenerale.

The main part of this work has been devoted to the development of digital filtersfor the PANDA nuclear particle experiment thanks to a collaboration between:

• the PANDA group and the Istituto Nazionale di Fisica Nucleare (INFN) Turinsection.

Will also be presented the analysis of data coming form collaborations between Di-partimento di Fisica Generale and:

• Istituto Nazionale di Ricerca Metrologica (INRiM) for the elaboration of singlephoton pulses.

• Corso di Studi in Ottica e Optometria for the analysis of human corneal im-ages.

• European Organization for Nuclear Research (CERN) for the processing ofreal data form ALICE experiment.

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1 – Introduction

This thesis deals with the digital treatment of signals coming from different fieldsof Physics, as reported in the Summary Section. Digital Signal Processing (DSP)can be applied whenever a signal is digitalized to:

• reduce noisy (unwanted) components;

• enhance desired ones;

• extract features of interest (amplitude, peak time, signal energy, ..);

• estimate indirect measurements (particle energy, energy resolution, ..);

• process on-line real data (with FPGA);

• process off-line measured and saved waveforms.

All analyses presented in the following Sections have been performed by meansof Matlab programs [1], Simulink block diagrams [1], and Xilinx tools [2] for theVHDL implementation of filter structures on FPGA devices.

1.2 Scientific publication

This introductory Chapter continues with a list of articles and presentations wheresome of the results here described have been published.

International Journals :

• Optical transition-edge sensors single photon pulses analysis,D.Alberto, M.Rajteri, E.Taralli, L.Lolli, C.Portesi, E.Monticone, Y. Jia,R.Garello, M.Greco, to be published on IEEE Transaction on AppliedSuperconductivity, June 2011, DOI: 10.1109/TASC.2010.2087736.

• Ti/Au Transition-Edge Sensors Coupled to Single Mode Op-tical Fibers Aligned by Si V-Groove, L.Lolli, E.Taralli, C.Portesi,D.Alberto, M.Rajteri, E.Monticone, to be published on IEEE Transactionon Applied Superconductivity, June 2011.

• FPGA implementation of digital filters for nuclear detectors,D.Alberto, E.Falletti, L.Ferrero, R.Garello, M.Greco, M.Maggiora, NU-CLEAR INST. AND METHODS IN PHYSICS RESEARCH, A, 611:99-104 October 2009, DOI: 10.1016/j.nima.2009.09.049.

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1 – Introduction

• Digital Filtering for Noise Reduction in Nuclear Detectors, D.Alberto, M.P.Bussa, E.Falletti, L.Ferrero, R.Garello, A.Grasso, M.Greco,M.Maggiora, NUCLEAR INST. AND METHODS IN PHYSICS RE-SEARCH, A, 594 (3): 382-388 September 2008, DOI: 0.1016/j.nima.2008.06.032 -2008.

• Effects of Extremely Low-Frequency Magnetic Fields on L- glu-tamic Acid Aqueous Solutions at 20, 40, and 60 µT Static Mag-netic Fields, D. Alberto, L.Busso, R.Garfagnini, P.Giudici, I.Gnesi,F.Manta, G.Piragino, L.Callegaro, G.Crotti, ELECTROMAGNETIC BI-OLOGY AND MEDICINE, pp. 241-253, 2008, Vol. 27, ISSN: 1536-8378,DOI: 10.1080/15368370802344052.

• Effects of Static and Low-Frequency Alternating Magnetic Fieldson the Ionic Electrolytic Currents of Glutamic Acid AqueousSolutions, D. Alberto, L.Busso, G.Crotti, M.Gandini, R.Garfagnini,P.Giudici, I.Gnesi, F.Manta, G.Piragino, ELECTROMAGNETIC BIOL-OGY AND MEDICINE, pp. 25-39, 2008, Vol. 27, ISSN: 1536-8378, DOI:10.1080/15368370701878788.

International Conferences :

• Ti/Au Transition-Edge Sensors Coupled to Single Mode Op-tical Fibers Aligned by Si V-Groove, L.Lolli, E.Taralli, C.Portesi,D.Alberto, M.Rajteri, E.Monticone, Y. Jia, IEEE Applied Superconduc-tivity Conference - Washington D.C., 1-6 August 2010.

• Optical transition-edge sensors single photon pulses analysis,D. Alberto, M.Rajteri, E.Taralli, L.Lolli, C.Portesi, E.Monticone, Y.Jia,R.Garello, M.Greco, IEEE Applied Superconductivity Conference - Wash-ington D.C., 1-6 August 2010.

• Epithelial, Bowman’s layer, Stroma and pachimetry changeswith FDOCT during orthokeratology, M.Frisani, D.Alberto, A.Ca-lossi, M.Greco, 1st EuCornea Congress - Venice, 17-19 June 2010.

• Digital Filtering of Particle Detector Signals, D.Alberto, M.Greco,M.Maggiora, S.Spataro, 17th IEEE Real Time Conference - Lisboa, 24-28May 2010.

• Epithelial, Bowman’s layer, Stromal and corneal thickness chang-es during orthokeratology by SD-OCT, M.Frisani, D.Alberto, M.Greco, EAOO - European Academy of Optometry and Optics 2010 -Copenhagen, 15-16 May 2010.

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1 – Introduction

• Digital Filters for Noise Reduction in Nuclear Detectors, D. Al-berto, M.P.Bussa, E.Falletti, L.Ferrero, R.Garello, A.Grasso, M.Greco,M.Maggiora, 16th IEEE Real Time Conference - Beijing, 10-15 May 2009.

National Conferences :

• Filtraggio adattativo su segnali da rivelatori di particelle, XCVICongresso Nazionale Societa Italiana di Fisica (S.I.F.) - Bologna, 20-24/09/2010.

• La tomografia a coerenza ottica FD-OCT per lo studio morfo-metrico delle diverse componenti della cornea, M.Frisani, D. Al-berto, M.Greco, A.Calossi, INOA - Istituto Nazionale Ottica ApplicataCNR, Vinci, 7-8 November 2009.

• Implementazione su FPGA di filtri digitali standard e adatta-tivi per il trattamento di segnali da rivelatori di particelle, XCVCongresso Nazionale Societa Italiana di Fisica (S.I.F.) - Bari, 28/09 -03/10/2009. Winner of the first prize as the best presentation in sec-tion: Sezione V a - Fisica applicata (Sect. V a -Applied Physics). On-line English version, Digital filters for nuclear particle detectors,D.Alberto, M.P.Bussa, E.Falletti, R.Garello, M.Greco, IL NUOVO CI-MENTO B - Basic topics: Special Issue, 125 B (5-6), p. 677-686 June2010, DOI:10.1393/ncb/i2010-10860-0.

• Comparazione tra filtraggio digitale standard ed adattativo perla riduzione del rumore nei rivelatori di particelle, XCIV Con-gresso Nazionale Societa Italiana di Fisica (S.I.F.) - Genova, 24/09/2008.

• Filtraggio Digitale di Segnali Generati in Rivelatori di ParticelleNucleari, XCIII Congresso Nazionale Societa Italiana di Fisica (S.I.F.)- Pisa, 24-29/09/2007.

PANDA collaboration meetings :

• Signal Processing for Nuclear Detectors, Panda DAQT Meeting -Bavarian Forest, 22-24/04/2009.

• Signal Processing on FPGA, Panda DAQT Meeting - Darmstadt,8-10/12/2008.

• Improvements in Digital Filtering for Nuclear Detectors, PandaDAQT Meeting - Torino, 18-19/06/2008.

• Digital Filtering in Nuclear Detectors, Panda DAQT Meeting -Juelich, 18-19/03/2008.

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1 – Introduction

1.3 Notations, Definitions and Acronyms

This Chapter ends with the most important notations, definitions and recurrentacronyms used in this work.

Antiproton: the antiparticle of the proton.

CP symmetry: it states that the laws of physics should be the same if a particlewere interchanged with its antiparticle (C - charge conjugation - symmetry),and left and right were swapped (P - parity - symmetry).

CP violation: a violation of the postulated CP symmetry.

electron Volt (eV): the energy acquired by an electron when accelerated by aelectric potential difference of 1 Volt and is equal to 1.602 · 10−19C.

Ergodic process: a stochastic process with statistical properties (such as its meanand variance) that can be deduced from a single, sufficiently long sample(extraction) of the process.

Exotic State: a state of matter not foreseen by usual QCD calculations that in-cludes quark and gluons (as hybrid hadrons) or only gluons (as glueballs).

Filter: whatever block able to receive an input signal and produce an outputwaveform, even not modified (in this case it could be a simple delayer).

• Standard F.: a filter with constant transfer function coefficients, oncethey have been calculated they do not change during the elaboration.

• Adaptive F.: a filter that self-adjusts its transfer function according toan optimizing algorithm.

Force: in physics, a quantitative description of the interaction between two physicalbodies and can be of four fundamental types:

• Gravitational: the force of attraction between all masses in the universe.

• Electro-Magnetic: associated with electric and magnetic fields and isresponsible for atomic structure, chemical reactions, the attractive andrepulsive forces associated with electrical charge and magnetism.

• Weak: the fundamental force that acts between leptons and is involvedin the decay of hadrons, it is also responsible for nuclear beta decay.

• Strong: it mediates interactions between quarks and gluons.

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1 – Introduction

Fourier Transform: the operation that decomposes a signal into its constituentfrequencies.

Hadron: a composite particle made of quarks held together by the strong force (i.e.protons and neutrons). Hadrons are categorized into two families: baryons(made of three quarks), and mesons (made of one quark and one antiquark).

Hyperon: any baryon containing one or more strange quarks, but no charm orbottom quarks.

Gluon: the particle that mediates the strong force.

Lepton: an elementary particle and a fundamental constituent of matter, the bestknown of all leptons is the electron.

Nucleon: a collective name for two particles: the neutron and the proton, theseare the two constituents of the atomic nucleus.

Orthokeratology: an ophthalmological treatment with contact lens that modifiesthe cornea’s shape changing its refraction properties.

Phonon: in physics, a quasiparticle representing the quantization of the modes oflattice vibrations of periodic, elastic crystal structures of solids.

Quark: an elementary particle and a fundamental constituent of matter, it can onlybe found within hadrons. There are six types of quarks, known as flavors: up,down, charm, strange, top, and bottom.

Standard Model: a physical theory concerning the electromagnetic, weak, andstrong nuclear interactions, which mediate the dynamics of the known sub-atomic particles.

Stochastic process: a non-deterministic process where there is some indetermi-nacy in its future evolution described by probability distributions.

Synchrotron: a particular type of cyclic particle accelerator in which the magneticfield (to turn the particles so they circulate) and the electric field (to acceleratethe particles) are carefully synchronised with the travelling particle beam.

WSS process: is a (Wide Sense Stationary) stochastic process whose joint proba-bility distribution does not change when shifted in time or space. As a result,parameters such as the mean and variance, if they exist, also do not changeover time or position.

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1 – Introduction

Here is provided a list of the acronyms that can be found throughout the text.

ADC: Analog to Digital ConverterALICE: A Large Ion Collider ExperimentALTRO: ALice Tpc Read Out chipASIC: Application Specific Integrated CircuitsATLAS: A Toroidal LHC ApparatuSCERN: Conseil Europeene pour la Recherche Nucleaire

(European Organization for Nuclear Research)CM: Center of MassCMS: Compact Muon SolenoidDAQ: Data AcQuisitionDF: Direct FormDFT: Discrete Fourier TransformDGT: DigitalDSP: Digital Signal ProcessingER Energy ResolutionFAIR: Facility for Antiproton and Ion ResearchFD: Fourier DomainFEC: Front-End CardFIR: Finite Impulse ResponseFPGA: Field Programmable Gate ArrayFWHM: Full Width at Half MaximumGSI: Gesellschaft fur SchwerionenforschungHESR: High Energy Storage RingIIR: Infinite Impulse ResponseINRiM: Istituto Nazionale di Ricerca MetrologicaIROC: Inner Read-Out ChamberISR: Intersecting Storage RingsKVL: Kirchhoff’s Voltage LawLEP: Large Electron-Positron ColliderLHC: Large Hadron ColliderLHCb: LHC beauty experimentLINAC: LINear ACceleratorLMS: Least Mean SquareLP: Low PassLSB: Least Significant BitLUT: Look-Up TableMMSE: Minimum MSEMSE: Mean Square ErrorMSPS: Mega Samples Per Second

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1 – Introduction

MWPC: Multi-Wire Proportional ChambersOCT: Optical Coherence TomographyPANDA: anti-Proton ANnihilation at DArmstadtPASA: Pre-Amplifier Shaping AmplifierPD: Peak DistortionPSD: Power Spectral DensityQCD: Quantum Cromo-DynamicsQGP: Quark Gluon PlasmaRCU: Readout Control UnitRMS: Root Mean SquareSG: Savitzky-GolaySNR: Signal-to-Noise RatioSPS: Super Proton SynchrotronSQUID: Superconducting Quantum Interference DeviceTD: Time DomainTES: Transition Edge SensorTPC: Time Projection ChamberVHDL: VHSIC Hardware Description LanguageVHSIC: Very High Speed Integrated CircuitsWGN: White Gaussian NoiseWSS: Wide Sense Stationary

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Chapter 2

Nuclear particle signals analysisfor PANDA experiment

2.1 Introduction

The PANDA (anti-Proton ANnihilation at DArmstadt) Experiment will be one ofthe key experiments at the Facility for Antiproton and Ion Research (FAIR) whichis currently being built on the area of the GSI Helmholtzzentrum fur Schwerionen-forschung in Darmstadt (Fig.2.1), Germany [3], [4]. FAIR is an extension of theexisting Heavy Ion Research Lab (GSI) and is expected to start its operation in2019.

The antiproton project was initiated by a large community of scientists outsideGSI, who had worked very successfully with antiprotons at LEAR/CERN and atthe Fermilab antiproton accumulator. Many of the physics ideas of PANDA werealready described in a Letter of Intent (Construction of a GLUE/CHARM-Factoryat GSI, Ruhr-University Bochum, 1999) and were extended afterwards in the FAIRConceptual Design Report (GSI, 2001), the Technical Progress Report (FAIR, 2005)and further PANDA specific reports. After the approval of FAIR, further projectsinvolving antiprotons were proposed (experiments with low energy and polarizedantiprotons) which are now in the preparatory phase.

The proposed project FAIR is an international accelerator facility of the nextgeneration. It builds on the experience and technological developments already madeat the existing GSI facility, and incorporates new technological concepts. At its heartis a double ring facility with a circumference of 1100 meters. A system of cooler-storage rings for effective beam cooling at high energies and various experimentalhalls will be connected to the facility.

The existing GSI accelerators, together with the planned proton-linac, serve asinjector for the new facility. The double-ring synchrotron will provide ion beams of

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2 – Nuclear particle signals analysis for PANDA experiment

Figure 2.1: FAIR facility overview: at present (blue) and in the near future (red).

unprecedented intensities as well as of considerably increased energy. Thereby in-tense beams of secondary particles (unstable nuclei or antiprotons) can be produced.The system of storage-cooler rings allows the quality of these secondary beams tobe drastically improved. Moreover, in connection with the double ring synchrotron,an efficient parallel operation of up to four scientific programs can be realized at atime. The central part of FAIR is a synchrotron complex providing intense pulsedion beams (from proton to Uranium). Antiprotons produced by a primary protonbeam will then be filled into the High Energy Storage Ring (HESR) which collidewith the fixed target inside the PANDA Detector (Fig.2.2).

The PANDA Collaboration with more than 450 scientists from 17 countriesintends to do basic research on various topics around the weak and strong forces,exotic states of matter, the structure of hadrons, gluonic excitations, the physics ofstrange and charm quarks. In order to gather all the necessary information from theantiproton-proton collisions a versatile detector will be build able to provide precisetrajectory reconstruction, energy and momentum measurements (in the range 1.5-15GeV/c) and very efficient identification of charged particles. Protons and neutrons,collectively called nucleons, belong to the family of hadrons. They are built of quarks(see Fig. 2.3) and bound by the strong force that is mediated via gluons. The forceacting between two quarks has an unusual behaviour: it is very small when thequarks are at close distance and increases as the distance grows.

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Figure 2.2: PANDA spectrometer 3-dimensional view.

Figure 2.3: Quarks and their confinement pictorial representation.

If one attempts to separate a quark-antiquark pair, the energy of the gluon fieldbecomes larger and larger. As a result, one does not end up with two isolatedquarks but with new quark-antiquark pairs instead. This absolute imprisonment ofquarks is called confinement. One of the greatest intellectual challenges of modernphysics is to understand confinement not just as a phenomenon but to comprehendit quantitatively from the theory of the strong force.

Another puzzle of hadron physics addresses the origin of the hadron masses, i.e.of the particles composed of quarks. In the nucleon, less than 2% of the mass can

11


be accounted for by the three valence quarks. Obviously, the bulk of the nucleonmass results from the kinetic energy and the interaction energy of the quarks con-fined in the nucleon. Physicists believe that new experiments exploiting high-energyantiproton and ion beams will also elucidate the generation of hadronic masses.

The PANDA scientific program includes several measurements [5], which addressfundamental questions of Quantum Cromo-Dynamics (QCD), mostly in the non-perturbative regime:

• Hadron spectroscopy up to the region of charm quarks. Here the search forexotic states like glueballs, hybrids and multiquark states in the light quarkdomain and in the hidden and open charm region is in the focus of interest.

• Study of properties of hadrons inside nuclear matter. Mass and width modifi-cations have been reported and will be investigated also in the charm region.

• Study of nonperturbative dynamics, also including spin degrees of freedom.

• Antiproton induced reactions are a very effective tool to implant strangebaryons in nuclei.

• Hard exclusive antiproton-proton reactions can be used to study the structureof nucleons (time-like form factors) and the relevance of certain models, like theHand Bag approach. Interesting aspects of Transverse Parton Distributionswill be studied in Drell-Yan production.

• In a later stage of the project, when all systematic effects are well studied, alsocontributions to electroweak physics can be expected, like direct CP violationin hyperon decays and CP violation and mixing in the charm sector.

All measurements will profit from the high yield of antiproton induced reac-tions and from the fact that, in contrast to e+e− reactions, all non-exotic quantumnumber combinations for directly formed states are allowed, whereas states withexotic quantum numbers can be observed in production. The achievable precision,as far mass and width measurements are concerned, is very high as was successfullydemonstrated by the Fermilab experiments.

The physics purpose of PANDA has been described above. However, the area inwhich our work will focuses is more engineering-oriented, it is the Data-Acquisition(DAQ) section of the experiment.

The standard architecture of a DAQ systems in nuclear detectors is based ona two-layer hierarchical approach. A subset of especially instrumented detectors isused to evaluate a first-level trigger condition. For the accepted events, the fullinformation of all detectors is then transported to the next higher trigger level or

12


to storage. The time available for the first-level decision is usually limited by thebuffering capabilities of the front-end electronics [6] - [8].

The next generation of experiments in the hadron facilities, like the FAIR oneat Darmstadt, will study rare events at a drastically improved sensitivity. Inter-esting signals will only become available by a combination of high interaction rates(normally higher than 10 MHz), fast detectors and broad bandwidth data acquisi-tion systems to select in a fitting way only the events of interest. These constraintsmake it necessary to go beyond the old two-layer hierarchical approach towards self-trigger systems. They autonomously detect signals and pre-process them to extractand transmit only the physically relevant information for further processing. Thismeans that they are able to discriminate how relevant the event is and, if requiredto select it, to select means to filter in the right way and dynamically the signals(see for example [9]).

We aim to develop a data acquisition system through the study of specific al-gorithms for the reliable detection of informative pulses (i.e., the pulses generatedby the interaction of a charged particle) partially buried in noise, as well as theirimplementation on electronic boards which use programmable devices.

In the following Sections a system model for the DAQ and a set of standardand adaptive filters aiming to the noise reduction of noisy simulated signals [10] willbe presented. These digital filters will be compared on their capability to extractfeatures of interest from acquired pulses (amplitude, peak distortion, Signal to Noiseratio - SNR ). The translation of the Matlab-simulated filtering structures intohardware-oriented ones and their implementation on FPGA devices for the on-lineprocessing [11] will also be presented. This Chapter will end with a statisticalapproach on the performed simulations [12].

13


2.2 System model and simulation scheme

When a charged particle is detected, the detector produces a current pulse that isprocessed through a transmission chain. This signal is affected by several causesof noise (thermal, shot, flicker, etc.) which impair the correct pulse detection. Weaim to develop a filtering system able to reduce this noise as much as possible. Theelaboration will be performed in the digital domain and for this reason the signalmust be processed by an Analog to Digital Converter (ADC). However, because ofthe reduced input bandwidth (20 MHz) and the sampling rate of the ADCs currentlyavailable, the bandwidth of the input signal (i.e., informative pulses plus noise) hasto be properly reduced. Thus, a Low-Pass (LP) analog transmission chain had to beintroduced before the ADC. A possible model is presented in Fig. 2.4, together witha pictorial representation of the impulse response. The analog section is composedof the following:

• Detector: it detects charged particles and produces informative pulses withamplitude proportional to the charge (Q).

• Preamplifier/integrator: it integrates the input signal and at the sametime reduces the bandwidth. When the current pulse is present, the integratorproduces a signal proportional to the charge. The gain kp is used to normalizethe peak to a fixed value; τp is the time constant of the preamplifier/integrator.

• PoleZero compensator: it introduces a faster pole erasing the preamplifierone, in order to enlarge the total pass-band and thus to avoid as much aspossible distortion of the informative pulses, which can cause a pile-up effectof the filtered pulses. The gain kpz is used to normalize the peak to a fixedvalue, while the time constant τpz τp.

• Analog shaper/antialias filter: it represents the final LP antialiasing filter,opportunely matched to the ADC input bandwidth and sampling rate. Inpractice, it enlarges the top of the signal, allowing the ADC to obtain morethan one significant sample for each informative pulse. The gain kAS is used tonormalize the peak to a fixed value, tAS = 1/fB is the time constant, where fB

is the desired signal band, and n is the denominator exponent. The factor ndetermines the slope of the transfer function in the transition bandwidth andthe top flatness of the signal. However, the higher the parameter n is, the moredifficult the hardware assemblies are. Hence, n was chosen as a compromisebetween the hardware complexity and the minimization of the sampling error.

14


Figure 2.4: Analog transmission sub-chain.

After these three blocks the shaped signal is ready to be converted by the digitalsub-chain (Fig. 2.5):

• ADC: it converts the analog signal into a digital one;

• Noise filter: it is digital and is designed to possibly reduce the noise thataffects the desired signal. It can be standard or adaptive, with Finite (FIR)or Infinite (IIR) Impulse Response.

In general, an analog signal and an ADC sampler are not synchronous in time.In the analog-to-digital conversion, focussing on the sampling process, the analogmaximum could not be sampled and a Peak Distortion (PD) could be introduced.This happens because the highest sampled value is considered the maximum and itcould not correspond to the analog maximum. Here, with PD we intend the relativedifference between the analog measured peak value and the sampled one (usuallyexpressed in percentage). It is important to remember that in this subsection we arenot considering the quantization, that is the process by which an analog quantity (i.e.a sampled value) is converted into a digital one (represented with a finite number of

Figure 2.5: Digital transmission sub-chain.

15


bits). The quantization error cannot be avoided but depends on the number of bitsthe ADC is capable, thus, choosing a more performant ADC, it can be reduced. Aswe can see in Fig. 2.6, different asynchronous samplings lead to different PD (∆Pi

in the Figure).The worst situation arises when we have two samples with the same highest value

(i.e. blue squares in Fig. 2.6). They correspond to the highest PD (∆P1) becausein any other case there is only one sample with the highest value, so the relateddistortion is lower (i.e. ∆P2 in the same Figure). The ideal case is represented bya distortion equal to zero (i.e. ∆P3 in Fig. 2.6) that can be obtained only when acasual synchronization between signal and sampler occurs.

In our case the sample time TS is 10 ns (being the sampling frequency of 100MHz) and represents the time distance between two consecutive samples.

Figure 2.6: Pictorial representation of different asynchronous samplings.

16


In order to obtain one sample as close as possible to the analog signal maximum,we have to enlarge the signal top. This operation is performed introducing theAnalog Shaper block and paying particular attention to the choice of the n term, thatis the Shaper (denominator) degree. Neglecting the hardware fabrication complexity,in Fig. 2.7 we have evaluated the PD (expressed as percentage error) as a functionof the Shaper order n. This distortion is expressed in percentage with respect tothe analog value and presents a decreasing behaviour because the higher the n is,the more flat the signals tops are, consequently the PDs are reduced. The Shaper’stime constant τAS has been set to 50 ns in according to a foreseen band of interestof 20 MHz (for this choice, see Sect. 2.3).

0 2 4 6 8 10 12 14 160

0.5

1

1.5

2

2.5

shaper order

% e

rror

Figure 2.7: Peak Distortion as a function of the Shaper order n..

17


However the shaper order n should be chosen also considering the pile-up effect.A high n introduces a low PD but also a high top signal enlargement, this enlarge-ment increases the separation time needed by our board to detect as separate twodifferent consecutive particles. In Fig. 2.8 the minimum inter-arrival time betweentwo particles, to be detected as different ones by our board, as a function of ourshaper order n is shown. For any value of n, if two particles are separated in timeless than the corresponding time value given by Fig. 2.8, a pile-up effect occurs.Also in this case the time constant τAS has been set to 50 ns in according to thevalue of the previous simulation.

0 2 4 6 8 10 12 14 16400

600

800

1000

1200

1400

1600

1800

shaper order

inte

r. ti

me

[ns]

Figure 2.8: Inter-arrival time between two particles as a function of the Shaper order n.

In order to evaluate the improvement obtained using a digital filter to partiallysuppress the noise, in our simulation we modelled the information pulses with a seriesof successive finite support waveforms, with very narrow time duration (0.5 ns) andrandom times of arrival. No specific assumptions have been made at this stage aboutthe mathematical model of the times of arrival, since experimental results are notstill available. However, superimposed input pulses have been explicitly avoided.

The pulses series are processed by the transmission chain model and the selectionof the noise model has been essentially driven by the sake of simplicity. Thus, an

18


additive White Gaussian Noise (WGN) model has been chosen and the additionstage has been placed right before the ADC device, as shown in Fig. 2.9.

We evaluated by simulation the behaviour of several digital filters (using Mat-

lab, Simulink [1]). In particular, we focused on:

• standard LP III order Butterworth filter;

• adaptive LMS filter.

Aiming at the best noise reduction, we compared the output of every digital filterto the digitalized output of the analog shaper. The results are presented in thefollowing Section.

Figure 2.9: Simulated transmission chain.

This Simulation Scheme has been implemented both in Matlab and in Simulink.In Figs. 2.10, 2.11, 2.12 Simulink schematics, analog and digital chains are pre-sented.

19


Figure 2.10: Simulink-implemented transmission chain schematics.

Figure 2.11: Simulink-implemented schematics: Analog Chain (green block in Fig. 2.10).

Figure 2.12: Simulink-implemented schematics: DGT Chain (blue block in Fig. 2.10).

20


2.3 Digital conversion and filtering algorithms

Let us consider the detection of two charged particles. The detector will producetwo short current pulses with the amplitude proportional to the charge depositedby every single particle. In our simulation the two amplitudes are different, alreadytransformed in voltage and expressed in normalized voltage unit (nvu). The firstpulse is equal to 1 and the second to 0.5 nvu; the pulse duration is 0.5 ns, while theinterarrival time is 0.7 µs.

The bandwidth used to digitally represent the analog section is 5 GHz, while thedigital bandwidth after the ADC is reduced to 50 MHz. The impulse response ofthe analog blocks simulated with Matlab is summed up in Fig. 2.13, while in Fig.2.14 the same result elaborated with Simulink is reported. Adding to the shaperoutput a WGN signal one-sixth less powerful at the same sampling rate, we obtaina simulation of a noisy analog measurement (Fig. 2.15).

0 500 1000 1500 20000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time [ns]

ampl

itude

[nvu

]

inputpreamplP/Z compshaper

PREAMPLIFIER

POLE / ZERO COMP

SHAPER

INPUT

Figure 2.13: Analog outputs obtained with Matlab simulation.

Putting our attention on the analog shaper output, Fig. 2.15a represents thenoisy measurement from which we would like to extract the desired signal (Fig.

21


Figure 2.14: Analog outputs obtained with Simulink simulation: input(red), pre-ampl(cian),PZcomp(purple), shaper(yellow).

2.15b), after sampling and quantization. Note that the sampling operation is asimple down-sampling in our simulation. Quantization is performed by the modelof ADC introduced in Sect. 2.2. Performing the Discrete Fourier Transform (DFT)of the digitized analog shaper output, we have its representation in the frequencydomain. Its square modulus provides the Power Spectral Density (PSD) of theconsidered signal; it is a useful mathematical tool that shows the most importantfrequency components in the signal spectrum.

In Fig. 2.16 we can see the PSD of the analog shaper desired output, where thesignal power has been normalized to the unit. The most significant frequencies arebounded in the lower part of the spectrum; as a consequence, the starting pointof our analysis is a LP digital filter. Furthermore, Shannon Theorem is satisfiedbecause the band involved is 20 MHz and we sample this signal with a samplingfrequency of 100 MHz (≥ 2×20 MHz).

The Matlab simulated noisy and desired digital signals are compared in Fig.2.17.

22


0 200 400 600 800 1000 1200 1400 1600 1800 2000−0.5

0

0.5

1

1.5NOISY analog signal

time [ns]

ampl

itude

[nvu

]

0 200 400 600 800 1000 1200 1400 1600 1800 2000−0.5

0

0.5

1

1.5DESIRED analog signal

time [ns]

ampl

itude

[nvu

]

a

b

Figure 2.15: Continuous shaper output: noisy(a) vs desired(b).

0 5 10 15 20 25 30

0.05

0.1

0.15

0.2

0.25

0.3

frequency [MHz]

PS

D

Band of 20 MHz

Figure 2.16: PSD of analog shaper output.

23


0 500 1000 1500 2000

−0.2

0

0.2

0.4

0.6

0.8

1

Noisy vs desired DGT signals

time [ns]

ampl

itude

[nvu

]

noisydesired

Figure 2.17: Matlab simulated noisy and desired digital signals.

24


2.3.1 Butterworth LP digital filter

The well known Bessel, Butterworth and Chebyshev standard filter families, withtransfer functions from the II to the V order, have been compared in terms ofSNR and PD estimations. The results are reported in Tabs. 2.1 and 2.2. In thisSection, the SNR is calculated as the ratio between the desired signal power and thenoise power, expressed in dB. The PD, or percentage error, is defined here as therelative difference between the desired signal peak and the filtered one, expressedin percentage. In case of several peaks, we decided to focus on the case with thehighest distortion.

Order II III IV V

Butterworth 5.50 5.76 5.88 5.95Bessel 4.22 3.52 3.05 2.71Chebyshev 2.88 4.29 4.95 5.37

Table 2.1: SNR improvement [dB], our choice in bold typeface.

Order II III IV V

Butterworth 7.72 8.34 10.45 11.40Bessel 5.89 7.19 7.93 8.38Chebyshev 7.45 9.60 12.66 12.58

Table 2.2: PD [% error], our choice in bold typeface.

The percentages shown in Tab. 2.2 all refer to the same worse condition. Fromthese results the best performances both in noise reduction and in peak reproducibil-ity are obtained with the Butterworth family. The highest noise reduction is ob-tained with the III order transfer function, while the highest peak reproducibility(lowest percentage error) is obtained with the II order. We chose the III order be-cause our aim was the noise reduction and we set the cut-off frequency to 20 MHz,i.e., to the bandwidth of the input signal. The normalized transfer function in thecontinuous complex frequency domain is:

H(s) =1

s3 + 2s2 + 2s + 1(2.1)

Denormalizing this transfer function to the cut-off frequency (Figs. 2.18, 2.19),we obtain:

25


Figure 2.18: Butterworth III order denormalized analog transfer function, 3D plot (right) andprojection on complex plain (left).

Figure 2.19: Butterworth III order denormalized analog transfer function, 2D project maskobtained from the projection of the 3D plot on the plain real positive freq. vs modulus.

26


H(s) =1.98 · 1024

s3 + 2.51 · 108s2 + 3.16 · 1016s + 1.98 · 1024(2.2)

To convert this analog transfer function in the digital domain, we used a digitaltransfer function H(z) obtained from bilinear transform of H(s) [13] with a cut-offfrequency according to the bandwidth of the desired input signal(20 MHz):

H(z) =0.075 + 0.226z−1 + 0.226z−2 + 0.075z−3

1 − 0.827z−1 + 0.515z−2 − 0.087z−3(2.3)

Once evaluated, the transfer function coefficients are fixed and time independentby definition of standard filter. The typical structures to implement a digital IIRfilter corresponding to H(z) will be presented in Sect. 2.5.

2.3.2 MMSE noise canceller

The standard III order LP Butterworth analog filter has fixed parameters (transferfunction coefficients) that are known and calculated with Butterworth III orderpolynomials [13], denormalized at a particular cut-off frequency, bilinear transformedand then implemented in the digital domain. However, they do not depend on thecharacteristics of the specific considered signal. In this Section, we introduce afiltering algorithm whose parameters are dynamically calculated and adapted to theinput signal in real time, the LMS algorithm (see [14] for a complete development).

We are interested in the Minimum Mean Square Error (MMSE) noise cancellerimplementation of this filter and in its Finite Impulse Response (FIR) form. If aprocess d(n) is to be estimated from an observed process x(n) corrupted by the noisev(n):

x(n) = d(n) + v(n) (2.4)

and if we do not have any kind of information about d(n) or v(n), it is not possibleto separate the signal from the noise. However, given a reference signal, this problemcan be solved [14]. In the nuclear detector applications, here considered, we cannothave a reference signal and so we can adopt a different approach. We simply delaythe process x(n) of n0 samples (Fig. 2.20), where x(n) is the measurement, d(n)

is the desired component, d(n) is an estimate of d(n), v(n) is the noise componentuncorrelated from d(n), v(n) is an estimate of v(n).

The typical structures to implement a digital LMS FIR filter will be introducedin Sect. 2.5. In our model we can assume that d(n) is a narrowband process (Fig.2.16) and that v(n) is a broadband process with:

E v(n) · v(n − k) = 0, |k| ≥ k0 (2.5)

27


Figure 2.20: MMSE noise canceller using a LMS filter.

where E· is the statistical expectation and, if v(n) is white, then k0 = 1.So shifting the reference of at least k0 samples, the noise component of the signalx(n − n0) is uncorrelated with the noise of the measured signal x(n). Therefore,if k0 ≤ n0 ≤ k1, the delayed process x(n − n0) will be uncorrelated with the noisev(n), but correlated with d(n) (from the condition n0 ≤ k1). Thus, the samples ofx(n − n0) may be used as a reference signal to estimate d(n) as illustrated in Fig.2.20.

The problem we want to solve is how to obtain an estimate of the current sampled(n) of the desired signal from a set of M = k1 − k0 + 1 previous samples of themeasured signal:

xM(n − n0) = [x(n − n0),x(n − n0 − 1), . . . ,x(n − n0 − M + 1)]T . (2.6)

To do this, the observation vector xM(n−n0) must be filtered by a proper linearpredictor, designed as a FIR filter with coefficients:

wM = [w0,w1, . . . ,wM−1]T (2.7)

so that the filtered signal, d(n), is written as:

d(n) = wHM · xM(n − n0) =

M−1∑

k=0

w∗

k · x(n − n0 − k) (2.8)

where wHM is the Hermitian transpose of wM . The optimum design of the filter

coefficients can be made through the minimization of the Mean Square EstimationError (MSE) [13], defined as:

E|e(n)|2

(2.9)

28


where e(n) = d(n) − d(n) is the estimator error. It is a known result of theMMSE filter design theory [14], [15] that the optimum set of filter coefficients forthe linear prediction problem, stated as before, is given by:

Rxx · wM = rxd (2.10)

where Rxx = ExM(n − n0 · xHM(n − n0) is the M × M autocorrelation matrix

of the input process, xHM is the Hermitian transpose of xM , and rxd = ExM(n −

n0) · x(n) is the cross-correlation vector between the past observation xM(n − n0)and the current one x(n), which contains the desired component d(n).

However, in the case of the considered experiments, the nonstationarity of theobserved process suggests to choose an iterative, adaptive implementation of theabove formulation, known as LMS adaptive filter [14], [15]. Using a one-point samplemean (for a more complete discussion of the LMS algorithm the reader is referredto [15]), the update equation assumes a simple form known as the LMS Algorithm:

wM(n + 1) = wM(n) + µ · e(n) · x∗

M(n − n0) (2.11)

where wM(n + 1) is a new vector of filter coefficients at time n + 1, wM(n) isthe filter coefficients vector at time n, e(n) is the error at time n, x∗

M(n− n0) is thecomplex conjugate of the measurement at time n − n0, n0 is the introduced delay,and µ is the stepsize. It is a positive number that affects the rate at which theweight vector wM(n) moves down towards a stable solution.

Since this Section is preliminary for the implementation of these filtering algo-rithms on FPGAs, we need to take into account their computational complexity.Let us consider the LMS complexity in terms of additions and multiplications: Eq.2.11 requires one addition to compute the error e(n) and one multiplication to formthe product µe(n), M multiplications, and M additions to update the filter coeffi-cients. Finally, M multiplications and M−1 additions are necessary to calculate theoutput, y(n) = d(n), of the adaptive filter. Thus, a total of 2M + 1 multiplicationsand 2M additions per output point are required.

The choice of the stepsize µ corresponds to a tradeoff among:

• the SNR, in order to have a rough estimation of the real noise reduction;

• the time dependence of the squared error function (e2(n)). The e(n) functionis a positive or negative quantity involved in Eq. 2.11 responsible for real timecorrection of the filter coefficients. If the algorithm converges to a stable setof coefficients the correction, as a function of time, and its squared estimate,should have a decreasing behaviour;

• the coefficient settlement during the measurement or the simulation in orderto understand if the algorithm has reached a stable solution.

29


2.4 Behaviour comparison of simulated results

In Fig. 2.17 the desired digital signal is plotted superimposed on the noisy digitalsignal. The filtered signal obtained with an IIR Standard Butterworth LP III or-der digital filter is presented in Fig. 2.21. The input SNR is 8.41 dB. Using theButterworth filter this quantity rises to 14.17 dB with an improvement of 5.76 dB.

0 500 1000 1500 2000

−0.2

0

0.2

0.4

0.6

0.8

1

time [ns]

ampl

itude

[nvu

]

noisydesiredButt. III

Figure 2.21: Butterworth III order filtered signal vs. noisy and desired signals.

However, this filter introduces a peak amplitude distortion that is worse for thefirst of the two processed pulses. In Fig. 2.21, the amplitude of the first peak forthe Butterworth filtered signal is greater than the desired signal, with a distortionof 8.34 %. The FIR LMS order n was fixed to 4 to introduce a medium level ofcomplexity with a reasonable elaboration time. Using a LMS filter the performancesare a function of the stepsize µ, as shown in Tab. 2.3 and Figs. 2.22 - 2.24. A quickconsideration must be done for Fig. 2.24: it represents a functional to be optimizedto find the best filter performances. Since we want to enhance the SNR and reducethe PD at the same time, multiplying the SNR by the inverse of the PD we obtaina quantity to be maximized, it is the result presented in the last aforementionedFigure.

The percentage error is calculated as for the Butterworth filter for the first peak,that is for the highest distortion condition.

30


LMS stepsize µ error [%] SNR improvement [dB]

0.05 36.88 5.510.10 15.54 6.350.15 5.70 6.580.18 0.06 6.570.20 2.63 6.520.25 5.93 6.330.30 6.67 6.09

Butterworth 8.34 5.76

Table 2.3: LMS performances vs µ, our choice in bold typeface.

0 0.1 0.2 0.3 0.4 0.54

4.5

5

5.5

6

6.5

7

stepsize µ

SN

R [d

B]

LMS

MAX: µ =0.16 −− SNR =6.59 dBButterw. SNR: 5.76 dB

Figure 2.22: SNR expressed in dB as a function of stepsize µ.

31


0 0.1 0.2 0.3 0.4 0.50

10

20

30

40

50

60

70

80

90

stepsize µ

Pea

k D

isto

rtio

n [%

]

LMS

MIN: µ =0.18 −− PD =0.06 %Butterw. PD: 8.34%

Figure 2.23: PD expressed in % as a function of stepsize µ.

0 0.1 0.2 0.3 0.4 0.50

2000

4000

6000

8000

10000

12000

µ

arbi

trar

y un

it

curvemax = (0.18, 11840)SNR = 6.57 dBPD = 0.06 %

Figure 2.24: SNR / PD functional to be maximized for the optimal solution retrieval.

32


In Fig. 2.25, the LMS filtered signal with µ = 0.18 is presented; the output SNRis 14.98 dB, so the enhancement is of 6.57 dB, and the PD is also enhanced (lowerpercentage error). Indeed, in Fig. 2.25 the amplitude of the first peak for the LMSfilter nearly matches the desired signal, the distortion being only of 0.06 %.

0 500 1000 1500 2000

−0.2

0

0.2

0.4

0.6

0.8

1

time [ns]

ampl

itude

[nvu

]

noisydesiredLMS

Figure 2.25: LMS filtered signal (µ = 0.18) vs. noisy and desired signals.

The choice of the stepsize corresponds to a tradeoff among the evaluated SNR en-hancement, squared-error function decreasing behaviour (Fig. 2.26), and coefficientssettlement (Fig. 2.27) introduced in the previous Section.

33


0 500 1000 1500 20000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

time [ns]

ampl

itude

[nvu

]

LMS Sq. Error

Figure 2.26: Squared error amplitude vs. time: decreasing behaviour

0 500 1000 1500 2000−0.2

0

0.2

0.4

0.6LMS w1

time [ns]

coef

f. va

lue

0 500 1000 1500 2000−0.2

0

0.2

0.4

0.6LMS w2

time [ns]

coef

f. va

lue

0 500 1000 1500 20000

0.2

0.4

0.6

0.8LMS w3

time [ns]

coef

f. va

lue

0 500 1000 1500 2000−0.1

0

0.1

0.2

0.3LMS w4

time [ns]

coef

f. va

lue

Figure 2.27: LMS four coefficients (µ=0.18) behaviour: a stable solution is reached.

34


2.5 FPGA-oriented implementations

After proper design and simulation, in order to perform digital signal processing onreal-time signals, filtering must be implemented on an ASIC (Application SpecificIntegrated Circuit) or on a programmable board, as a FPGA (Field ProgrammableGate Array). A Xilinx Virtex 4 ML402 FPGA [2] will be considered in this Section.For this purpose, filter structures suitable for hardware implementation must bedeveloped and optimized.

The Matlab and Simulink simulations of Butterworth III order LP and adap-tive LMS filtering algorithms has been discussed in the previous Section. However,the direct translation of those structures in VHDL leads to board consumptions toohigh for an FPGA implementation. For this reason, some changes have been in-troduced and in this Section they will be discussed and compared with the originalsimulated schematics.

2.5.1 Butterworth LP digital filter

For sake of simplicity the digital Butterworth III order transfer function is reportedhere:

H(z) =0.075 + 0.226z−1 + 0.226z−2 + 0.075z−3

1 − 0.827z−1 + 0.515z−2 − 0.087z−3(2.12)

It is important to remember that, once evaluated, the transfer function coeffi-cients are fixed and time independent by definition of standard filter. In our simula-tions H(z) was implemented with a Direct Form (DF) II structure, as shown in Fig.2.28. The numerator coefficients are the multipliers of the rightern subchain, thedenominator ones of the leftern. This second half is fed back and its contribution isadded to the input signal.

Translating this structure in VHDL language for the implementation on theVirtex 4, signals coming from the lowest register, passing into the fed back (leftern)subchain, encounter 2 multipliers and 4 adders (Fig. 2.28). This path is the so-calledcombinatorial path and represents the distance that a signal has to cover betweentwo consecutive registers, or between one register and the filter output [6]. Longcombinatorial paths take more time to execute, so they limit the maximum compilerate of the FPGA. The maximum working frequency is the inverse of this time.

As an example, in the case of the Butterworth III order filter, the implementationwith a DF II structure leads to a combinatorial path that imposes a maximumworking frequency of 34 MHz. Since this rate could not be enough to cope with theclock rate of some detectors of a nuclear experiment (as expected for Panda [3]),

35


the filtering structures must be optimized to increase the FPGA maximum workingfrequency.

To improve this feature we have to break the longest combinatorial path. Thiswas done by moving to a DF I transposed structure for translating the same transferfunction (Fig. 2.29). In this case the maximum combinatorial path is composed ofonly 1 multiplier and 2 adders (Fig. 2.29), thus the maximum working frequencyrises up to 63 MHz.

36


Figure 2.28: IIR Butter. III ord. filter with DF II structure, Simulink schematics. Longestcombinatorial path highlighted in dark green.

Figure 2.29: IIR Butter. III ord. filter with DF I transposed structure, Simulink schematics.Longest combinatorial path highlighted in dark green.

37


2.5.2 MMSE noise canceller

The structure used for the LMS simulation, translated in VHDL, presents a very longmaximum combinatorial path (3 multipliers and 5 adders) leading to a maximumworking frequency of 22 MHz (LMS1, Fig. 2.30). Also in this case, it is possibleto optimize the hardware structure and enhance this rate. In order to break thelongest combinatorial path, some extra registers were added.

The optimized adaptive filter structure (LMS2) is shown in Fig. 2.31. With thesechanges in the structure, the maximum combinatorial path starts after the registerunder the first coefficient subchain, passes through 4 adders and the multiplier withthe stepsize µ as second input. This path is shorter thus the maximum workingfrequency increases to 58 MHz.

Since the LMS filter structure was changed, we found, as expected, a differentvalue for the stepsize µ that optimizes our requirements on SNR maximization andPD reduction. It is worth noting that for both LMS1 and LMS2 structures everyadaptive coefficient is implemented with a dedicated subchain, fed by the multipli-cation of the stepsize µ with the difference between the noisy signal and the adaptivefilter output (Figs. 2.30, 2.31).

38


Figure 2.30: FIR LMS 4-coefficients structure Matlab simulated (LMS1), Simulink schematics.The longest combinatorial path is highlighted in dark green, mu is the stepsize.

Figure 2.31: FIR LMS 4-coefficients structure implemented for FPGA board (LMS2), Simulink

schematics. The longest combinatorial path is highlighted in dark green, mu is the stepsize, theadded registers are highlighted in light green.

39


2.6 Discussion of FPGA-oriented results

In this Section the performance of the FPGA digital filtering implementation is dis-cussed with reference to nuclear detector requirements. For the sake of comparisonwith the simulation results (see Sect. 2.2 and [10]), the same noisy signal has beenused for the FPGA filtering. The information pulses are modelled with a series ofsuccessive finite support waveforms, with very narrow time duration (0.5 ns) andrandom times of arrival. No specific assumptions had been made about the math-ematical model of the times of arrival, since it was intended to perform a generalanalysis not referred to a specific detector. White Gaussian Noise (WGN) had beenadded to the desired signal.

2.6.1 Filter performance

In Fig. 2.32 and in Tab. 2.4 the most relevant simulation results are presented. Asreported, the LMS filter introduces a much lower PD and a slightly higher SNR thanButterworth III order filter. The PD is evaluated for the filtered peak featuring thehighest distortion (the first in this analysis) (see Sect. 2.2 and [10]).

0 500 1000 1500 2000

−0.2

0

0.2

0.4

0.6

0.8

1

time [ns]

ampl

itude

[nvu

]

NoisyDesiredButt IIILMS1

Figure 2.32: Matlab simulated filter comparison.

The direct VHDL implementation of these structures leads to the same values of

40


Filter type Stepsize µ PD [%] SNR improvement [dB]

Butt. III DF II 8.34 5.76LMS1 0.18 0.06 6.57

Table 2.4: Performance of Matlab-simulted Butterworth and LMS filters.

SNR and PD of Matlab simulated filtering algorithms. The plot of the filtered sig-nals is presented in Fig. 2.33. In the VHDL translation both Butterworth and LMSfiltered signals present a delay of 2 sampling periods with respect to the Matlab

simulated ones. Indeed, in every structure of the previous Section the two registersfor input and output storage are not shown. This corresponds to a usual FPGAimplementing rule chosen to avoid a wrong numerical representation of signals whenhigh working frequencies are involved.

0 500 1000 1500 2000

−0.2

0

0.2

0.4

0.6

0.8

1

time [ns]

ampl

itude

[nvu

]


Figure 2.33: Matlab simulated structures implemented on FPGA.

As pointed out in the previous Section, a direct VHDL translation of the But-terworth DF II structure (Fig. 2.29) led to a low maximum working frequency.Thus, we adopted the translation of the DF I transposed structure (Fig. 2.28).The processing and the performances of PD and SNR do not change for the But-terworth filter because the same transfer function has been translated. A different

41


consideration has to be made for the VHDL translation of the LMS filter. Since theLMS1 structure worked at low frequency, some registers were added to break thetoo long combinatorial path. This insertion changed the structure (LMS2) givinghigher working frequency. Therefore, the LMS2 structure leads to a different beststepsize value, but also to a higher PD and a lower SNR than the Matlab simu-lated LMS1 structure (Tab. 2.4). This is the price that must be paid to work ata frequency two times and a half higher. Anyway, LMS filter continues to matchbetter the requirements related to the energy resolution and detector efficiency thanthe Butterworth one also in the implemented FPGA form (Tab. 2.5, Figs. 2.34 -2.36). The functional to be maximized has been calculated as reported in Sect. 2.4.

In Fig. 2.37 the optimized Butterworth (DF I transposed) and LMS (LMS2)filter outputs are shown. The LMS2 filter shows a delay of 8 sampling periods, withrespect to the LMS1 structure (Fig. 2.33), due to the added registers that shortenthe combinatorial path. The Butterworth FPGA filtered signal has no delay, becauseno register is added.

Filter type Stepsize µ PD [%] SNR improvement [dB]

Butt. III DF I 8.35 5.76LMS2 0.27 0.49 5.91

Table 2.5: Performance of FPGA-optimized Butterworth and LMS filtering structure.

42


0 0.1 0.2 0.3 0.4 0.53

3.5

4

4.5

5

5.5

6

6.5

stepsize µ

SN

R [d

B]

LMS

MAX: µ =0.39 −− SNR =6.22 dBButterw. SNR: 5.76 dB

Figure 2.34: SNR expressed in dB as a function of stepsize µ.

0 0.1 0.2 0.3 0.4 0.50

10

20

30

40

50

60

70

80

90

100

stepsize µ

Pea

k D

isto

rtio

n [%

]

LMS

MIN: µ =0.27 −− PD =0.49 %Butterw. PD: 8.35%

Figure 2.35: PD expressed in % as a function of stepsize µ.

43


0 0.1 0.2 0.3 0.4 0.50

100

200

300

400

500

600

700

800

900

µ

arbi

trar

y un

it

curvemax = (0.27, 886)SNR = 5.91 dBPD = 0.49 %

Figure 2.36: SNR / PD functional to be maximized for the optimal FPGA-oriented filtering solutionretrieval.

0 500 1000 1500 2000

−0.2

0

0.2

0.4

0.6

0.8

1

FPGA Filtering Comparison

time [ns]

ampl

itude

[nvu

]


Figure 2.37: FPGA-optimized filtering structures output.

44


2.6.2 Power consumption

In the implementation of the filtering structure on FPGA, it is very important toevaluate the consumption of the board internal components (i.e. the percentageof utilized memory slices and digital signal processors, the number of needed logiclevels). This analysis is indeed critical if we consider the total number of boards thatshould be used in a nuclear particle experiment. We have compared the differentfiltering structures described in the previous Section also under this point of view.

As reported in Tabs. 2.6 and 2.7, the DF I transposed and the LMS2 struc-tures optimized for FPGA reach not only higher maximum working frequency, butalso need less components for their implementation than the simulated and directlyVHDL translated filtering structures (DF II and LMS1).

Butterworth # Memory # DSP (%) Logic levels MAX workslices (%) freq. (MHz)

DF II < 1 14 30 34DF I transp. < 1 10 24 63

Table 2.6: Compared performance for different structures of Butterworth filter implemented on aXilinx Virtex 4 ML402.

LMS Stepsize µ # Memory # DSP (%) Logic levels MAX workslices (%) freq. (MHz)

LMS1 0.18 < 1 18 37 23LMS2 0.27 < 1 18 24 58

Table 2.7: Compared performance for different structures of LMS filter implemented on a XilinxVirtex 4 ML402.

2.6.3 Maximum working frequency

The structures were optimized to obtain the maximum working frequency with Xil-inx Virtex 4 board, doubling at least the rate. However, some detectors can work ata frequency higher than 60 MHz. The same structures were then implemented ona Virtex 5 FPGA using Xilinx ISE tool [2] to verify the possibility to increase theFPGA maximum working frequency. The results are summarized in Tab. 2.8, forcomparison. The maximum rate can be easily doubled using the Virtex 5, a boardthat has already been in FPGA marketplace for some years and is relatively low

45


cost. This means that the FPGA optimized structures are ready to be implementedand work at rates higher than 100 MHz.

filter FPGA # Memory # DSP (%) Logic levels MAX workslices (%) freq. (MHz)

Butt. III DF I tr. V4 < 1 10 24 63V5 < 1 5 40 130

LMS2 V4 < 1 18 24 58V5 < 1 18 28 116

Table 2.8: FPGA consumptions on different boards: Xilinx Virtex 4 XC4VSX3-10 (V4 ) and Virtex5 X5VSX35T-3 (V5 ).

2.7 A statistical approach

Up to now we have considered one single extraction of WGN noise to simulate a realmeasurement. To compare in a more significant way the behaviour of the filteringstructures here described, a statistical approach is required [12].

A thousand of waveforms were generated with Matlab from a WGN process.An ensemble of 1000 noisy signals has been built adding every single noise signal tothe same desired one. This ensemble was processed both with a Butterworth III ord.and the LMS1 (see Fig. 2.30) filters as described in the previous Sections. For everysimulated measurement, the PD and the SNR were evaluated. The SNR and PDdistributions of Butterworth and LMS filtered signals are shown in Figs. 2.38 and2.39, while the PD distributions are reported in Figs. 2.40 and 2.41, respectively.

It is worth to point out that the estimated SNR values follow a Gaussian distri-bution according to a Pearson’s chi-square test with confidence level of 5% for boththe considered filter outputs. Moreover, according to the PD values distribution,the 97% shows a distortion lower than 14% for the Butterworth and 1% for the LMSfilters.

On average we can conclude that, also from a statistical point of view, the LMSadaptive filter works better than the standard Butterworth III order LP filter bothon the SNR enhancement and on PD reduction.

46


Figure 2.38: Butterworth SNR values (blue dots), Gaussian distribution (white bars), mean value(red line) and our single simulation (green line, 5.76 dB).

Figure 2.39: LMS SNR values (blue dots), Gaussian distribution (white bars), mean value (redline) and our single simulation (green line, 6.57 dB).

47


Figure 2.40: Butterworth PD values (blue dots), distribution (white bars), mean value (red line)and our single simulation (green line, 8.34 %).

Figure 2.41: LMS PD values (blue dots), distribution (white bars), mean value (red line) and oursingle simulation (green line, 0.06 %).

48


2.8 Conclusions

Two classes of digital filters, the Butterworth one and the adaptive LMS filter, havebeen compared by simulation. This study was originated by the need to extractthe signal features for further on-line processing. The field of application is a newdata acquisition system for nuclear physics experiments where the selection of theaccepted events is no more performed by hardware triggers but is based on a so-phisticated software system working on pre-processed data. The requirements onPD and SNR are matched better by the LMS filter, thanks to the capability toadapt the parameters to the typically non-stationary environment of the nuclearphysics detectors. The simulation showed that a fast settlement of the coefficientscan be reached with an algorithm of medium-level complexity, with a processingtime scaling linearly with the filter order.

We have translated and implemented standard and adaptive digital filters on aFPGA for real time processing of nuclear detector signals. A direct VHDL transla-tion of the filtering structures previously simulated led to a high FPGA consump-tion and a low maximum working frequency. We optimized the implementations toenhance the working frequency in order to cope with the foreseen high rate dataacquisition of nuclear detectors. These different implementations introduced no sig-nificant variations in terms of PD reduction and SNR enhancement for the standardIIR Butterworth III order LP filter, while the adaptive LMS with our new struc-ture (LMS2, Fig. 2.31) presents slightly lower performances but a higher maximumworking frequency than the simulated implementation (LMS1, Fig. 2.30).

Finally, the average behaviour of the aforementioned filters has been studied.Also after this analysis, the LMS worked better than the Butterworth in the evalu-ations of the extracted features.

We can conclude that the requirements on PD and SNR are matched better bothin simulation and in FPGA implementation by the LMS filter.

As future work, we aim to process real signals to test the filters presented hereand to study and optimize new adaptive filtering structures as the sign-LMS (seeAppendix A) and the variable-stepsize (VSS) LMS.

49

Chapter 3

Transition-Edge Sensor singlephoton pulse analysis

3.1 Introduction

Transition-Edge Sensors (TESs) are extremely sensitive microcalorimeters capableof counting single photons from x-ray to infrared [16]. They are superconductingthin films with a sharp transition between the superconducting and the normalphase with very low transition temperatures (100-500 mK). The absorption of aphoton (γ) moves the bias point through its transition with a change in electricalresistance, see Fig. 3.1. The main advantage of TESs is their intrinsic energy

Figure 3.1: TES sharp transition.

resolution, i.e. their response is proportional to the absorbed photon energy, that

50

3 – Transition-Edge Sensor single photon pulse analysis

allows photon-number resolving capability when the photon energy is known. Theenergy resolution is a very important parameter that strongly depends on how it isevaluated and what kind of signal processing is carried out. Some solutions to obtainan optimal filter that take into account the different noise source contributions havebeen presented in [17] - [19] and also a bit simpler algorithm to quickly estimate thepulse parameters [20]; nevertheless, generally little information is given on how theenergy resolution is computed.

TESs can be applied in optical science and technology as in:

• Quantum metrology : absolute calibration of detectors (redefinition of qu-candela), nano-positioning.

• Quantum imaging : sub-shot-noise, weak image detection, ghost imaging.

• Quantum information: coding information, elaboration.

• Structure of matter : application to mass spectrometers.

• Astronomy : pulsar stars.

In this work we report the results of an off-line analysis applied to pulses mea-sured with a TiAu TES for the optical region [21]. The pulses are acquired with adigital oscilloscope and analyzed with numerical software using a Matlab platform[1], [14].

3.2 Experimental Setup

The experimental data analyzed in this work have been obtained at INRiM with aTiAu TES (see Figs. 3.2, 3.3), whose fabrication process has been described in [22].The photons are absorbed directly by the TiAu layer with an active area of 20µm x 20 µm and Tc=128 mK. The current signals, with a response time of 3.5µs, are read out with a 6 MHz bandwidth dc-SQUID (Superconducting QuantumInterference Device), and transformed to voltage waveforms. The TES is operatedinside a dilution refrigerator and coupled to a 1310 nm pulsed diode laser with anoptic fibre [23]. Therefore, the energy for the single detected photon Eλ is 0.95 eV(being Eλ = hν). The laser was pulsed at a repetition rate of 50 kHz with a pulsewidth of 100 ns, well below the response time of our detector, and the intensity wascontrolled with an optical attenuator.

The traces of the triggered pulses produced from the absorbed photons are mon-itored and saved in a binary format by means of a digital oscilloscope, without anykind of filtering. A collection of pulses of the order of tens of thousands traces aretypically saved for different experimental conditions, i.e. light intensity and biaspoint.

51


Figure 3.2: Experimental Setup, from the left: a laser diode (λ=1310 nm), the optical fiber coupledto the laser diode, the TES sensor, the SQUID and the Oscilloscope.

Figure 3.3: The TES Laboratory at INRiM.

52


3.3 Signal Analysis

The nature of the TES signal noise (mainly Johnson, phonon, etc.) makes it prefer-able to carry out a signal analysis after data acquisition to improve the most impor-tant TES waveform features. It can be an on-line processing, performed for exampleby an FPGA, or an off-line treatment. In this Section we present the results of anoff-line analysis that aims on the single pulse to:

• estimate the amplitude;

• evaluate the time jitter;

• calculate the SNR.

And on the complete dataset to:

• obtain the amplitude histogram;

• estimate the energy resolution (ER) and its uncertainty;

• construct reference signals ( 1γ, 2γ,..);

• calculate single photon energy on real signals;

• verify the range of the linear trend in TES amplitude responses.

To estimate the amplitude of every acquired signal (see Fig. 3.4) it is necessaryto define a baseline as reference for the measurement. Since these waveforms aretriggered, we know when a photon can be detected and, when the trigger fires, timestarts from 0 s. Therefore the signal samples with negative time values representthe dark, the noise, because the time interval between two consecutive events hasbeen set long enough to let every previous detected photon to extinguish its tail.

A quick way to have a rough idea of the amplitude distribution is to consider thepersistence of all dataset on the oscilloscope’s display superimposing all waveforms.Unluckily this result is available only in the case the oscilloscope has been pre-programmed to do so, instead, saving all signals in a format loadable by Matlab,we can rebuild the persistence information via software (see Fig. 3.5). However, thislevel of precision is not sufficient to calculate any signal’s feature. For this reason,we need a technique to evaluate every single signal amplitude.

In Fig. 3.6 we can see an example of the procedure used to estimate the am-plitude. It is the absolute difference between the noisy signal minimum and thebaseline. It is worth to mention that the baseline has been calculated as the meanof all samples before the trigger (with negative time values). This assumption isjustified considering Gaussian the distribution of this samples subset by means ofthe Central Limit theorem [24] (see Fig. 3.7).

53


−2 0 2 4 6 8 10 12 14−25

−20

−15

−10

−5

0

5

10

time [µs]

ampl

itude

[mV

]

1γ2γ3γ

Figure 3.4: Example of TES noisy signals.

Figure 3.5: Persistence representation of an acquisition of more than 34000 signals. Resolution:Time 0.02 µs - Amplitude 0.5 mV. The colorbar is expressed in percentage.

54


−2 0 2 4 6 8 10 12 14−20

−15

−10

−5

0

5

10

time [µs]

ampl

itude

[mV

]

2γ noisy sgnbaseline:2.38 mVmin:−18.78 mV,0.73 µs

A

Figure 3.6: Example of TES noisy signals carrying 2γ in blue solid line, the baseline is in greensolid line and the absolute minimum is represented with the red asterisk. Estimated amplitude:21.16 mV.

−4 −2 0 2 4 6 80

5

10

15

20

25

occu

rren

ces

[%]

amplitude [mV]

µ =2.2 mV

σ =1.8 mV

σ/µ =0.85

χ2red

=2.23

deg. free = 8

Figure 3.7: Gaussian amplitude distribution behaviour of samples before trigger for one signal ofthe complete analysed dataset (according to a Pearson’s chi-square test with confidence level of5%).

55


By applying this amplitude estimation procedure to the entire dataset, we canbuild the histogram reported in Fig. 3.8. Every peak represents the mean amplitudevalue of the carried photons. Signals carrying no photons can be considered purenoise, however, since their values are different from a flat line (fixed to 0 mV), theiramplitude is distributed around a 0γ peak higher than 0 mV. All the peaks follow aPoisson distribution, in according to the emittance of a laser diode [25], as reportedin the following Section.

On the same histogram we can distinguish up to 5 detected photons, however,the regions between two consecutive peaks are significantly higher than zero andthere are too many cases of uncertainty on the precise number of carried photons.In order to clarify this point, a signal analysis is required. To do so, we need tounderstand the behaviour of each peak in terms of Energy Resolution, SNR andTime Jitter.

0 10 20 30 40 50 60 700

500

1000

1500

2000

2500

amplitude [mV]

occu

rren

ces

4γ5γ

3γ

2γ

1γ0γ

Figure 3.8: Example of TES noisy signals amplitude histogram and carried photons.

56


3.3.1 Energy Resolution

Since the incident laser pulses are highly attenuated, the photon number distributionfollows a Poisson statistic [25]. Therefore, the aforementioned histogram (Fig. 3.8)can be fitted with a Poisson distribution convolved with Gaussian functions:

yfit =Ae−µ

√2πσ2

[exp

(−(x − x0)

2

2σ20

)σ

σ0+

n∑

i=1

µi

i!exp

(−(x − xi)

2

2σ2

)](3.1)

where A is a normalization factor; µ is the average number of photons detectedper every pulse of incident light; i is the number of absorbed photons detected inthe same pulse that generates the signal centered on amplitude equal to xi; σ is thestandard deviation for peaks representing more than zero carried photons; x0 is themean value of the first peak (0γ) and σ0 is its standard deviation.

We have assumed a common standard deviations σ of all peaks carrying morethan zero γ, while, for the zero photon peak a dedicated Gaussian has been intro-duced and is represented, in the fit equation, by the first quantity after the squareparenthesis.

All peak mean values can be calculated using the histogram (Fig. 3.8) and intro-duced in the Matlab fitting program to reduce the free variables to be estimated.Thus, the values obtained by the fit are: the normalization factor A, the averagenumber of detected photons µ, and the standard deviations σ and σ0.

As sigma (σ) is related to the Energy Resolution (∆E) a possible way to definethis quantity is:

∆E =2√

2ln2σ

x2 − x1

· Eλ (3.2)

where Eλ is the energy of the incident photon, in our case known (see Sect. 3.2) andequal to 0.95 eV for this dataset. The numerator is the Full Width at Half Maximum(FWHM) and can be easily calculated as 2.355σ, while at the denominator we havethe amplitude distance of the first two peaks representing signals carrying 2 and 1photons, respectively.

By using this procedure for the Energy Resolution estimation (see Fig. 3.9), wehave obtained a value of 0.46 eV. For the uncertainty calculation of this quantity,see Appendix B. We would like to stress that on this set of signals no processinghas been applied.

57


0 10 20 30 40 50 600

500

1000

1500

2000

2500

amplitude [mV]

occu

rren

ces

∆E =0.46±0.01 eV

histogram noisyfit

0γ

4γ5γ

3γ

2γ

µ1

µ2

FWHM

1γ

Figure 3.9: I dataset - 34000 pulses, λ = 1310 nm - Noisy signal amplitudes histogram (blue dots)and fit (red line). This is the procedure to evaluate the Energy Resolution. No processing hasbeen applied to any signal to reduce noise.

3.3.2 Signal-to-Noise Ratio

The SNR is an index that expresses the ratio between the signal and the noisepower, it is used to control the capability of a filter to reduce noise components ina measured signal, the higher it is, the easier the signal features extraction is (seeFig. 3.10). In this Section it has been calculated as:

SNRdB = 10 · log10A2

(4σ)2 (3.3)

where A is the amplitude of the considered signal and σ is the RMS of noisy values ofevery acquired waveform before trigger. Note that the noise amplitude distributionhas been considered Gaussian (see Fig. 3.7) and the denominator corresponds to the95% amplitude band of the aforementioned distribution. The SNR results of noisyand processed signals will be presented and discussed in Sect. 3.4 together with theenergy resolution estimation comparisons.

In order to start our filtering analysis, we could consider and investigate thefrequency signal components by means of a PSD analysis. The corrupting noisepresents an amplitude that is Gaussian distributed, as presented in Fig. 3.11, where

58


−2 0 2 4 6 8 10 12 14−20

−15

−10

−5

0

5

10

time [µs]

ampl

itude

[mV

]


A

4 σ

Figure 3.10: Example of SNR evaluation on the same signal of Fig. 3.6.

1000 amplitude values evaluated from 1000 different traces at the same time coordi-nate have been processed. In this example, the time coordinate has been considerednegative (before trigger) to take into account the signal samples before the pho-ton arrival. Therefore, we can average some thousands of signal carrying the samenumber of photons and construct the reference waveform for that specific numberof carried photons (see Appendix C). In Figs. 3.12 and 3.13 are presented thecomparisons between a noisy signal carrying 3γ and its reference signal in time andfrequency domains respectively.

This average shows an enhancement of 30 dB in the SNR evaluation since thisconsidered noisy signal presents a SNR of 12.0 dB while the reference of 42.3 dB.In the time domain (Fig. 3.12) we can appreciate a very high reduction in theamplitude fluctuations, whereas in the frequency domain (Fig. 3.13) two importantfeatures come out: first, the desired signal has the predominant contributes at lowfrequency (comparable with the noisy waveform) and, second, the noise componentis more powerful and evident at middle and high frequency (with respect to theconsidered bandwidth).

For this reason the filters chosen were basically Low-Pass (LP) filters. In thefollowing Sections 3.3.4, 3.3.5, the Savitzky-Golay and the Wiener filters will bedescribed.

59


−10 −5 0 5 10 150

2

4

6

8

10

12

14

16

18

20

22

occu

rren

ces

[%]

amplitude [mV]

Entries = 1000

µ = 4 mV

σ = 3 mV

σ/µ = 0.75

χ2red

= 2.04

deg. free = 10

Figure 3.11: Gaussian amplitude distribution of samples before trigger, according to a Pearson’schi-square test with confidence level of 5%. These samples have the same negative time value andcome from 1000 noisy signals that carry the same number of photons (1γ in this example).

−2 0 2 4 6 8 10 12 14−25

−20

−15

−10

−5

0

5

time [µs]

ampl

itude

[mV

]

noisy 3γref 3γ

Figure 3.12: Comparison of a noisy signal carrying 3γ and of its 3γ reference waveform (obtainedwith the averaging technique)

60


10−2

10−1

100

101

−50

−40

−30

−20

−10

0

10

20

Frequency [MHz]

PS

D [d

B]

noisy 3γref. 3γ

Figure 3.13: Detailed PSD behaviour of a noisy signal carrying 3γ and of its 3γ reference waveform(obtained with the averaging technique)

61


3.3.3 Time Jitter

In a TES signal detection, another important feature is the time jitter. It has beencalculated as the time fluctuation (RMS) of time values at half amplitude (HA) ina subset of waveforms corresponding the same number of photons. In Fig. 3.14an example of the procedure applied for the estimation of this feature is shown; acomplete comparison on all datasets will be presented in Sect. 3.4.

−2 0 2 4 6 8 10 12 14−25

−20

−15

−10

−5

0

5

10

time [µs]

ampl

itude

[mV

]


A / 2

HA time: 0.05 µs

Figure 3.14: Example of HA time evaluation.

3.3.4 Savitzky-Golay filter

One very common use for LP filters is to smooth noisy data. The premise of datasmoothing is that one is measuring a variable that is both slowly varying and alsocorrupted by random noise. Then it can sometimes be useful to replace each datapoint by some kind of local average of surrounding data points. Since nearby pointsmeasure very nearly the same underlying value, averaging can reduce the level ofnoise without (much) biasing the value obtained. In this Section we discuss a partic-ular type of LP filter, well-adapted for data smoothing, and called Savitzky-Golay(SG) filter [26], [27]. Rather than having their properties defined in the Fourier

62


domain, and then translated to the time domain, SG filters derive directly from aparticular formulation of the data smoothing problem in the time domain, as wewill now see.

A digital filter is applied to a series of equally spaced data values fi ≡ f(ti), whereti ≡ t0+i∆ for some constant sample spacing ∆ and i = . . . ,−2,−1,0,1,2, . . . One ofthe simplest type of digital filter (the non-recursive or finite impulse response filter)replaces each data value fi by a linear combination gi of itself and some number ofnearby neighbours:

gi =

nR∑

n=−nL

cnfi+n (3.4)

where nL is the number of points used to the left of a data point i, i.e., earlier thanit, while nR is the number used to the right, i.e., later. A so-called causal filterwould have nR = 0.

As a starting point for understanding SG filters, let us consider the simplestpossible averaging procedure: for some fixed nL = nR, we have to compute each gi

as the average of the data points from fi−nL to fi+nR. This is called moving windowaveraging and corresponds to Eq. (3.4) with constant cn = 1/(nL + nR + 1). Ifthe underlying function is constant, or is changing linearly with time (increasing ordecreasing), then no bias is introduced into the result. Higher points at one endof the averaging interval are on the average balanced by lower points at the otherend. A bias is introduced, however, if the underlying function has a non-zero secondderivative. At a local maximum, for example, moving window averaging alwaysreduces the function value.

The idea of SG filtering is to find filter coefficients cn that preserve higher mo-ments. Equivalently, the idea is to approximate the underlying function withinthe moving window not by a constant (whose estimate is the average), but by apolynomial of higher order, typically quadratic or quartic: for each point fi, weleast-squares fit [14] a polynomial to all nL + nR + 1 points in the moving window,and then set gi to be the value of that polynomial at position i. We make no useof the value of the polynomial at any other point. When we move on to the nextpoint fi+1, we do a whole new least-squares fit using a shifted window. Since theprocess of least-squares fitting involves only a linear matrix inversion, the coefficientsof a fitted polynomial are themselves linear in the values of the data. That meansthat we can do all the fitting in advance, for fictitious data consisting of all zerosexcept for a single 1, and then do the fits on the real data just by taking linearcombinations. There are particular sets of filter coefficients cn for which Eq. (3.4)automatically accomplishes the process of polynomial least-squares fitting inside amoving window. For a detailed description of the coefficient derivation, the readeris referred to [26].

63


Using Matlab, we needed only to choose the window dimension (expressed innumber of samples) and the order of the polynomial fitting curve to interpolatedata points. Since in all our analysed datasets the waveform length was fixed for allpulses and ranged, choosing an acquisition session, from 1000 to 2500 samples each,the window dimensions were 11, 51 and 101 samples. They were odd, as requiredby theory [26] to build proper matrixes for the coefficients calculation, and of lengthfrom one hundredth to one tenth of a noisy signal waveform. In Fig. 3.15 an exampleof a SG filtering (window: 101 samples, polynomial order: 3) of a noisy 2γ signal ispresented. The SNR of the noisy waveform is 9.8 dB, while for the SG filtered oneis 11.8 dB, with an improvement of 2 dB.

−2 0 2 4 6 8 10 12 14−20

−15

−10

−5

0

5

time [µs]

ampl

itude

[mV

]

noisy 2γSG 2γ

Figure 3.15: Comparison of a noisy signal carrying 2γ and of the Savitzky-Golay filteringelaboration performed on the same signal. The reference 2γ signal is superimposed.

A comparison of the improvements operated by SG filters on the complete datasetboth on the SNR, the Time Jitter and the Energy Resolution will be presented inSect. 3.4.

64


3.3.5 Wiener filter

Another alternative to blind smoothing is the so-called optimal or Wiener filter-ing [27], [14]. We are interested in the removal of noise from a corrupted signal.The particular situation we consider is this: there is some underlying, uncorruptedsignal u(t) that we want to measure. The measurement process is imperfect andwhat comes out of our measurement device is a corrupted signal c(t). First, theapparatus may not have a perfect delta-function response, so that the true signalu(t) is convolved with some known response function r(t) to give a smeared signals(t):

s(t) =

∫ +∞

−∞

r(t − τ)u(τ)dτ, or S(f) = R(f)U(f) (3.5)

where S,R and U are the Fourier transforms of s, r and u, respectively. Second, themeasured signal c(t) may contain an additional component of noise n(t):

c(t) = s(t) + n(t) (3.6)

To deconvolve the effects of the response function r, as in the absence of anynoise, we just divide C(f) by R(f) to get a deconvolved signal. Our task is to findthe optimal filter, φ(t) or Φ(f), which, when applied to the measured signal c(t) or

C(f), and then deconvolved by r(t) or R(f), produces a signal u(t) or U(f) thatis as close as possible to the uncorrupted signal u(t) or U(f). In other words weestimate the true signal U by:

U(f) =C(f)Φ(f)

R(f)(3.7)

We ask that U and U be close in the least-square sense minimizing:∫ +∞

−∞

|u(t) − u(t)|2dt =

∫ +∞

−∞

∣∣∣U(f) − U(f)∣∣∣2

df (3.8)

Substituting Eqs. (3.6) and (3.7), the right-hand side of (3.8) becomes:

∫ +∞

−∞

∣∣∣∣[S(f) + N(f)]Φ(f)

R(f)− S(f)

R(f)

∣∣∣∣2

df =

=

∫ +∞

−∞

|R(f)|−2|S(f)|2|1 − Φ(f)|2 + |N(f)|2|Φ(f)|2

df (3.9)

If the signal S and the noise N are uncorrelated their cross product, when integratedover frequency f , gives zero. Eq. (3.9) is a minimum if and only if the integrand

65


is minimized with respect to Φ(f) at every value of f . Let us search for such asolution where Φ(f) is a real function. Differentiating with respect to Φ, and settingthe result equal to zero gives:

Φ(f) =|S(f)|2

|S(f)|2 + |N(f)|2 (3.10)

This is the formula for the optimal filter Φ(f). Note that Eq. (3.10) involves S,the smeared signal, and N , the noise. These two components add up to be C, themeasured signal. Eq. (3.10) does not contain U , the true signal. This represents animportant simplification: the optimal filter can be determined independently of thedetermination of the deconvolution function that relates S and U .

To determine the optimal filter from Eq. (3.10) we need some way of separatelyestimating |S|2 and |N |2. There is no way to do this from the measured signalC alone without some other information, or some assumption. Luckily, the extrainformation is often easy to obtain. For example, we can sample a long stretch ofdata c(t) and plot its Power Spectral Density (PSD). This quantity is proportionalto the sum |S|2 + |N |2, so we have:

|S|2 + |N |2 ≈ Pc = |C(f)|2, 0 6 f < fc (3.11)

The resulting plot (see Fig. 3.16) often immediately show the spectral signature ofa signal sticking up above a continuous noise spectrum. If we draw a smooth curvethrough the noise spectrum, we can extrapolate it into the region dominated by thesignal as well. Now drawing a smooth curve through the signal plus noise powerwe can estimate the difference between these two curves: this is our smooth modelof the signal power. The quotient of our model of signal power to our model ofsignal plus noise power is the optimal filter Φ(f). Note that Φ(f) is close to unitywhere the noise is negligible, and close to zero where the noise is dominant.Theintermediate dependence given by Eq. (3.10) just turns out to be the optimal wayof going in between these two extremes. For the interested reader, a more detaileddescription of the Wiener filter can be found in [27].

Unfortunately, we don’t have long acquisitions of noisy measurements c(t), there-fore the needed extra-information to calculate S(f) (to be used in Eqs. 3.10,3.11)can come from an estimate of the desired component. Furthermore, we have morethan one desired (or reference) signal. In practise, we need a reference for every pos-sible ensemble of detected photons. A possible way to build this set of waveformscomes from the aforementioned averaging technique.

Considering the amplitude histogram reported in Fig. 3.17, in a region of ±1.5standard deviations around the mean value of the third peak (2γ signals), a subsetof noisy signals carrying confidently 2γ can be isolated. Since the amplitude valuedistribution of the same sample on different signals can be considered Gaussian, as

66


Figure 3.16: Optimal (Wiener) filtering. The power spectrum of signal plus noise shows a signalpeak added to a noise tail. The tail is extrapolated back into the signal region as a noise model.The signal model is obtained by subtraction.

reported in Fig. 3.11, the average of this subset can conduce to a reference signal(Fig. 3.18) significantly representative of all noisy signals carrying 2γ.

Calculating the maximum of each cross-correlation between this built waveformand all acquired signals, the histogram reported in Fig. 3.19 can be obtained.The relative minima among all peaks are highly reduced. By means of this newdistribution a set of reference signals can be easily produced with the averagingtechnique (Fig. 3.20, see Appendix C). Since the TES response may be slightlynon-linear, we chose to identify different reference pulse signals (instead of usingmultiple of the elementary one). Reference signals presented in Fig. 3.20 are usedto feed the Wiener filter.

Working on a Matlab platform, our best results have been obtained with a 30th

order Wiener filter. In Fig. 3.21 this filter’s output is reported and compared witha SG 3rd order (window 101 samples) filter elaboration and the original unfiltered2γ trace.

The results of the elaborations with the Wiener filter will be presented anddiscussed in the following Section.

67


0 10 20 30 40 50 60 700

500

1000

1500

2000

2500

mV

occu

rren

ces

4γ5γ

3γ

2γ

1γ0γ

Figure 3.17: Amplitude histogram of the TES noisy signals already reported in Fig. 3.8 with, inthe red ellipse, the subset of noisy signals that carry confidently 2γ.

−2 0 2 4 6 8 10 12 14

−16

−14

−12

−10

−8

−6

−4

−2

0

2

time [µs]

ampl

itude

[mV

]

ref. 2γ

Figure 3.18: Reference signal obtained averaging the subset of noisy signals that carry confidently2γ.

68


0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090

1000

2000

3000

4000

5000

6000

7000

8000

cross−correlation [ V2 ]

occu

rren

ces

0γ

5γ4γ

1γ

2γ

3γ

Figure 3.19: Histogram of cross-correlation between the 2γ reference signal and all noisy waveforms.

−2 0 2 4 6 8 10 12 14−45

−40

−35

−30

−25

−20

−15

−10

−5

0

5

time [µs]

ampl

itude

[mV

]

ref 1γref 2γref 3γref 4γref 5γ

Figure 3.20: All 5 reference signals for the Wiener filter. These waveforms are obtained with theaveraging technique, based on the cross-correlation histogram.

69


−2 0 2 4 6 8 10 12 14−20

−15

−10

−5

0

5

time [µs]

ampl

itude

[mV

]

SNRnoisy

:9.8 dB − SNRSG

:11.8 dB − SNRWien

:13.0 dB

noisy

ref 2 γSG

Wien.30th ord

Figure 3.21: Example of a TES noisy signal (cian), reference signal (green), SG (blue) and Wiener(red) filter’s outputs. SNR comparisons are also reported.

70


3.4 Results and discussion

Until now examples of filtering have been given in the time domain, it is worth topresent the performances of these filters also in the frequency domain. An exampleof PSD comparison of a TES noisy raw signal with the results of SG and Wiener fil-tering is reported in Fig. 3.22. Both filters reduce the noise components in the range1-5 MHz while Wiener enhances better the desired components at low frequency (<1 MHz). The SNR values reported in Table 3.1 indicate that Wiener works better

10−2

10−1

100

101

−50

−40

−30

−20

−10

0

10

20

frequency [MHz]

PS

D [d

B]

noisy

ref. 2γSGWiener

Figure 3.22: PSD of the same noisy signal (cian) presented in Fig. 3.21, of the 2γ reference signal(green), of SG (blue) and of Wiener (red) filter outputs.

than SG in the SNR improvement. The reason is that the SNR evaluation hereconsidered (Eq. 3.3) highly depends on peak-to-peak amplitude measurement andon the enhancement of the desired low-frequency components.

SNR [dB] 1γ 2γ 3γ 4γ 5γ

Noisy 3.3±2.0 8.5±1.6 11.8±1.5 14.1±1.5 15.8±1.4SG 3.2±1.9 8.5±1.6 11.7±1.5 14.1±1.5 15.9±1.4

Wiener 6.3±2.4 11.3±2.0 14.0±1.8 16.0±1.7 17.4±1.6

Table 3.1: SNR evaluation on noisy and filtered signals as a function of the number of carriedphotons.

71


As filters introduce a delay, it is important to estimate the time jitter of theresulting waveforms (Sect. 3.3.3). From the results presented in Tab. 3.2, we canconclude that neither SG nor Wiener reduce the time jitter and that, fixed the filter,the time jitter estimate is sensitive to the SNR.

Jitter [µs] 1γ 2γ 3γ 4γ 5γ

Noisy 0.10 0.04 0.03 0.02 0.02SG 0.13 0.07 0.05 0.03 0.03

Wiener 0.11 0.04 0.02 0.02 0.01

Table 3.2: Time jitter estimates on noisy and filtered signals as a function of the number of carriedphotons.

With the Energy Resolution estimation and the procedure described in Sect.3.3.1 by using the cross-correlation information presented in Fig. 3.19, the 0γ his-togram bars can be isolated and neglected. Therefore, for all fits shown from nowon, signals with 0γ will not be considered. In this way fitting elaboration andconvergence are easier and faster.

In Fig. 3.23 original noisy data have been used to build a histogram whose fitallowed us to estimate a ∆E of 0.46 eV with the aforementioned procedure (Eq. 3.2).Applying the same technique to amplitude estimates after a SG filtering (Fig. 3.24),the ER is 0.39 eV. The best result is given by the Wiener filter, the correspondingamplitude histogram is reported in Fig. 3.25 together with its fit. The ER is 0.22eV and is less than half of the original value obtained from noisy measurements.

The uncertainties on these Energy Resolution estimations are small enough toconclude that the improvements obtained with signal processing are statisticallysignificant.

By taking into account the noisy signals fit (Fig. 3.23) and the one evaluatedafter Wiener filtering (3.25), we can see that, on the second fit, the peak mean valuesare spread in a symmetric way with respect to the 3γ peak. This feature dependson the ratio between the power of the desired component (reference) and the noiseone for the analysed pulse. In this way, SNR modifies the Wiener transfer function(Eq. 3.10) and the filtering returns the aforementioned spread of peak mean values(for a detailed explanation see Appendix D).

Finally, it can be interesting to compare the persistence of noisy signals reportedon Fig. 3.5 with the one obtained on signals processed with Wiener filter on Fig.3.26. Only signals of one or more photons have been retained and separation isappreciable: white areas appear between two consecutive curves in the plot of Fig.3.26. Some other examples of Wiener filtering will be given in Sect. 3.5.

72


0 10 20 30 40 50 600

500

1000

1500

2000

2500

amplitude [mV]

occu

rren

ces

∆E =0.46±0.01 eV

histogram noisyfit

1γ

3γ

5γ

2γ

4γ

Figure 3.23: I dataset - 34000 pulses, λ = 1310 nm - Noisy signal amplitudes histogram (dots) andfit (line). No processing has been applied. The 0γ peak has been isolated and neglected using thecross-correlation information.

0 10 20 30 40 50 600

500

1000

1500

2000

2500

amplitude [mV]

occu

rren

ces

∆E =0.39±0.01 eV

histogr SGfit

2γ

3γ

1γ

4γ5γ

Figure 3.24: I dataset - 34000 pulses, λ = 1310 nm - Savitzky-Golay filtered signal amplitudeshistogram (dots) and fit (line). 0γ peak is neglected using the cross-correlation information.

73


0 10 20 30 40 50 600

500

1000

1500

2000

2500

3000

amplitude [mV]

occu

rren

ces

∆E =0.22±0.01 eV

histogram Wienerfit

1γ

3γ

2γ

5γ4γ

Figure 3.25: I dataset - 34000 pulses, λ = 1310 nm - Wiener filtered signals amplitude histogram(dots) and fit (line). 0γ peak is neglected using the cross-correlation information.

Figure 3.26: Persistence representation of Wiener filtered signals of 1 to 5γ. The uncertaintyregions are reduced and among them white areas become appreciable. Resolution: Time 0.02 µs -Amplitude 0.5 mV. The colorbar is expressed in percentage.

74


3.5 Other examples of analysed datasets

0 10 20 30 40 50 60 70 80 900

500

1000

1500

2000

2500

3000

amplitude [mV]

occu

rren

ces

∆E =0.45±0.01 eV

histogram noisyfit

1γ

2γ

4γ

3γ

5γ 6γ

Figure 3.27: II dataset - 45000 pulses, λ = 1550 nm - Noisy signal amplitudes histogram (dots)and fit (line). The 0γ peak has been isolated and neglected using the cross-correlation information.

0 20 40 60 80 1000

500

1000

1500

2000

2500

3000

3500

4000

4500

amplitude [mV]

occu

rren

ces

∆E =0.25±0.01 eV

histogram Wienerfit

70 80 90 1000

20

40

60

80

100

amplitude [mV]

occu

rren

ces

5γ 6γ4γ

3γ

2γ

1γ

5γ

6γ

Figure 3.28: II dataset - 45000 pulses, λ = 1550 nm - Wiener filtered signals amplitude histogram(dots) and fit (line). 0γ peak is neglected using the cross-correlation information.

75


20 40 60 80 100 120 140 160 180 2000

500

1000

1500

2000

2500

3000

amplitude [V]

occu

rren

ces

∆E =0.59±0.01 eV

histogram noisyfit

1γ

2γ

3γ

4γ

5γ6γ

Figure 3.29: III dataset - 50000 pulses, λ = 1310 nm - Noisy signal amplitudes histogram (dots)and fit (line). The 0γ peak has been isolated and neglected using the cross-correlation information.

50 100 150 2000

500

1000

1500

2000

2500

3000

3500

4000

amplitude [V]

occu

rren

ces

∆E =0.29±0.01 eV

histogram Wienerfit

180 200 220

0

10

20

30

40

amplitude [mV]

occu

rren

ces

2γ

3γ

4γ

5γ

1γ

6γ

7γ

8γ

Figure 3.30: III dataset - 50000 pulses, λ = 1310 nm - Wiener filtered signals amplitude histogram(dots) and fit (line). 0γ peak is neglected using the cross-correlation information.

76


3.6 Energy evaluation and TES amplitude response

Another important feature that can be estimated directly on real data is the energyof the single photon. It is interesting to compare this estimate with the valuecorresponding to the laser wavelength. A discrepancy hints to a system inefficiency.

To perform this analysis, we have to consider the TES bias circuit reported in Fig.3.31. When a photon arrives, the current inside TES changes and is transformedinto a voltage signal by the dc-SQUID (see Fig. 3.2).

Figure 3.31: TES polarization circuit.

We can evaluate the signal energy by integrating the signal power over time:

E(t) =

∫ +∞

0

PTES(t) dt =

=

∫ +∞

0

VTES(t) · ∆ITES(t) dt (3.12)

where we write the KVL (Kirchhoff’s Voltage Law) for the loop on the left in Fig.3.31:

VTES(t) = Rbias · Isensor(t) =

= Rbias · [Ibias − (ITES + ∆ITES(t))] (3.13)

The 1γ averaged voltage signal is reported in Fig. 3.32 for the III dataset. Thiswaveform can be transformed into a current pulse dividing it by a trans-resistancefactor G ( ∆ITES(t) = V (t)/G ).

77


−6 −4 −2 0 2 4 6 8 10 12 14−20

−15

−10

−5

0

time [µs]

ampl

itude

[mV

]

Figure 3.32: One photon average signal.

For the III dataset, the bias circuit parameters were:

• G = 327317 V/A

• Ibias = 315 µA

• ITES = 6.2 µA

• Rbias = 2.38 mΩ

• dt = 10 ns (time sampling)

Eq. 3.12 gives an energy for the single photon of 0.86 eV. At a laser wavelengthof 1310 nm, the theoretical energy for the single photon is 0.95 eV, as described inSect. 3.2 and Appendix B. This difference is mainly due to phonons absorption(layer dispersion) and electrodes dispersion at the TES borders.

By Eq. 3.12 we estimate the TES amplitude response for the energy of theaverage signals carrying 1 to 8 photons, see Figs. 3.33 and 3.34.

We conclude that the III dataset presents a linear trend for the first 7 photons.

78


0 1 2 3 4 5 6 7 8 9−1

0

1

2

3

4

5

6

7

8

# photons

Ene

rgy

[eV

]

datalin. trend

7.95 8.00 8.05

7.20

7.25

7.30

7.35

7.40

# photons

Ene

rgy

[eV

]

Figure 3.33: Linear trend of the TES amplitude response of the first 7 detected photons, thenon-linear region starts form 8γ. The inset shows the estimated error in the 8γ energy evaluation.

0 1 2 3 4 5 6 7 8 90.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

# photons

rel.

Am

pl v

s re

l. E

nerg

y

AmplitudeEnergylin. trend

Figure 3.34: Linear trend of the TES amplitude response of the first 7 detected photons. Normal-ized to the unit amplitude and energy values as a function of the number of carried photons. Theonly mean values are reported.

79


3.7 Conclusions and outlook

An analysis in both frequency and time domains has been performed. Signal pro-cessing is necessary to significantly enhance the most important features in triggeredTES acquisitions with known wavelength. On the single pulse to:

• estimate the amplitude;

• evaluate the time jitter;

• calculate the SNR.

And on the complete dataset to:

• obtain the amplitude histogram;

• estimate the energy resolution (ER) and its uncertainty;

• construct reference signals ( 1γ, 2γ,..);

• calculate single photon energy on real signals;

• verify the range of the linear trend in TES amplitude responses.

The filtering techniques we used to improve these features were a Savitzky-Golayand an Optimum Wiener filters.

Several datasets of noisy signals have been processed and, on them all, Wienerfilter worked better than SG, both in SNR enhancement and in ∆E improvementwith negligible superposition between two consecutive peaks in the amplitude his-tograms. Time Jitter was reduced neither by SG nor by Wiener filters. Wienerfiltering allowed in all cases a gain close to a factor 2 in ∆E.

As a future development, it is highly desirable to improve data resolution by usinga 12-14 bits ADC as present data acquisition was performed by an oscilloscope withan 8-bit ADC. In this way, avoiding slow oscilloscope acquisitions, we could alsoacquire signals in less time. Furthermore, on-line filtering implemented on FPGA,will not require to save all noisy waveforms because only processed values will besaved, drastically reducing the dimension of the storage memories.

80

Chapter 4

Amplitude estimation of realsignals for the ALICE experiment

4.1 Introduction

This Chapter describes the signal analyses conduced by the author, as visiting PhDstudent at CERN (European Organization for Nuclear Research - Geneva [28]),during a period of three months from March to May 2010. There, the authorhad the possibility of studying new DSP algorithms dealing with real data froman LHC (Large Hadron Collider [28]) experiment. In particular, he worked in theALICE (A Large Ion Collider Experiment [29]) TPC (Time Projection Chamber)data acquisition group, with Dr. Luciano Musa as supervisor.

In the following paragraph a brief introduction to CERN will be given, while adescription of the ALICE experiment will be provided in Sect. 4.2.

The creation of a European Laboratory was recommended at a UNESCO meet-ing in Florence in 1950, and less than three years later a Convention was signed by12 countries of the Organisation Europeene pour la Recherche Nucleaire [28], [30].CERN was born, the prototype of a chain of European institutions in space, astron-omy and molecular biology. CERN exists primarily to provide European physicistswith accelerators that meet research demands at the limits of human knowledge.In the quest for higher interaction energies, the Laboratory has played a leadingrole in developing colliding beam machines. The Intersecting Storage Rings (ISR)proton-proton collider was commissioned in 1971, and the proton-antiproton col-lider at the Super Proton Synchrotron (SPS), which came on the air in 1981 andproduced the massive W and Z particles two years later, confirming the unified the-ory of electromagnetic and weak forces. The main impetus came from the LargeElectron-Positron Collider (LEP), where measurements unsurpassed in quantity andquality tested our best description of subatomic Nature, the Standard Model.

81

4 – Amplitude estimation of real signals for the ALICE experiment

All evidence indicates that new physics, and answers to some of its most pro-found questions, lie at energies around 1 TeV. To look for this new physics, the nextresearch instrument in Europe’s particle physics armory is the LHC, Large HadronCollider. It is installed in the 27 Km LEP tunnel and is fed by existing particlesources and pre-accelerators. The new accelerator, however, is using the most ad-vanced superconducting magnet and accelerator technologies available today. Thefour detectors of the LHC project are ATLAS, CMS, ALICE and LHCb, depicted inFig. 4.1. These experiments are different in nature and have been designed to sensevarious theoretically predicted phenomena. They must, nevertheless, be preparedfor unexpected effects and, possibly, a novel type of physics.

Figure 4.1: LHC overview and the 4 experiments.

The aim of high-energy heavy-ion physics is to study matter interactions atextreme energy densities. In ordinary matter, quarks are bound together to formprotons and neutrons inside a nucleus. Physicists believe high-energy heavy-ioncollisions will compress and heat nuclei to a point where individual protons andneutrons will collapse, creating an enormous concentration of energy in a localizedarea. For a brief time, a large number of free quarks and gluons can then exist

82


unbounded to any other particle. This state of matter is called quark-gluon plasma(QGP) and, like its name suggests, it is a soup, or plasma, of quarks and gluons.

The understanding of QGP enable us not only to uncover the mysteries of the firstmoments of our Universe, but to enlarge our knowledge of neutron stars and also toinspect the smallest constituents of matter known today. The LHC and the ALICEexperiment are at the same time, the most sensitive microscope, the most powerfultelescope and most accurate time machine. Such ambitious project imposes harshconstraints but also stimulating challenges in the design of front-end electronics ableto readout, digitize, process and ship physics data to a collaboration of more than1000 scientists from 94 institutes in 28 countries working only for ALICE.

4.2 ALICE experiment

ALICE is a 26 m long, 16 m diameter cylinder-like detector with a total weightof approximately 10000 t (see Fig. 4.2) where lead and calcium ions will collidehead-on at energies at the center-of-mass in the order of 5.5 TeV/nucleon [29], [31].Around 20.000 charged particles will be created in each heavy ion collision within thedetector’s acceptance. Each particle follows a different trajectory (track) accordingto its charge, momentum, mass and possible decay. ALICE has various detectorsable to measure these physical quantities, decrypting in a myriad of tracks eachidentified particle. One of the main detectors inside the ALICE experiment is calledTPC, Time Projection Chamber [31].

This detector has an array of large volume gas-drift-chambers, which record elec-tronically the track of ionization electrons left behind in the gas by each traversingparticle. Owing to the large number of expected tracks, the detector system musthave an extreme spatial resolution to separate each particle track. At the TPCend-plates, conventional Multi-Wire Proportional Chambers (MWPC) provide thecharge amplification and readout by means of a cathode plane segmented in about560.000 pads. Each pad is connected to an electronics chain designed to amplify,integrate, shape, digitize and pre-process the detector signal before transmission tothe DAQ.

The front-end electronics system (see Fig. 4.3) has to satisfy several detec-tor constraints while meeting the required performance specifications. Namely, theconstraints are related to the temperature stability of the electronics, available me-chanical space on the detector, large number of readout pads, reliability, very largedata volume and radiation hardness. The specifications of the frontend electronicscame as a consequence of the detector constraints and refer to the level of cross-talk, signal-to-noise ratio, dead-time introduced by the electronics, conversion gain,shaping time, sampling frequency, tail cancellation and data reduction.

83


Figure 4.2: ALICE spectrometer schematic reported in [31].

Figure 4.3: An overview of the ALICE TPC front end electronics reported in [31].

84


These specifications and constraints are more stringent than any other detectorof this type, presenting increasing challenges to engineers and physicists alike. Anew concept of front-end electronics was studied and implemented. Each detectorpad is connected to a charge sensitive amplifier that integrates the induced chargeand converts it into a voltage. This signal is subsequently shaped by a shapingamplifier. These functions are implemented in a dedicated analog 16-channel chip,PASA (Pre-Amplifier Shaping Amplifier), block diagram and chip layout in Figs.4.4 and 4.5 respectively.

Figure 4.4: A simplified block diagram of the PreAmplifier-ShAper (PASA) signal processing chainreported in [31].

Figure 4.5: Layout of the PASA chip presented in [31].

85


Instead of storing the signal in analog memories, as in previous solutions, theoutput of the PASA is directly digitized using an ADC per channel. The unavail-ability of commercial components that integrate a high number of A/D converterswith a custom data processor have driven the design of the ALTRO (ALice TpcRead Out) chip, see Figs. 4.6 and 4.7.

Figure 4.6: ALTRO chip block diagram reported in [31].

This mixed-signal ASIC integrates 16 channels, each of them consisting of a 10-bit, 25-MSPS ADC, a pipelined digital processor and a multi-event data memorywith a capacity of 800 Kbit. The digital processor performs several functions andstores the data in a memory minimizing the electronics system dead-time. A front-end card, FEC, host 8 PASAs chips and 8 ALTROs acquiring a total of 128 channels.Each channel is readout individually by a higher level module called RCU, ReadoutControl Unit, that is responsible to address 25 FECs. Close to 200 RCUs are neededto read the full 560.000 channels of the ALICE TPC detector.

A fundamental data reduction technique present in the ALTRO chip is zero-suppression, which simply discards all data samples below a pre-defined thresholdas shown in Fig. 4.8.

86


Figure 4.7: ALTRO chip layout presented in [31].

Figure 4.8: Zero Suppression scheme reported in [31].

87


An efficient zero-suppression can reduce the data volume by a factor of 5 or moreand it is under these assumptions that the system level architecture was designed.Beside decreasing the amount of hardware, reducing the data volume implies areduction in the dead-time of the system. The efficiency of the zero-suppressiondepends mainly on occupancy (i.e. rate), on the behaviour of the baseline on whichthe pulses are sitting and on any long tail super-imposed to the pulses. Whereasoccupancy is given by physics, various elements can introduce degradation of thebaseline: temperature variations causing a slow signal drift; the detector triggersignal inducing systematic perturbations, AC coupling effects or even pick-up noise.Process variations of the analog circuits induce slightly different gains and pulseshapes, which preferably would be equalized on-line on a per channel basis in orderto comply with the resolution requirements. Gain equalization and correction ofbaseline perturbations (see Fig. 4.9) can be accurately suppressed by dedicateddigital circuitry present in the ALTRO chip. The signal induced in the detector

Figure 4.9: Baseline Correction II block operation principle presented in [31].

pads is characterized by a fast rise time (less than 1 ns) and a long tail induced bythe ions. The shape of the tail is rather complex and depends on the gas mixtureand the geometry of the chamber. This tail, causing pile-up effects and eventuallysaturation, sets the main limitation to the maximum track density at which a MWPCcan be exposed, same is to say that zero-suppression becomes ineffective under highoccupancy scenarios. Therefore, an accurate and reconfigurable tail cancellationand baseline restoration of the detector signal are fundamental requirements forthe front-end electronics. In previous experiments, analog tail cancellation filters

88


were designed using continuous-time analog filters. Owing to the poor precisionin the matching of the passive components, available by today’s integrated circuittechnology, the tail cancellation filter would not reach the high accuracy needed:0.1% at 1 µs after the peak. In order to cope with these specifications and thelimitations of the analog circuits, a digital tail cancellation filter is proposed. Thisnew digital system allows reconfiguration of the filter coefficients, which is vital sincethe exact shape of the signal is difficult to predict and it is known with high precisiononly when the detector starts its operation.

In order to perform baseline corrections, gain equalization and tail cancellation,several flexible and programmable discrete-time algorithms were developed and im-plemented in the digital processor of the ALTRO chip. It was found, however, thatthe synthesis of some of these circuits was insufficiently studied in the literature.The unavailability or inadequacy of algorithms to design and synthesize digital fil-ters with the specific time-domain requirements of the ALICE TPC triggered thedevelopment of a general framework for such synthesis. This methodology reflectsrequirements on the shape of a desired output rather than requirements on thefrequency-domain. This new definition of the problem, results in a faster, morestructured and more accurate coefficient estimation based on polynomial equations.Moreover, the algorithm can take into account other effects such as long decay,long/short undershoots/overshoots.

Ultimately, the goal is to extract the amplitude and time information of eachpulse. While in the ALICE TPC case this is performed off-line, there are a variety ofmethods [32] - [34] that are able to extract these features on-line in isolated pulses.Their performance, however, is degraded in environments of high occupancy withextensive pile-up effects such as the ALICE TPC. In this sense, more performingalgorithms to accurately extract the pulse features in such harsh conditions arerequired.

In the following Section a known algorithm, developed by Dr. V. Buzuloiu [32],will be investigated to be applied for the extraction of the amplitude and the timeinformation of each real pulse to be implemented in the ALICE TPC.

4.3 Self-adaptive piecewise linear filter

A method of fast and precise estimation of arrival time and amplitude of a pulse willbe presented in this Section first with simulations and then in a possible applicationto real data. The procedure works with few signal samples (no more than 5) and,with a self-adaptive piecewise linear filter; it returns the signal features of interest.This method was presented by Dr. V. Buzuloiu in [32].

89


4.3.1 Simulation results: ideal case

To understand this linear piecewise procedure, it is worth to start with a simu-lation in accordance with the description reported in [32]. Therefore, consideringa pulse-shaped signal of finite duration, mathematically it can be represented bya real-valued function. Among all possible signal shapes of pulses generated by anuclear detector (Lorentzian, Breit-Wigner, Γ4,...), since ALICE TPC signals areparametrized as Γ4 functions (see Fig. 4.10), we will choose Γ4 curve to representour desired components:

Γk (B,A,t0,τ,k; t) = B + Aek

(t − t0

τ

)k

e−k

t − t0τ (4.1)

where B is the baseline amplitude, in our case equal to 0 amplitude arbitraryunits (a.u.); A is the amplitude peak, for us normalized to the unit; t0 is the starttime, 9.5 time a.u. in our simulation; τ is the peak time, set in our simulation equalto 2 a.u; k is the function order, for the ideal TPC impulse response equal to 4.Therefore under these conditions, the simulated reference signal becomes:

Γ4 = Ae4

(t − 9.5

2

)4

e−4

t − 9.5

2 (4.2)

The peak coordinate are (t0 + τ ,A) = (11.5 a.u.,1 a.u.), the simulated time intervalis 50 a.u., the number of digital samples to build the analog curve simulated withMatlab is 1000. In this way, 1 a.u. contains 20 samples (or sub-samples). A digitalwaveform sampled p times corresponds to a point in a p-dimensional space, see Fig.4.11 where the sampling period T has been set equal to 1 a.u. This representativepoint depends on the shift between the clock pulses and the analog signal. If weaim to extract a feature of the signal we must search an invariant of the whole set ofrepresentative points when the shift extends over all possible values. Let us considera uniform sampling, being T the sampling period. Then the shift runs over theinterval [0,T ]. The representative point in the same space will describe a curve, therepresentative curve (Fig. 4.12). Any invariant of this curve can be used to describea feature of the signal. We want an invariant that can be easily computed (a linearone if it exists). On the other hand, we have to deal not with one signal but with afamily of signals having the same shape and of various amplitudes. We would likean algorithm that computes in a simple way both parameters: the amplitude andthe shift. If the representative curve of our signal is a plane curve, then all its pointssatisfy a relation of the form:

p∑

i=1

aisi = v (4.3)

90


0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

time [a.u.]

ampl

itude

[a.u

.]

Figure 4.10: Ideal Γ4 function. The amplitude scale is normalized, while both scales are expressedin arbitrary units.

this is a dot product where p is the number of samples; v and ai are constant; andsi are the samples in temporal order and also the coordinates in the sample space.Apparently, v is an invariant of the curve and it is of the simplest kind (linear),suited for fast computation.

In general, the pulses do not possess the property described by Eq. 4.3. However,if we try to find a plane that best fits all the points of the representative curve, wefind that the errors in v are very small (a fraction of a percent, see Fig. 4.13) forpulses of various shapes even for very few samples (3 to 5). Nevertheless a way toreduce the error, while preserving a simple computation, is needed. We could saythat the representative curve is an e-neighborhood of a plane if the relative error inv does not exceed e. Let ε be the maximum admissible error. If the best fitted planeleads to an error higher then ε, we only need to relax the demand that the wholerepresentative curve must be in an ε-neighborhood of a single plane: we shall try tofit a few planes, each on a segment of the curve. In this way, for each segment, thecurve remains in a ε-neighborhood of the corresponding plane. At the same time,we have to be able to characterize in a simple way each segment. For example, inthe case of a pulse shape like that in Fig. 4.11, s1 is an increasing quantity and s3

a decreasing one when the shift varies from 0 (corresponding to the initial p-tuple,when s1 takes the smallest value accepted as non-negligible) to T.

Therefore, the commuting points for the approximation planes can be defined in

91


8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

time [a.u.]

ampl

itude

[a.u

.]

analog Γ4

DGT Γ4

s2

s3

s1

Figure 4.11: Detailed example of ideal Γ4 sampling. The first 3 DGT samples with amplitudehigher then 0 (a.u.) are s1, s2 and s3.

terms of quotient of s1 and s3 or, equivalently, using comparison:

• if s1 < k1s3, then use dot product n. 1;

• if k1s3 < s1 < k2s3, then use dot product n. 2;

• if k2s3 < s1 < k3s3, then use dot product n. 3;

• ...

where k1, k2 and k3 are defined by the corresponding segments of the representativecurve. In this way the amplitude is the output of a (self-) adaptive piecewise linearfilter. We calibrate the output through ai, for a standard unitary amplitude, then,for a different one, the samples will be proportional if the shape is the same. Thedesigning steps of this feature extractor are as follows: we consider a pulse shaperepresentative of a family of pulses we can detect and we sample it uniformly witha sampling period of T. Let p be the number of samples. We vary the shift betweenthe clock pulses and, say, the instant of the peak of the signal, in small steps τ , overan interval of T. We will have N = T/τ values (or sub-samples). In our example,we have set one sampling period T (equal to 1 a.u., see Fig. 4.11) made by 20sub-samples, or different values of τ .

92


0.7

0.8

0.9

1

0.4

0.6

0.8

10.2

0.3

0.4

0.5

0.6

0.7

s1 [ T ]s

2 [ T ]

s 3 [ T

]

Figure 4.12: Representative curve using three (higher than 0) consecutive samples of the ideal Γ4

pulse. T is the sampling period.

0 0.2 0.4 0.6 0.8 1−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

shift [ T ]

ampl

itude

err

or [

% ]

Figure 4.13: Amplitude error (percentage of the maximum value) as a function of the shift (from0 to the value of the sampling period T ). The procedure has been applied to the first 3 sampleswith amplitude higher then 0, no segmentation has been considered.

93


For the k value of the shift we have the p-dimensional vector of the samplessk = (sk

1, · · · ,skp). We fit in the optimal way one plane solving the (over-determined)

system:

p∑

i=1

αiski = v (4.4)

where k = 0, · · · ,N − 1 and we compute the error in each point. If the error insome points exceeds the maximum value, we divide the [0, T ] interval and resumethe procedure. The implemented algorithm performs subdivisions by a factor 2,therefore the period can be divided only in 1, 2 and 4 segments. The value 8 is notallowed since, even in the case of 3 first samples, we would need 3 · 8 = 24 equationsto calculate the coefficients. However, being the estimated period made of 20 sub-samples, we can have only 20 equations and the system would be under-determined.In Figs. 4.14 and 4.15 the result of this procedure is shown when a maximum errorof 0.05% is imposed: 4 planes (segments) are necessary.

Another possibility to reduce the maximum error is to increase the number ofsamples on which we apply the method. In Fig. 4.16 there have been used the first5 samples (whose amplitude was higher than 0 a.u.) and, to reach the maximumerror target (0.10 %, in this case), no segmentation was requested. Using more than3 samples Eqs. 4.3 and 4.4 describe hyperplanes.

For computing the shift we must first normalize the samples (to compute theamplitude). Given the normalized amplitude, the equations to evaluate the shiftdot product coefficients are:

p∑

i=1

βiski = t0 + kτ (4.5)

where k = 0, · · · ,N − 1. Again we get an over-determined system, which will bebroken into sub-systems to reduce the error. Fig. 4.17 shows an example of thevariation of this error with the shift value.

An example of treatment of a 2-dimensional pulse with this procedure can befound in Appendix E.

94


9 10 11 12 13 14 15 16 17 180

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time [a.u.]

ampl

itude

[a.u

.]

s2

s1

s3

Figure 4.14: Example of sampling period subdivision: on the first 3 samples we have 4 segmen-tations. The first is in green colour, the second in cian, the third in magenta and the fourth inyellow.

0 0.2 0.4 0.6 0.8 1−0.015

−0.01

−0.005

0

0.005

0.01

0.015

shift [ T ]

ampl

itude

err

or [

% ]

Figure 4.15: Amplitude error (percentage of the maximum value) when 4 planes (segments) areused on the first 3 samples. The red asterisk indicates the maximum error: 0.01 %, target 0.10 %.

95


0 0.2 0.4 0.6 0.8 1−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

shift [ T ]

ampl

itude

err

or [

% ]

Figure 4.16: Amplitude error (percentage of the maximum value) using the first 5 samples withno segmentation. The red asterisk indicates the maximum error: 0.05 %, target 0.10 %.

0 0.2 0.4 0.6 0.8 1−0.06

−0.04

−0.02

0

0.02

0.04

0.06

shift [ T ]

ampl

itude

err

or [

% ]

Figure 4.17: Shift error (percentage of the maximum value) using the first 4 samples with 2hyperplanes (segments). The red asterisk indicates the maximum error: 0.05 %, the target was0.10 %.

96


4.3.2 Simulation results: ideal case + noise

The results presented until now can be considered ideal since they were in absenceof noise. Before starting to analyse real ALICE TPC data, we can corrupt our idealΓ4 signal (Fig. 4.10, Eq. 4.2) and evaluate the performance of the same filteringprocedure.

A set of 1000 extractions of a White Gaussian Noise (WGN) process has beenadded to this desired analog reference. The mean value µ has been set equal to 0 a.u.and the Root Mean Square (RMS) σ to 2−10 a.u., by assuming a noise amplitude of1 Least Significant Bit (LSB) of the TPC 10-bit ADC [31]. In this way, 1000 noisyanalog signals have been built to study the distribution of the original amplitudeestimation applying the Buzuloiu’s procedure.

Also for the examples considered in this Section, the sampling period T is 1 a.u.and every digital signal has 50 digital samples (with comprehensive time durationof 50 a.u.). Furthermore, being the analog signals made by 1000 samples, each timea.u. is made by 20 analog samples (or sub-samples). Therefore T is made by 20sub-samples.

Another consideration concerns the ADC clock, it is not synchronous with thetime of arrival of every detected particle nor with its peak, thus in our simulationwe have introduced a random shift (or position of the first DGT sample) amongall the simulated DGT signals. This random shift has been considered uniformlydistributed in the sampling period T, and, for every analog signal (to be sampled),it is different and constant. As T is equal to 20 analog sub-samples, the randomshift can assume only 20 different values (from 0 to 19). In the two examples shownin Fig. 4.18 we can see the effect of this random shift. The relative difference of 8sub-samples corresponds to 0.4 T.

Our purpose is the estimation of this shift in order to build a reference signal.On this ideal re-built signal we have to apply the procedure for the coefficientscalculation, then we can store them. This represents what we will do with real data.Once a reference is at disposal, the coefficients can be evaluated and then stored inFPGA Look-Up Tables (LUTs) to be read in real time for the filtering procedure.This elaboration aims to the estimation of the amplitude values. Therefore, theforeseen on-line processing for the amplitude estimation will only consist of a set ofsums and multiplications among few real samples (3-5) and the coefficients saved inLUTs from a previous training DAQ session.

Finally, to take into account the zero-suppression procedure described in Sect.4.2, we have set a threshold of 5σ (0.005 a.u.) as a data reduction technique (seeFig. 4.19). In this way, every simulated noisy signal has samples higher than zeroonly in the peak region.

97


8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

time [a.u.]

ampl

itude

[a.u

.]

Noisy signal # 157, random shift: 11 sub−samples

analog noisyDGT noisy

8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

time [a.u.]

ampl

itude

[a.u

.]

Noisy signal # 352, random shift: 19 sub−samples

analog noisyDGT noisy

Figure 4.18: Example of random shift on 2 different noisy built signals over an ensemble of 1000waveforms. It is expressed in terms of sub-samples. The relative difference is of 0.4 T.

8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

time [a.u.]

ampl

itude

[a.u

.]

analog noisyDGT noisyDGT noisy ZS

15.5 16 16.5 17 17.5 18−0,005

0

0,005

0,010

0,015

0,020

0,025

0,030

time [a.u.]

ampl

itude

[a.u

.]

threshold

Figure 4.19: Example of the zero-suppression procedure. All noisy samples under threshold areset to 0 a.u. The inset shows a detail of the suppression.

98


Now we are ready to describe the way in which the reference signal (necessaryfor the coefficients calculation) has been built. For every simulated zero-suppressednoisy signal, the Center of Mass (CM) has been estimated as:

CM =

nf∑

i=1

tisi (4.6)

where i runs from the first sample higher than zero (not zero-suppressed) tothe number of first nf considered samples (3-5); ti is the time value correspondingto the sample si. CM gives an estimate of the analog peak time: it is importantsince it represents the information needed to realign all noisy signals. With thisrealignment the ADC jitter can be highly reduced and signals can be averagedto return the reference signal. It is worth noting that the granularity along thesimulated time axis is of one twentieth of sample period T ; thus by realigning noisysignals we collect several noisy samples for each fraction of period and the averagegives a mean value for every sub-sample time-slot. To be more precise, since we haveconsidered a time jitter uniformly distributed over the period (T = 20 sub-samples),having 1000 noisy DGT signals, on average we have 50 noisy amplitude values thatfill every sub-sample time-slot on which the mean is evaluated. Furthermore, sincethe simulated noise process was a WGN one, the average converges to the meanvalue µ = 0 a.u. (of added corruption).

Fig. 4.20 on the left side shows the difference between the estimated CMs (peaktimes) on noisy signals and the known desired ones. The distribution of these valuesis presented in the right side of the same figure. The figure represents the error thatthis method can introduce in the realignment and is expressed in sub-samples.

This procedure introduces a negligible bias between desired and estimated peaktimes (no more than 1 sub-sample, 0.05 T ), therefore we can proceed averaging allthe realigned signals, the resulting plot is presented in Fig. 4.21.

99


Figure 4.20: Right: difference between desired and estimated peak time. The maximum error isof one sub-sample (one twentieth of T ). Left: distribution of the aforementioned difference.

9 10 11 12 13 14 15 16 17 180

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time [a.u.]

ampl

itude

[a.u

.]

analog re−built sgn

Γ4 ideal curve

Figure 4.21: Analog re-built signal and superimposed ideal Γ4. Example of a good match accordingto a Pearson’s chi-square test with confidence level of 5%.

100


By sampling the analog re-built waveform starting from the first value abovethreshold, the DGT reference for the filtering algorithm can be built (see Fig. 4.22).The coefficients have been calculated with the procedure reported in the previous

9 10 11 12 13 14 15 16 17 180

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time [a.u.]

ampl

itude

[a.u

.]

analog re−built sgnDGT re−built sgn

Figure 4.22: Analog re-built signal and DGT reference for the filtering procedure.

Section and are presented in Tabs. 4.1,4.2. The number of the first consideredsamples run from 3 to 5 and the period subdivisions were 1, 2 and 4.

The processing of the aforementioned sample of 1000 noisy simulated DGT sig-nals has permitted to calculate all sets of coefficients presented in Tabs. 4.1 and 4.2.To evaluate the best performance, the mean value of the absolute amplitude errorestimates has been calculated, see Tab. 4.3.

Absolute values have been taken since the amplitude error can be negative. Theworst and the best results are plotted in Figs. 4.23 and 4.24 respectively.

It is worth to note that the best result is obtained by using the first 4 samples with4 period subdivisions and not with 5 samples, even in the case of 4 segmentations.The reason is that a bit of over-determination is needed to give better performances,as mentioned in Sect. 4.3.1.

101


We can conclude this Section by outlining a procedure that can be useful toanalyse real data:

• collect a big number of DGT real signals;

• for every signal evaluate the time-shifts with respect to the analog (unknown)maximum by using the CM procedure;

• realign them and populate the periods of the analog re-built signal;

• apply on it Buzuloiu’s algorithm to estimate the coefficients;

• perform all possible combinations of first-considered-samples and period sub-divisions and calculate the corresponding set of coefficients;

• save these coefficients in a LUT to be used in a real time data processing.

102


3 samples 4 samples

1 segm. 0.4303 0.1922 0.8909 0.4597 0.0944 1.1576 -0.33692 segm. 0.4303 0.1925 0.8902 0.3217 0.4253 0.3082 0.7111

0.4963 0.0888 0.9959 0.5177 -0.0168 1.3280 -0.44964 segm. 0.5459 0.1706 0.9046 0.4854 0.2545 0.7030 0.2426

0.3889 0.2234 0.8656 0.3713 0.2800 0.7068 0.20410.4354 0.1718 0.9140 0.3968 0.3226 0.4707 0.58160.5840 -0.0788 1.1807 0.5346 0.2090 0.2193 1.3439

Table 4.1: Coefficients for the filtering procedure calculated using the reported number of firstsamples (3-4) and period segmentations.

5 samples

1 segm. 0.4604 0.0779 1.2849 -0.8119 0.70452 segm. 0.3174 0.4168 0.4087 0.2980 0.6389

0.5234 -0.0426 1.3953 -0.4840 -0.13274 segm. 0.2188 0.1679 1.4961 -2.5338 3.8895

0.1015 0.9137 -0.6229 0.4786 3.13890.3784 0.3974 0.2772 0.7017 0.32030.3118 1.5471 -3.6309 3.3133 8.5347

Table 4.2: Coefficients for the filtering procedure calculated using the first 5 samples and reportedperiod segmentations.

[ % ] 3 samples 4 samples 5 samples

1 segm. 0.137 0.135 0.1332 segm. 0.092 0.106 0.1284 segm. 0.085 0.078 0.094

Table 4.3: Amplitude mean error estimations expressed in percentage. Best and worst results inbold typeface.

103


100 200 300 400 500 600 700 800 900 1000

−0.4

−0.2

0

0.2

0.4

0.6

0.8

# simulated signal

Am

plitu

de e

rror

[%]

Figure 4.23: Amplitude percent. error between noisy simulated and ideal signals using the first 3coefficients and no period subdivision. Mean abs. error: 0.137 %.

0 200 400 600 800 1000

−0.4

−0.2

0

0.2

0.4

0.6

0.8

# simulated signal

Am

plitu

de e

rror

[%]

Figure 4.24: Amplitude percent. error between noisy simulated and ideal signals using the first 4coefficients and 4 period subdivisions. Mean abs. error: 0.078 %.

104


4.4 TPC real-data analysis

A set of 13 millions real digital signals from ALICE TPC proton-proton first colli-sions at 7 TeV was available in the middle of March 2010, see Fig. 4.25. Therefore,we tried to apply Buzuloiu’s procedure described in the previous Sections to thisdataset. As the algorithm can treat only signals with constant amplitude and shape,only waveforms with comparable amplitude from detector regions near to each other,since the shape depends on the distance from the TPC center. To do so, we focusedour attention on signals with the features reported in Tab. 4.4.

Figure 4.25: Reconstructed trajectories of particles from first proton-proton collisions at 7 TeVdetected by ALICE TPC.

A sub-set of 500 signals satisfied these strong constraints. However, by using theCM technique to realign the waveforms we can see in Fig. 4.26 that the numberof samples that populate the same time-slot is not uniform over a 20 sub-samplesperiod subdivision. As the TPC sampling frequency fS is 10 MHz, the samplingperiod T is 100 ns. Subdividing T into 20 time-slots makes each sub-sample equalto 5 ns. For clarity, in the following Figures, we preferred expressing the time axisin sub-samples (and not in seconds).

On account of the non-uniform population, the reference signal obtained byaveraging all values in a time slot (presented in Fig. 4.27) is clearly wrong anduseless.

105


Feature Range Units Global Range

Amplitude 40-60 ADC counts [0-1000] −→ 50 ± 1% ADC countsSector 0-35 [0-71] −→ IROC (∗)

Row 42-52 [0-95] −→ 47 ± 5%Pad 63-77 [0-140] −→ 70 ± 5%Start Time 0-25 µs [0-100] −→ 25 %Bunch Length 8-10 samples [3-324] −→ 9 ± 1 samples

Table 4.4: Range of max. amplitude, TPC region, start time and length values used to selectsignals candidate for the filtering procedure. (∗) Inner Read-Out Chamber, see [31].

0 50 100 150 2000

5

10

15

20

25

30

time slot [sub−samples]

occu

rren

cies

[# s

igna

ls]

Figure 4.26: Not uniform time-slot filling. Period of 20 sub-samples, 1 sub-sample is equal to 5 ns.Result obtained with the CM technique.

By changing the period subdivision to only 10 time-slots, each one of 10 ns,the distribution remains not uniform (see Fig. 4.28) and the reference signal againwithout any possible application (Fig. 4.29). We can conclude that the CM tech-nique is not sufficiently sensitive for a realignment with the granularity requiredby the Buzuloiu’s procedure since we need at least 10-20 sub-samples to write anover-determined system of equation to calculate Buzuloiu’s coefficients, as discussedin Sect. 4.3.1.

106


0 50 100 150 2000

10

20

30

40

50


ampl

itude

[AD

C c

ount

s]

Figure 4.27: Reference signal obtained averaging all realigned values that fill the same time-slot.Period of 20 sub-samples. Result obtained with the CM technique.

0 20 40 60 80 1000

10

20

30

40

50

60


occu

rren

cies

[# s

igna

ls]

Figure 4.28: Not uniform time-slot filling even in the case of 10 sub-samples period, 1 sub-sampleis equal to 10 ns. Result obtained with the CM technique.

107


0 20 40 60 80 1000

5

10

15

20

25

30

35

40

45

50


ampl

itude

[AD

C c

ount

s]

Figure 4.29: Reference signal obtained averaging all realigned values that fill the same time-slot inthe case of 10 sub-samples period. Result obtained with the CM technique.

Another possibility is to fit every single DGT real signal and use the estimatedpeak time to realign all selected signals with the aim that by averaging on thepopulated time-slots a reference could be built. An example of fitting of ALICETPC noisy signals is shown in Fig. 4.30. By applying this procedure to all selectedsignals, we can try to realign all waveforms by using the estimated peak times τ .By subdividing the interval (τmin ÷ τmax) into 20 segments, we can populate eachone for the averaging. Also in this case the constructed signal cannot be used toapply the Buzuloiu’s procedure, as shown in Fig. 4.31. Slightly better results wereobtained by using a coarser granularity, i.e. by subdividing the time period T into10 slots (see Fig. 4.32), but, again, the curve is not regular enough to extract areference.

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Another approach comes from the following consideration: the selected signalshave roughly the same number of samples (8-10) and come from the same TPCregion (see Tab. 4.4). We could average all selected real signals without any re-alignment and fit the obtained mean DGT waveform. In Fig. 4.33 the result of thischoice is presented and the fitted Γ4 curve can be considered a reference curve onwhich extract the filter coefficients for the Buzuloiu’s procedure. With coefficientsevaluated in this way, real data can be processed. In Fig. 4.34 we can see an exampleof filtered data when a number of first samples equal to 3 and no period subdivisionhas been chosen. The maximum DGT value of each signal (selected in the range40-60 ADC counts) and its filtered value are compared: clearly, there are too manywrong estimations too far from the DGT amplitude range. Furthermore, all analogmaxima should present values at least equal to (or higher than) the DGT maxima:this is not the case, since 15 % of the values are lower than 40 ADC counts.

The situation does not improve by introducing period segmentation nor by work-ing with a higher number of first considered DGT samples, as reported in Fig. 4.35.In this case, we have 25 % of values lower than 40 ADC counts.

However, we cannot conclude that this algorithm does not work since the com-plete treatment is based on the assumption that the reference constructed curveis representative for all real selected signals. To investigate this issue, a study ofresiduals among real pulses and two fitting models has been performed. A simple Γ4

model and a linear combination of a Γ4 and a Gaussian function (LC(Γ4,G)) have

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Figure 4.33: Analog candidate for the Buzuloiu’s procedure obtained fitting averaged real data.Example of a good match according to a Pearson’s chi-square test with confidence level of 5%.

been compared. For each real signal the residuals on the first 5 DGT samples forboth fitting models have been estimated, see Fig. 4.36.

In Tab. 4.5 are reported the mean value and the RMS of the residuals on thefirst 5 DGT samples for both the fitting models, obtained by averaging the resultscoming from the complete dataset. Since the LC fitting method minimizes both themean value and RMS of the residuals, we can conclude that a possible explanationfor the failure of Buzuloiu’s procedure on the considered dataset could be the choiceof the Γ4 model for the reference signal estimation.

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Figure 4.34: Real TPC filtered data with Buzuloiu’s procedure working on the first 3 samplesand no period segmentation. Red dots represent the maximum DGT amplitude value, the blueasterisks the analog maximum estimates.

Figure 4.35: Real TPC filtered data with Buzuloiu’s procedure working on the first 4 samplesand 2 period segmentations. Red dots represent the maximum DGT amplitude values, the blueasterisks the analog maximum estimates.

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Figure 4.36: Example of a comparison between two different fitting models: a Γ4 and a LC(Γ4,G).There are reported the results of the χ2 test together with the residual estimation on the thirdsample for one fitting model.

sample Γ4 [ADC counts] LC(Γ4,G) [ADC counts]

s1 -0.02 ± 0.08 0.03 ± 0.03s2 0.06 ± 0.07 -0.03 ± 0.03s3 -0.02 ± 0.11 0.02 ± 0.03s4 -0.04 ± 0.07 -0.02 ± 0.03s5 0.03 ± 0.06 -0.02 ± 0.02

Table 4.5: Mean values and RMS of the residuals estimated on the first 5 samples with a Γ4 anda LC(Γ4,G) models.

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4.5 Conclusions

The aim of this work was the study and possibly the validation of Buzuloiu’s proce-dure to estimate, in real time, amplitude and time information of real ALICE TPCsignals. As a first approach, in March 2010 this algorithm has been written to treata set of 1000 simulated signals with a Γ4 shape of constant amplitude and widthwith uniformly distributed phase shift. Each ideal signal was corrupted by one of1000 noisy signals extracted from a WGN process.

In our simulation it was possible to find the analog re-built curve with the CMmethod and to estimate Buzuloiu’s coefficients. The results were promising, since,after filtering, the error on the analog maximum estimated amplitude was really low,less than 0.08 % in the best combination of period subdivisions and first consideredsamples.

At the end of March 2010 a set of 13 millions real DGT signals from the ALICETPC was available for testing Buzuloiu’s procedure previously simulated. Becausethe algorithm can be applied only to signals with constant amplitude and shape, only500 waveforms out of the whole set were selected, from adjacent detector regionsbecause their shape and amplitude were similar (but not equal). However, the CMmethod was not efficient for building an analog reference, since the estimated timeshift was too frequently wrong. Without good time-shift estimates the sub-set ofsignals cannot be realigned to build the analog reference and evaluate the coefficientsfor the filtering session. As an alternative approach to evaluate the time-shift, ananalytical Γ4 fitting has been applied on every selected DGT signal, but also in thiscase it was not possible to obtain the analog re-built curve.

Finally, an average of all selected 500 signals has been proposed, since the re-sulting waveform was very well fitted by the theoretical Γ4 curve and the χ2 testconfirmed this assumption. The coefficients were calculated, but the estimated max-imum amplitude values obtained with the filtering were clearly wrong in too manycases.

In May 2010 the work continued focussing on the distributions of the residuals ofthe first 5 DGT samples around the peak region to understand if a reference signalcould be found and, in the case, if it could be really representative of the completesub-set. As confirmed by the study of the residuals on the two proposed fittingmodels, the Γ4 one lacks accuracy to represent the entire dataset. Since the LCmodel fits better the dataset, a possible future work could be the evaluation of thealgorithm with this new reference shape. Only after this study Buzuloiu’s procedurecould be definitively discarded.

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Chapter 5

Digital processing of OCT cornealimages

5.1 Introduction

Optical coherence tomography (OCT) is an optical signal acquisition and process-ing method [35]. It captures micrometer-resolution, three-dimensional (3D) imagesfrom within optical scattering media. OCT is an interferometric technique typicallyemploying near-infrared light. The use of relatively long wavelength light allowsit to penetrate into the scattering medium. Confocal microscopy, another similartechnique, typically penetrates less deeply into the sample.

OCT bases itself upon low coherence interferometry [35] - [38]. In conventionalinterferometry with long coherence length (laser interferometry), interference of lightoccurs over a distance of meters. In OCT, this interference is shortened to a distanceof micrometers, thanks to the use of broadband light sources (sources that canemit light over a broad range of frequencies). Light with broad bandwidths canbe generated by using superluminescent diodes (superbright LEDs) or lasers withextremely short pulses (femtosecond lasers). White light is also a broadband sourcewith lower powers.

Light in an OCT system is broken into two arms (see Fig. 5.1): a samplearm (containing the item of interest) and a reference arm (usually a mirror). Thecombination of reflected light from the sample arm and reference light from thereference arm gives rise to an interference pattern, but only if light from both armshave travelled the same optical distance (same meaning a difference of less than acoherence length). By scanning the mirror in the reference arm, a reflectivity profileof the sample can be obtained (this is the so-called time domain (TD) OCT). Areasof the sample that reflect back a lot of light will create greater interference than areasthat do not. Any light that is outside the short coherence length will not interfere.

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5 – Digital processing of OCT corneal images

Figure 5.1: Typical optical setup of single point OCT.

This reflectivity profile, called an A-scan, contains information about the spatialdimensions and location of structures within the item of interest. A cross-sectionaltomograph (B-scan) may be achieved by laterally combining a series of these axialdepth scans (A-scan).

A relatively recent implementation of optical coherence tomography, frequency-domain (FD) OCT, provides advantages in SNR, permitting faster signal acquisi-tion. Commercially available optical coherence tomography systems are employedin diverse applications, including art conservation, to analyse different layers in apainting (Fig. 5.2), and diagnostic medicine (Fig. 5.3), notably in ophthalmology(Fig. 5.4) where it can be used to obtain detailed images from within the retina(Fig. 5.5).

A first two-dimensional in vivo depiction of a human eye fundus along a horizon-tal meridian based on white light interferometric depth scans has been presented atthe ICO-15 SAT conference in 1990 [39]. Further developed 1990 by N. Tanno [40]and in particular since 1991 by Huang et al. [41], OCT with micrometer resolutionand cross-sectional imaging capabilities has become a prominent biomedical tissue-imaging technique. First in vivo OCT images, displaying retinal structures, werepublished in 1993 [42], [43]. OCT has critical advantages over other medical imaging

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Figure 5.2: Example of OCT application to art conservation.

Figure 5.3: Example of OCT application to study of biological tissues: a fingertip.

systems. Medical ultra-sonography, magnetic resonance imaging (MRI) and confo-cal microscopy are not suited to morphological tissue imaging: the first two havepoor resolution; the last lacks millimeter penetration depth [44], [45].

OCT is a technique for obtaining subsurface images of translucent or opaque ma-terials at a resolution equivalent to a low-power microscope. It is attracting interestamong the medical community, because it provides tissue morphology imagery atmuch higher resolution (better than 10 µm) than other imaging modalities such asMRI or ultrasound.

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Figure 5.4: Image of human eye anatomy. Figure 5.5: Retina: 3D OCT detail of opticnerve connection over retinal internal plane.

The key benefits of OCT are:

• live subsurface images at near-microscopic resolution;

• instant, direct imaging of tissue morphology;

• no preparation of the sample or subject;

• no ionizing radiation.

OCT delivers high resolution because it is based on light, rather than sound orradio frequency. An optical beam is directed at the tissue, and a small fraction ofthis light that reflects from subsurface features is collected. Note that most light isnot reflected but, rather, scatters. The scattered light has lost its original directionand does not contribute to forming an image but rather contributes to glare. Theglare of scattered light causes optically scattering materials (i.e. biological tissue,candle wax, or certain plastics) to appear opaque or translucent even while they donot strongly absorb light. Using the OCT technique, scattered light can be filteredout, completely removing the glare. Even the very tiny proportion of reflected lightthat is not scattered can then be detected and used to form the image.

The physics principle allowing the filtering of scattered light is optical coherence.Only the reflected (non-scattered) light is coherent (i.e. retains the optical phasethat causes light rays to propagate in one or another direction). In the OCT instru-ment, an optical interferometer is used in such a manner as to detect only coherentlight. Essentially, the interferometer strips off scattered light from the reflected light

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needed to generate an image. In the process depth and intensity of light reflectedfrom a subsurface feature is obtained. A 3D image can be built up by scanning, asin a sonar or radar system.

Within the range of non-invasive 3D imaging techniques that have been intro-duced to the medical research community, OCT as an echo technique is similar toultrasound imaging. Other medical imaging techniques such as computerized axialtomography, magnetic resonance imaging, or positron emission tomography do notutilize the echo-location principle.

The technique is limited to imaging 1 to 2 mm below the surface in biologicaltissue, because at greater depths the proportion of light that escapes without scat-tering is too small to be detected. No special preparation of a biological specimen isrequired, and images can be obtained non-contact or through a transparent windowor membrane. It is also important to note that the laser output from the instru-ments is low (eye-safe near-infra-red light is used) and no damage to the sample istherefore likely.

In the following paragraphs an application of DSP to OCT corneal images willbe presented, see Fig. 5.6.

Figure 5.6: Cornea: 3D OCT acquisition.

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5.2 Experimental setup

We had available a FD-OCT RTVue-100 Optovue device [46], with a laser diodeworking at a wavelength of 870 nm. Each tomogram is the average of 16 images,with 26000 scans per second. The OCT was connected to a pc to visualize and storecorneal images. In a second time, the acquired tomograms have been processed toextract the features of interest.

The average time duration per patient of the medical analysis was 10 min, whilethe DSP processing of few seconds, see Fig. 5.7.

Figure 5.7: Example of OCT corneal medical examination, courtesy of Dr. M. Frisani.

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5.3 Our purpose

A sample of 52 records form healthy patients without any eye disorder has beenanalysed with a FD-OCT system described in the previous Section. For every pa-tient, three images of each eye have been recorded on a corneal area of 6 mm x 4mm (1016 x 640 pixel), for a total number of more than 300 images (Fig. 5.8).

Figure 5.8: OCT corneal tomogram and our region of interest (ROI) on the corneal axis.

On the complete set of images our purpose was to estimate the thickness ofcorneal inner layers, whose subdivision is known from histological evaluations [47],[48] (see Fig. 5.9).

5.4 Our approach

A customized Matlab program has been developed to automatically process allthe images and segment the inner corneal layers. Since each image is available ingray-scale format and is uploaded as a 640 x 1016 matrix, every pixel intensity canbe represented in the third dimension, as reported in Fig. 5.10. This informationcan be considered as the amplitude of the signals we are going to construct.

On the apex of every investigated meridian (image), a ROI has been chosen aspresented in Figs. 5.8, 5.9. Particular attention has been paid to the dimensionof the ROI, in order not to consider on the same row different pixels coming fromdifferent layers. With respect to Fig. 5.11, in each tomogram, the Epithelium pixelshave been interpolated to find the maximum angle (referred to the corneal curvaturecenter and defined as α in the figure) after which the same layer would have beenrepresented on a different, close and lower pixel row. This procedure returned values

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Figure 5.9: Comparison between histological reference representation of corneal layers (left) andour OCT ROI (right) to be processed.

Figure 5.10: 3D representation of one single gray-scale tomogram, the pixel intensity is used asthird dimension and ranges form 0 to 255.

of ROI maximum width equal to 85-90 columns, two lagMAX in the figure. Thisrange corresponds to 8.4-8.9 % of the image total width (1016 pixel columns).

In particular, for every acquisition, the ROI area has been divided in three slicesone next to each other (Fig. 5.12). We expect the three slices, being adjacent, notto be affected by strong differences of thickness. Each slice is composed of the samenumber of pixel columns (20-30).

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Figure 5.11: ROI width calculation: each pixel row must contain the same corneal layer to producea consistent treatment. lagMAX represents the half of the maximum number of columns that canfill the ROI.

Figure 5.12: Detailed example of ROI subdivision in three slices one next to each other.

The following procedure is applied to each slice. Referring to the histologicalmodel (Fig. 5.9), corneal layers can be identified from the different reflectivity ofanatomical boundaries. Suppose that the slice is composed of one (central) pixelcolumn. Different reflectivity means different pixel intensity (between 0 and 255 in

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the gray scale). If the pixel intensity profile is plotted as a function of pixel rows, thesearch for minima and maxima corresponds to localizing the beginning of corneallayers. In order to reduce noise (speckle, flicker and so on), the pixel intensity profileis indeed averaged on the number of (20-30) columns composing the slice (for theimprovement using the averaging technique, see Appendix C).

In Figs. 5.13 and 5.14, the results of this procedure are shown. Epithelium,Bowman’s membrane and Stroma layers are easily identified. The global maximumcorresponds to the beginning of Epithelium; the global minimum is the Epitheliumend; from the next pixel the Bowman layer starts and it ends at the following(second) global minimum; Stroma starts hereafter and ends at the last maximumon the right side of the pixel intensity profile; Endothelium starts after the lastmaximum and ends when the signal goes under two RMS of intensity values comingfrom the rightern region outside cornea (noise). Descemet ’s membrane is too thinto be detected with one pixel width.

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The procedure has been carried out on three slices (Fig. 5.12) to evaluate theuncertainties on the layer thickness estimations by means of weighted averages. Itis worth noting that using the pixel automated procedure, the original image isnot altered. As a further step, a subpixeling technique has been applied to reduce

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Figure 5.14: Detail of the pixel analysis. In the box, thickness estimates are reported both in pixeland subpixel scale.

the uncertainty in thickness estimations when it was expressed in pixel scale. Thechosen linear ratio between pixel and subpixel has been set one to eight. As a result,the subpixel intensity profile is smoothed if compared with the pixel averaged one(Fig. 5.13). The complete set of images has been processed with this customizedalgorithm. A calibration procedure has been applied to convert thickness valuesfrom pixel (and subpixel) to micron. By examining OCT images of contact lenswith known thickness, the conversion factor pixel-µm has been found: 1 pix = 4.13µm, 1 subpix = 0.52 µm. The resulting thickness of all layers and of the completecornea are reported in Tab. 5.1, no significant difference occurred in the comparisonbetween right and left eyes except for the Stroma layer, where the discrepancycorresponds to two pixel (1.8 % of the layer thickness).

layer left eye [µm] right eye [µm]

Epithelium 42.3 ± 0.5 42.8 ± 0.5Bowman 17.0 ± 0.5 17.0 ± 0.5Stroma 445.0 ± 0.5 453.0 ± 0.5

Endothelium 9.8 ± 0.5 9.8 ± 0.5Global cornea 514 ± 1 523 ± 1

Table 5.1: Thickness estimation of the four main corneal layers obtained averaging the evaluationsperformed on more than 150 images per eye.

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5.5 OCT images and SNR

An interesting feature in the OCT image processing is the evaluation of the SNRimprovement obtained when using the averaging technique (see Appendix C). Ifwe consider a ROI 90 pixel wide (maximum dimension estimated in the previousSection) the pixel intensity distribution is Gaussian, as reported in Fig. 5.15 for apixel row in the Stromal region of a tomogram.

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Figure 5.15: Pixel intensity Gaussian distribution in the Stromal region of a pixel row with mediumpoint centred on the corneal axis and 90 pixel wide, being lagMAX equal to 45 pix. This result isin according to a Pearson’s chi-square test with confidence level of 5%.

Also in this case of real noisy signals, we can define the SNR as in Eq. 3.3 ofSect. 3.3.2 :

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(4σ)2(5.1)

where A is the maximum pixel intensity (peak value) and σ is the RMS of the noise,the factor 4 implies a 95% noise band. An example of SNR evaluation has beenpresented in Fig. 5.16, where the noise has been investigated in the Stromal region,since it is the thickest among all corneal layers with average estimated thickness of100 pixels.

The pixel intensities inside the Stroma layer are Gaussian distributed, as reportedin Fig. 5.17 for the central column (corneal axis) and in Fig. 5.18 for the averaged

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Figure 5.16: Pictorial representation of SNR evaluation on the central column (green) of an OCTtomogram. Dashed purple lines show the extremal values of amplitude (A) and 95% noise band(4σ). The blue solid line is the linear trend. The noise has been considered in the only Stromalregion.

signal. In this last figure, the RMS reduction obtained by the averaging procedureis appreciable. It is worth mentioning that we decided to subtract the linear trendin the Stromal region to perform a more accurate RMS estimation.

By evaluating the SNR on the subset of images coming from the III acquisition ofthe right eye for all 52 patients, we can compare the SNR distributions of the centralpixel column (Fig. 5.19) and of the averaged signal (Fig. 5.20). For the mean valuesof these two distributions, the improvement obtained with the averaging techniqueis of 5.4 dB. The theoretical improvement can be calculated from the Eq. C.5 withthe number of N noisy signal to be averaged equal to 30 and the lag factor equal to15 pixel. Therefore, in principle we would have found an improvement of 14.8 dB.In practise, we found a lower value since we did not have available some thousandsnoisy signals (columns) on which apply the procedure. The first assumption spelledout in Appendix C is not satisfied. However, this technique allowed us to drasticallyenhance the corneal layer boundaries detection.

Finally, in this Section we presented the result of the SNR evaluation in pixelresolution since no significant difference has been found performing this analysis insubpixel scale.

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Figure 5.17: Pixel intensity Gaussian distribution in the Stromal region of a pixel column centredon the corneal axis. This result is in according to a Pearson’s chi-square test with confidence levelof 5%. There has been subtracted the linear trend to perform the analysis, therefore, negativepixel intensities correspond to values lower than the trend.

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Figure 5.18: Pixel intensity Gaussian distribution in the Stromal region of a pixel averaged column

(lag = 15 pix) centred on the corneal axis. This result is in according to a Pearson’s chi-squaretest with confidence level of 5%. As in Fig. 5.17, negative pixel intensities correspond to valueslower than the trend.

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Figure 5.19: Gaussian distribution of SNR estimations evaluated on the corneal axis pixel column.Sample of 52 patients, right eyes, third acquisition. This result is in according to a Pearson’schi-square test with confidence level of 5%.

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Figure 5.20: Gaussian distribution of SNR estimations evaluated on the averaged pixel column.The lag has been chosen equal to 15 pix, therefore the average has been performed on 30 pixelcolumns centred on the corneal axis. Sample of 52 patients, right eyes, third acquisition. Thisresult is in according to a Pearson’s chi-square test with confidence level of 5%.

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5.6 Treatment of corneal marginal regions

The procedure reported in the previous sections can be repeated at different anglesin a vertical plane containing the corneal axis to estimate the thickness profile ofeach layer (see Fig. 5.21).

Figure 5.21: Example of analysis applied to marginal corneal regions. Blue lines represent thedirections on which layer thicknesses have been estimated. Pink dots are positioned on layerboundaries per each considered direction. The inset presents a detail of this procedure.

With respect to the estimated center of curvature, a range of ± 20 degrees aroundthe corneal axis has been investigated. In this area, the procedure has been appliedon 21 directions separated by 2 degrees.

Fig. 5.22 shows a case report of a patient treated for one month with orthoker-atology [49], [50]. This is a treatment with contact lens that modifies the cornea’sshape hence changing its refraction properties. Thickness profiles of Epithelium,Bowman, Stroma and the global cornea before and after the treatment are compared.The most significant change can be observed in the Epithelium layer.

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Figure 5.22: Detailed representation of thickness profile estimation on a patient before (red) andafter (blue) an orthokeratology treatment, courtesy of Dr. M. Frisani.

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5.7 Conclusions

A customized procedure has been applied to segment corneal layers on more than300 FD-OCT corneal images. Initially it has been applied to estimate layer thicknesson a sample of 52 healthy patients without any eye disorder. From this analysis anaverage value for all layers has been provided and no significant difference betweenright and left eyes occurred.

In addition, an averaging technique has been introduced (see Appendix C) to con-struct reference signals to be used for the thickness estimations. The improvementin SNR has been introduced and discussed.

In a second time, the procedure has been improved to be applied to cornealmarginal regions. This second version of the algorithm allowed the study of med-ical cases to quantify the effective change in the corneal layers produced by theophthalmological treatment.

As future work we are considering the processing of several tomograms of thesame eye to build a 2D layer surface (a simulation of this reconstruction can be foundin Appendix F). Finally, further development is in progress for studying contact lensthickness estimations.

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Chapter 6

Final conclusions

Several examples of the application of DSP to problems arising in physics, rangingfrom nuclear particle to biomedical applications have been provided in this work ofthesis.

In Chapter 2, for the processing of signals coming from the PANDA experiment,two classes of digital filters have been studied. In a first time, the Butterworthand the adaptive LMS filters have been compared by simulation on Matlab andSimulink platforms. This study was conduced to extract the signal features forfurther on-line processing. In the new data acquisition system for nuclear physicsexperiments, the selection of the accepted events is based on a sophisticated softwaresystem working on pre-processed data and no more performed by hardware triggers.This is the field of application where our work can be applied.

In our simulations, the requirements on PD and SNR matched better using theLMS filter, thanks to the capability to adapt the parameters to the typically non-stationary environment of the nuclear physics detectors. Our results showed that afast settlement of the coefficients can be reached with an algorithm of medium-levelcomplexity, with a processing time scaling linearly with the filter order.

In a second time, we focussed on the FPGA implementation of these standardand adaptive digital filters to apply these algorithms to real data processing. Adirect VHDL translation of the filtering structures previously simulated led to ahigh FPGA consumption and a low maximum working frequency. We optimizedthe implementations to enhance the working frequency in order to cope with theforeseen high rate data acquisition of nuclear detectors.

The standard IIR Butterworth III order LP filter introduced no significant varia-tions in terms of PD reduction and SNR enhancement in the FPGA-oriented with re-spect to Matlab-simulated implementations. However, the adaptive LMS with ournew structure (LMS2) presented slightly lower performances but a higher maximumworking frequency than the simulated implementation (LMS1).

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6 – Final conclusions

Finally, to give a statistical validation of our work, 1000 noisy simulated sig-nals have been processed both with Matlab and FPGA-oriented filter structures,and the results confirmed that the requirements on SNR improvement and PDminimization have been obtained with the LMS adaptive filter.

We can conclude that the requirements on PD and SNR are matched better bothin simulation and in FPGA implementation by the LMS filter. A future work is thestudy and implementation of other adaptive filter structures, as the sign-LMS (seeAppendix A), and the processing of real signals to test the filters here presented.

In TES pulse analysis, reported in Chapter 3, an off-line processing in both timeand frequency domains has been performed. In this case, the features of interest ona complete dataset of waveforms were: amplitude distribution, Energy Resolution,its uncertainty, reference signals construction, energy evaluation and linear trendbehaviour of TES working point. And on a single pulse: amplitude estimation,Time Jitter, SNR. These TES acquisitions were triggered, with known wavelengthand the signal processing has been necessary to significantly enhance these importantfeatures. The filtering techniques we used to elaborate these signals were a Savitzky-Golay and an Optimum Wiener filters.

In all the analysed datasets of noisy signals Wiener filter worked better thanSG both in the SNR enhancement, and in ∆E improvement making negligible thesuperposition between two consecutive peaks in the amplitude histograms. NeitherSG nor Wiener filters were able to reduce Time Jitter. Furthermore, Wiener allowedin all our processed cases a gain close to a factor 2 in ∆E evaluations.

A future development is the introduction of an ADC in the DAQ chain, possiblysupported by an FPGA, to increase the data resolution, reduce the time acquisitionand perform the processing here presented directly on-line. In this way, only thefeature values will be saved and the storage of all waveforms avoided, highly reducingthe dimension of the memories dedicated to data storage.

The processing of ALICE TPC real data, discussed in Chapter 4, permitted thestudy and the development of an algorithm to search invariants inside an ensemble ofDGT signals coming from a nuclear particle detector. This algorithm was introducedby Dr. Buzuloiu [32] to estimate amplitude and time information from DGT signalacquisitions.

In our case, the features of interest were the analog maximum amplitude esti-mation and the time shift evaluation of DGT signals coming from ALICE TPC.As a first approach, this procedure has been applied to set of 1000 noisy simulatedsignals. The reference signals were Γ4 curves with constant amplitude and uniformlydistributed phase shift. Each ideal signal was corrupted by one of 1000 noisy sig-nals extracted from a WGN process. Having simulated the same Γ4 shape, it waspossible to find the analog re-built curve with the CM method and to estimate theBuzuloiu’s coefficients. The results were promising, since, after the filtering, the

134


error on the analog maximum estimated amplitude was really low, being less than0.08 % in the best combination of period subdivisions and first considered samples.

Afterwards, this procedure has been chosen for the processing of real data comingfrom the first proton-proton collisions at an energy of 7 TeV from ALICE TPC.An amount of 13 millions real DGT signals has been processed. The Buzuloiu’salgorithm can be applied only on signals with constant amplitude and shape. Sincethe signal shape highly depends on the detector regions in which the signals areproduced, inside the aforementioned ensemble only 500 waveforms were selected.

Unfortunately, it has not been possible to construct a reference signal signifi-cantly representative of all the subset of analysed pulses. For this reason the Buzu-loiu’s algorithm failed in the amplitude estimation and the CM technique in thetime shift evaluation.

Another approach to evaluate the time-shift was represented by an analyticalΓ4 fit to be applied on every selected DGT signal, but also in this case it was notpossible to obtain the analog re-built curve.

The work continued analysing two proposed fitting models: the Γ4 and the LCof a Γ4 and a Gaussian functions. We focussed on the distributions of the residualsof the first 5 DGT samples around the peak region to understand if a referencesignal could be found and, in the case, if it could be really representative of thecomplete sub-set. The study of the residuals confirmed that the Γ4 model was toolow accurate to represent all dataset. Since the LC model fits better the dataset, apossible future work could be the evaluation of the algorithm with this new referenceshape.

However, before definitively discarding this procedure, we have to consider amore detailed study on a theoretical model that bestfits real data. It is currentlyunder investigation. A future work will be an improvement of the same algorithmwith a more representative reference signal obtained with the new fitting model.

The last example of DSP application, reported in Chapter 5, has been providedby the OCT corneal images elaboration. A customized procedure has been writtento estimate corneal layer thicknesses on its axial region for a sample of 52 healthypatients without any eye disorder. More than 300 FD-OCT corneal images have beenprocessed and the results have been reported. In addition, an averaging techniquehas been introduced (see Appendix C) to construct reference signals to be usedfor the thickness estimations. The improvement in SNR has been evaluated anddiscussed.

From this analysis an average value for all layers has been provided and no sig-nificant difference between right and left eyes occurred. This algorithm has beenenhanced to be applied on corneal marginal regions to perform a layer profile esti-mation. Also in this case the results were promising and the procedure has beenused to study and quantify the effective change in the corneal layers in a case reportof a patient treated with ophthalmological therapies.

135


A possible development of this analysis is the estimation of the layer thicknessin central and marginal regions of different tomograms of the same analysed eye toconstruct a 2D layer surface (as reported in the simulation presented in AppendixF).

In the four cases of analysis reported here, the DGT processing of data allowedextraction of the requested features of interest (in ALICE experiment, it suggesteda more detailed investigation on the fitting model). This kind of processing can bealways made off-line, our purpose is its VHDL translation for FPGA on-line analysis,in particular for PANDA, TES and ALICE experiments.

The results and the examples of this thesis showed that DSP can be usefullyapplied to a great number of problems arising in physics. Its importance is clearnot only within any data acquisition session, but also inside the analytical methodsused to solve the problems. Since the future of science will be more and moreinterdisciplinary, DSP will play a key role also within physical research.

136

Appendix A

The sign-LMS adaptive filter

A possible adaptive filter able to process data in real time with a computational effi-ciency higher than the LMS filter is the sign-LMS filter [14]. A Simulink schematicsof this adaptive filter is presented in Fig. A.1. The adaptive coefficient subchainsare fed by the product between the estimated error sign and the data sign.

Figure A.1: Sign-LMS filter structure Simulink schematics, the green squares represent the signevaluation of the estimated error e[n] and of the data x[n]. The red circles are the filter input andthe stepsize, while the blue circle is the filter output.

137

A – The sign-LMS adaptive filter

The update coefficient equation 2.11 of the LMS algorithm assumes a simpleform known as the sign-LMS algorithm:

wM(n + 1) = wM(n) + µ · sgne(n) · sgnx∗

M(n − n0) (A.1)

where wM(n + 1) is a new vector of filter coefficients at time n + 1, wM(n) isthe filter coefficients vector at time n, sgne(n) is the sign of the error at time n,x∗

M(n − n0) is the complex conjugate of the measurement at time n − n0, n0 is theintroduced delay, and µ is the stepsize. It is a positive number that affects the rateat which the weight vector wM(n) moves down towards a stable solution. Also forthe weight vector there has been evaluated the sign.

The sign function can be defined as:

sgnf(n) =

1 if f(n) < 00 if f(n) = 0

−1 if f(n) < 0(A.2)

In this algorithm, the coefficients wn(k) are updated by either adding or sub-tracting a constant µ. Usually, the sign algorithm is slower to converge then theLMS adaptive filter and presents a larger excess mean-square error. On FPGA itallows higher maximum working frequency since in the multiplication there are in-volved the only sign bits of the two factors. If they are represented on N bits each,this operation is reduced of a factor N in time elaboration.

138

Appendix B

The uncertainty on TES EnergyResolution

The uncertainty on the Energy Resolution estimation has been calculated propagat-ing the errors on the ∆E formula (Eq. 3.2) that we can rewrite as:

∆E = Eλ · 2.355σ

µ2 − µ1(B.1)

where Eλ is the energy of one incident photon, the numerator is the FWHM (FW),

while at the denominator we have the distance of the first two amplitude peaksrepresenting signals carrying 2 and 1 photons, respectively.

We present now an example of the evaluation of this uncertainty applied to theII dataset (45000 pulses, λ = 1550 nm) of Wiener filtered signals (see Fig. 3.28,Sect. 3.5). The factor Eλ, in this case equal to 0.8 eV, can be calculated as:

Eλ =c · h

λ · γeV

(B.2)

where c is the speed of light in vacuum (3 · 108 m/s), h is the Planck constant

(6.62 · 10−34 J·s), λ is the laser wavelength (in this example equal to 1550 nm), γeV

is the unit of energy equal to approximately 1.602 · 10−19 J (we want to express theresult in eV).

Substituting Eq. B.2 in Eq. B.3 we obtain:

∆E =c · h

λ · γeV

· FW

µ2 − µ1(B.3)

On this equation we can propagate the error, as we know the uncertainties from

the Experimental Setup (Sect. 3.2) and from the Particle Physics Booklet [51]:

139

B – The uncertainty on TES Energy Resolution

• σc: 0 m/s (by definition)

• σh: 0.3·10−42 J·s

• σγeV: 10−28 J

• σλ: 2·10−8 m

• σFW : 2.355·σfit V

• σµ1: os/

√N1 V

• σµ2: os/

√N2 V

where:

• σfit = 0.46 ·10−4 V, obtained from the fit and specific of this example

• os = 0.7 · 10−3 V, is the oscilloscope sensitivity

• N1 is the number of signals carrying 1 photon (15500 pulses)

• N2 is the number of signals carrying 2 photons (10000 pulses)

Therfore, the (square) uncertainty on ∆E is:

σ2∆E =

(∂∆E

∂c

)2

σ2c +

(∂∆E

∂h

)2

σ2h +

(∂∆E

∂λ

)2

σ2λ +

(∂∆E

∂γeV

)2

σ2γeV

+

+

(∂∆E

∂FW

)2

σ2FW +

(∂∆E

∂µ2

)2

σ2µ2

+

(∂∆E

∂µ1

)2

σ2µ1

=

=

(Eλ · FW

c · (µ2 − µ1)

)2

σ2c +

(Eλ · FW

h · (µ2 − µ1)

)2

σ2h +

(− Eλ · FW

λ · (µ2 − µ1)

)2

σ2λ +

+

(− Eλ · FW

γeV · (µ2 − µ1)

)2

σ2γeV

+

(Eλ

(µ2 − µ1)

)2

σ2FW +

(− Eλ · FW

(µ2 − µ1)2

)2

σ2µ2

+

+

(Eλ · FW

(µ2 − µ1)2

)2

σ2µ1

=

= 0 + 1.4 · 10−20 + 1.1 · 10−5 + 2.6 · 10−20 + 2.5 · 10−5 + 0.7 · 10−14 +

+ 1.1 · 10−14 (B.4)

140

B – The uncertainty on TES Energy Resolution

And finally:

σ∆E = 0.006 eV (B.5)

This value has been rounded to 0.01 eV when reported in Fig. 3.28. As we can see,

the predominant contributions are due to the uncertainty on the laser wavelength λand the σfit estimation.

141

Appendix C

The averaging procedure andSNR improvement

To apply the averaging technique to a complete subset of signals, the followinghypotheses have to be verified:

• the signal is deterministic and it is repeated several times;

• the noise is random, Wide Sense Stationary (WSS) and ergodic (ideally Gaus-sian);

• noise and desired components are uncorrelated.

We describe here the application of this procedure to TES signals as a case study,where we have some thousands of signals carrying the same number of photons (1γin this example) and the noise has been proved to be Gaussian among all selectedwaveforms (see Sect. 3.3.2, Fig. 3.11). However, the noise and the desired com-ponents are slightly correlated, as shown in Fig. C.1 by the overlapping bands.

A noisy signal s(t) can be written as:

s(t) = d(t) + n(t) (C.1)

where d(t) represents the desired component, while n(t) the corrupting one (un-wanted noise). By averaging over an ensemble of N waveforms, the averaged signalhas the form:

s(t) =1

N

N∑

i=1

si(t) =

= d(t) +1

N

N∑

i=1

ni(t) (C.2)

142

C – The averaging procedure and SNR improvement

102

103

104

−90

−80

−70

−60

−50

−40

−30

−20

−10

0

10

Normalized Power Spectrum behaviour in dB

Frequency [KHz]

PS

D [d

B]

noisynoise

1γ

Figure C.1: PSD comparison among a noisy signal carrying one photon (blue), pure noise (green)and the signal obtained as the average of 10000 pulses carrying one γ (red).

Therefore, the desired component is preserved and the procedure works on the noise.The power of noise can be calculated as:

E [s(t) − d(t)]2 =1

N2E

[N∑

i=1

ni(t)

]2

=

=1

N2

N∑

i=1

E [ni(t)]2 =

=1

N2Nσ2

N =σ2

N

N(C.3)

In this way, the SNR can be defined as:

SNR =d(t)

σn/√

N=

d(t)

σn

·√

N (C.4)

And in dB:

SNRdB = 10 · log10

(d(t)

σn/√

N

)2

=

143

C – The averaging procedure and SNR improvement

= 20 · log10

(d(t)

σn

)+ 20 · log10

√N =

= SNRdB−orig + 10 · log10N (C.5)

where SNRdB−orig is the SNR evaluated on one single noisy pulse (defined in theformula as original). If N is equal to 10000, the SNR theoretical improvement isof 40 dB. In Tab. 3.1, the 1γ noisy signal SNR presented a mean value of 3.3 dB.Since in the I dataset we had 10000 pulses carrying one photon, the 1γ average (orreference) signal reported in Fig. 3.20 obtained a SNR of 43 dB, confirming ourassumption.

We would like to stress that this improvement does not depend on the numberof σ chosen to define the noise amplitude interval in the SNR formula (Eq. 3.3),since this constant quantity occurs only in the first term of the addition in Eq. C.5,while the improvement depends on the second term.

144

Appendix D

Wiener filter and SNR

In Sect. 3.4 we have seen that the histogram of Wiener filtered pulses presented aspread in the mean peak values distribution with respect to the one calculated fornoisy signals, as shown in Fig. D.1. The amplitude peak values as a function ofcarried photons, for both noisy and Wiener filtered pulses, are reported in Fig. D.2.

The reason for this particular behaviour is that the Wiener filter transfer function(Eq. 3.10) depends on the SNR value of the specific signal to be processed. Asreported in Tab. 3.1, the SNR values of different subsets of detected photons changeas a function of the number of carried photons since the noise component remainsthe same (in power, amplitude distribution and colour), while the desired componentgrows up with the number of detected photons. In this way, when changing the ratiobetween noise and desired power components, the Wiener transfer function changesand, as a consequence, the amplification factor is not constant during the completedataset processing.

In order to investigate this behaviour, we present in this Appendix the processingof a sample of 10000 simulated pulses with fixed SNR value. The first 2000 waveformscarry 1γ, the second 2γ,..., pulses from 8001 to 10000 carry 5γ. Therefore, everysubset carries the same number of photons (i.e. 2000) and the amplitude of eachpeak (occurrence) is not Poisson distributed (see Sect. 3.3.1). The reason of thissimplification is that, in this context, we are only interested in the evaluation ofthe Wiener filtered peak position changes as a function of the SNR. The desiredcomponents (from 1 to 5γ) are real and come form the I dataset analysed in Chap.3, while the noisy ones are extracted from a WGN process with zero mean and RMScalculated to maintain constant the SNR.

With reference to the SNR estimates presented in Tab. 3.1, 1γ noisy signalsreported an average SNR equal to 3.3 dB and treatment with the Wiener filterattenuated their signal amplitudes, see Figs. D.1, D.2. By simulating a constantSNR of 3 dB on the sample of 10000 noisy pulses mentioned above, we obtained theamplitude trend shown in Fig. D.3.

145

D – Wiener filter and SNR

0 10 20 30 40 500

500

1000

1500

2000

2500

3000

amplitude [mV]

occu

rren

ces

noisy fitwiener fit

1γ

2γ

2γ

3γ

4γ 4γ5γ5γ

1γ

Figure D.1: Superimposition of noisy and Wiener filtered fitted histograms. There is present aspread in the mean peak values. Wiener attenuates 1γ, 2γ while amplifies 4γ,5γ peak amplitudes.

1 2 3 4 55

10

15

20

25

30

35

40

45

50

# γ

ampl

itude

[mV

]

noisywiener

4.6 4.8 5.0

47.0

47.2

47.4

47.6

# γ

ampl

itude

[mV

]

4.8 4.9 5.0

42.8

43.0

43.2

43.4

# γ

ampl

itude

[mV

]

Figure D.2: Amplitude trend for noisy and Wiener filtered waveforms. Points represent meanamplitude values of a specific subset of carried photons, bars are the corresponding errors calculatedon the mean values.

146


All amplitude peaks are attenuated in the Wiener treatment. It is worth notingthat the peaks width (FWHM) increases with the number of photons both in noisyand in Wiener filtered pulses, as required by the SNR Eq. 3.3, since the SNR ismaintained fixed.

With reference to the results for 5γ noisy signals reported in Tab. 3.1, the SNRmean value was 15.8 dB and the Wiener behaviour was clearly of amplification, aspresented in Figs. D.1, D.2. By simulating a constant SNR of 15 dB, the resultshown in Fig. D.4 has been obtained. Wiener filtered signals present an amplitudepeak value enhancement, while, for both noisy and elaborated pulses, the FWHMsare reduced in according to the higher fixed SNR value (and to Eq. 3.3).

The amplitude peak values as a function of carried photons, for both noisy andWiener filtered pulses at the two different SNR values, are reported in Fig. D.5.The amplification (attenuation) factor, fixed the SNR, is constant on all peaks.

Finally, it is interesting to consider the evolution of the Wiener amplificationfactor (fA) as a function of the SNR of the simulated dataset. To perform thiscomparison, we have focussed on the difference between the average noisy 1γ peakand the Wiener filtered one. This difference has been normalized to the averagenoisy 1γ peak value. This result is reported in Fig. D.6. A negligible amplificationcan be obtained at a SNR close to 12 dB, that is the SNR mean value of the Idataset 3γ peak for which there was no enhancement nor attenuation, see 3.1 andFigs. D.1, D.2. This last result expresses the good accordance of our simulationwith real data.

147


0 10 20 30 40 500

200

400

600

800

1000

1200

1400

1600

amplitude [mV]

occu

rren

ces

noisy fitWiener fit

3γ

3γ

4γ

4γ

5γ

1γ 2γ

1γ

5γ

2γ

SNR = 3 dB

Figure D.3: Superimposition of noisy and Wiener filtered fitted histograms on a sample of 10000noisy simulated pulses. The SNR has been maintained fixed to 3 dB. All Wiener mean peakamplitudes are attenuated with respect to noisy ones.

0 10 20 30 40 500

500

1000

1500

2000

2500

3000

amplitude [mV]

occu

rren

ces

noisy fitWiener fit

3γ

5γ

5γ4γ3γ

1γ

1γ2γ

4γ

SNR = 15 dB

2γ

Figure D.4: Superimposition of noisy and Wiener filtered fitted histograms on a sample of 10000noisy simulated pulses. The SNR has been maintained fixed to 15 dB. All Wiener mean peakamplitudes are amplified with respect to noisy ones.

148


1 2 3 4 55

10

15

20

25

30

35

40

45

50

# γ

ampl

itude

[mV

]

Wien15dB

noisyWien

3dB

Figure D.5: Amplitude trend for noisy signals (blue), Wiener filtered at SNR 15 dB (red) andWiener filtered at SNR 3 dB (green) waveforms. Points represent mean amplitude values of aspecific subset of carried photons. Error bars are not appreciable at this scale.

0 5 10 15 20−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

SNR [ dB ]

f A =

(γ w

− γ

n ) /

γ n

3 dB, fA= −0.49

15 dB, fA= +0.09

12 dB, fA ≈ 0

Figure D.6: Estimated amplification factor fA of the Wiener transfer function expressed as afunction of the SNR. fA has been calculated as the amplitude difference between 1γ noisy andWiener filtered peak normalized to 1γ noisy peak.

149

Appendix E

The Buzuloiu’s algorithm for 2Dsignals

The Buzuloiu’s procedure can be also applied to multi-dimensional signals. If wehave a 2D pulse with a Γ4 shape, as presented in Fig. E.1, the estimation of theamplitude error can be obtained as described in Sect. 4.3.1 and is here presented inFigs. E.2, E.3.

Figure E.1: Simulated 2D normalized to unit Γ4 pulse to be analysed. Red dots represent the firstDGT sample to apply the procedure and evaluate the coefficients.

150

E – The Buzuloiu’s algorithm for 2D signals

00.2

0.40.6

0.81

0

0.5

1−0.4

−0.2

0

0.2

0.4

0.6

shift x [ T ]shift y [ T ]

ampl

itude

err

or [%

]

Figure E.2: Amplitude error (percentage of the maximum value) as a function of the shift alongx and y time directions (from 0 to the value of the sampling period T). The procedure has beenapplied to the first 3x3 samples with amplitude higher then 0, no segmentation has been considered.

00.2

0.40.6

0.81

0

0.5

1−0.4

−0.2

0

0.2

0.4

shift x [ T ]shift y [ T ]

ampi

tude

err

or [%

]

Figure E.3: Amplitude error (percentage of the maximum value) in the same conditions of theprevious figure but with two segmentations on both x and y time axes.

151

Appendix F

Simulation of a 2D corneal layerreconstruction

Starting from one single tomogram from one single eye, we can simulate the acquisi-tion of other tomograms by introducing a WGN that corrupts the original (and real)one. We have made the assumption that the RMS of the WGN was of the orderof the fluctuations between right and left eye corneal layer thickness for the samepatient. By applying the procedure improved for marginal regions (Sect. 5.6) to allsimulated tomograms, we can perform a 2D corneal layer surface reconstruction, asshown in Fig. F.1.

Figure F.1: Simulated 2-dimensional surface reconstruction of corneal layers. On the top, theEpithelium-Bowman surface is appreciable, while on the bottom, the Stroma-Endothelium one.

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156

List of figures

2.1 FAIR facility overview: at present (blue) and in the near future (red). 102.2 PANDA spectrometer 3-dimensional view. . . . . . . . . . . . . . . . 112.3 Quarks and their confinement pictorial representation. . . . . . . . . 112.4 Analog transmission sub-chain. . . . . . . . . . . . . . . . . . . . . . 152.5 Digital transmission sub-chain. . . . . . . . . . . . . . . . . . . . . . . 152.6 Pictorial representation of different asynchronous samplings. . . . . . 162.7 Peak Distortion as a function of the Shaper order n.. . . . . . . . . . 172.8 Inter-arrival time between two particles as a function of the Shaper

order n. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.9 Simulated transmission chain. . . . . . . . . . . . . . . . . . . . . . . 192.10 Simulink-implemented transmission chain schematics. . . . . . . . . 202.11 Simulink-implemented schematics: Analog Chain (green block in

Fig. 2.10). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.12 Simulink-implemented schematics: DGT Chain (blue block in Fig.

2.10). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.13 Analog outputs obtained with Matlab simulation. . . . . . . . . . . 212.14 Analog outputs obtained with Simulink simulation: input(red), pre-

ampl(cian), PZcomp(purple), shaper(yellow). . . . . . . . . . . . . . . 222.15 Continuous shaper output: noisy(a) vs desired(b). . . . . . . . . . . . 232.16 PSD of analog shaper output. . . . . . . . . . . . . . . . . . . . . . . 232.17 Matlab simulated noisy and desired digital signals. . . . . . . . . . . 242.18 Butterworth III order denormalized analog transfer function, 3D plot

(right) and projection on complex plain (left). . . . . . . . . . . . . . 262.19 Butterworth III order denormalized analog transfer function, 2D project

mask obtained from the projection of the 3D plot on the plain realpositive freq. vs modulus. . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.20 MMSE noise canceller using a LMS filter. . . . . . . . . . . . . . . . . 282.21 Butterworth III order filtered signal vs. noisy and desired signals. . . 302.22 SNR expressed in dB as a function of stepsize µ. . . . . . . . . . . . . 312.23 PD expressed in % as a function of stepsize µ. . . . . . . . . . . . . . 322.24 SNR / PD functional to be maximized for the optimal solution retrieval. 32

157

LIST OF FIGURES

2.25 LMS filtered signal (µ = 0.18) vs. noisy and desired signals. . . . . . 33

2.26 Squared error amplitude vs. time: decreasing behaviour . . . . . . . . 34

2.27 LMS four coefficients (µ=0.18) behaviour: a stable solution is reached. 34

2.28 IIR Butter. III ord. filter with DF II structure, Simulink schematics.Longest combinatorial path highlighted in dark green. . . . . . . . . . 37

2.29 IIR Butter. III ord. filter with DF I transposed structure, Simulink

schematics. Longest combinatorial path highlighted in dark green. . . 37

2.30 FIR LMS 4-coefficients structure Matlab simulated (LMS1), Simulink

schematics. The longest combinatorial path is highlighted in darkgreen, mu is the stepsize. . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.31 FIR LMS 4-coefficients structure implemented for FPGA board (LMS2),Simulink schematics. The longest combinatorial path is highlightedin dark green, mu is the stepsize, the added registers are highlightedin light green. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.32 Matlab simulated filter comparison. . . . . . . . . . . . . . . . . . . 40

2.33 Matlab simulated structures implemented on FPGA. . . . . . . . . 41

2.34 SNR expressed in dB as a function of stepsize µ. . . . . . . . . . . . . 43

2.35 PD expressed in % as a function of stepsize µ. . . . . . . . . . . . . . 43

2.36 SNR / PD functional to be maximized for the optimal FPGA-orientedfiltering solution retrieval. . . . . . . . . . . . . . . . . . . . . . . . . 44

2.37 FPGA-optimized filtering structures output. . . . . . . . . . . . . . . 44

2.38 Butterworth SNR values (blue dots), Gaussian distribution (whitebars), mean value (red line) and our single simulation (green line,5.76 dB). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.39 LMS SNR values (blue dots), Gaussian distribution (white bars),mean value (red line) and our single simulation (green line, 6.57 dB). 47

2.40 Butterworth PD values (blue dots), distribution (white bars), meanvalue (red line) and our single simulation (green line, 8.34 %). . . . . 48

2.41 LMS PD values (blue dots), distribution (white bars), mean value(red line) and our single simulation (green line, 0.06 %). . . . . . . . 48

3.1 TES sharp transition. . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.2 Experimental Setup, from the left: a laser diode (λ=1310 nm), theoptical fiber coupled to the laser diode, the TES sensor, the SQUIDand the Oscilloscope. . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.3 The TES Laboratory at INRiM. . . . . . . . . . . . . . . . . . . . . . 52

3.4 Example of TES noisy signals. . . . . . . . . . . . . . . . . . . . . . . 54

3.5 Persistence representation of an acquisition of more than 34000 sig-nals. Resolution: Time 0.02 µs - Amplitude 0.5 mV. The colorbar isexpressed in percentage. . . . . . . . . . . . . . . . . . . . . . . . . . 54

158

LIST OF FIGURES

3.6 Example of TES noisy signals carrying 2γ in blue solid line, the base-line is in green solid line and the absolute minimum is representedwith the red asterisk. Estimated amplitude: 21.16 mV. . . . . . . . . 55

3.7 Gaussian amplitude distribution behaviour of samples before trig-ger for one signal of the complete analysed dataset (according to aPearson’s chi-square test with confidence level of 5%). . . . . . . . . . 55

3.8 Example of TES noisy signals amplitude histogram and carried photons. 563.9 I dataset - 34000 pulses, λ = 1310 nm - Noisy signal amplitudes

histogram (blue dots) and fit (red line). This is the procedure toevaluate the Energy Resolution. No processing has been applied toany signal to reduce noise. . . . . . . . . . . . . . . . . . . . . . . . . 58

3.10 Example of SNR evaluation on the same signal of Fig. 3.6. . . . . . . 593.11 Gaussian amplitude distribution of samples before trigger, according

to a Pearson’s chi-square test with confidence level of 5%. Thesesamples have the same negative time value and come from 1000 noisysignals that carry the same number of photons (1γ in this example). . 60

3.12 Comparison of a noisy signal carrying 3γ and of its 3γ referencewaveform (obtained with the averaging technique) . . . . . . . . . . 60

3.13 Detailed PSD behaviour of a noisy signal carrying 3γ and of its 3γreference waveform (obtained with the averaging technique) . . . . . 61

3.14 Example of HA time evaluation. . . . . . . . . . . . . . . . . . . . . . 623.15 Comparison of a noisy signal carrying 2γ and of the Savitzky-Golay

filtering elaboration performed on the same signal. The reference 2γsignal is superimposed. . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.16 Optimal (Wiener) filtering. The power spectrum of signal plus noiseshows a signal peak added to a noise tail. The tail is extrapolatedback into the signal region as a noise model. The signal model isobtained by subtraction. . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.17 Amplitude histogram of the TES noisy signals already reported inFig. 3.8 with, in the red ellipse, the subset of noisy signals that carryconfidently 2γ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.18 Reference signal obtained averaging the subset of noisy signals thatcarry confidently 2γ. . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.19 Histogram of cross-correlation between the 2γ reference signal and allnoisy waveforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.20 All 5 reference signals for the Wiener filter. These waveforms areobtained with the averaging technique, based on the cross-correlationhistogram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.21 Example of a TES noisy signal (cian), reference signal (green), SG(blue) and Wiener (red) filter’s outputs. SNR comparisons are alsoreported. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

159

LIST OF FIGURES

3.22 PSD of the same noisy signal (cian) presented in Fig. 3.21, of the 2γreference signal (green), of SG (blue) and of Wiener (red) filter outputs. 71

3.23 I dataset - 34000 pulses, λ = 1310 nm - Noisy signal amplitudeshistogram (dots) and fit (line). No processing has been applied. The0γ peak has been isolated and neglected using the cross-correlationinformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.24 I dataset - 34000 pulses, λ = 1310 nm - Savitzky-Golay filtered signalamplitudes histogram (dots) and fit (line). 0γ peak is neglected usingthe cross-correlation information. . . . . . . . . . . . . . . . . . . . . 73

3.25 I dataset - 34000 pulses, λ = 1310 nm - Wiener filtered signals ampli-tude histogram (dots) and fit (line). 0γ peak is neglected using thecross-correlation information. . . . . . . . . . . . . . . . . . . . . . . 74

3.26 Persistence representation of Wiener filtered signals of 1 to 5γ. Theuncertainty regions are reduced and among them white areas becomeappreciable. Resolution: Time 0.02 µs - Amplitude 0.5 mV. Thecolorbar is expressed in percentage. . . . . . . . . . . . . . . . . . . . 74

3.27 II dataset - 45000 pulses, λ = 1550 nm - Noisy signal amplitudeshistogram (dots) and fit (line). The 0γ peak has been isolated andneglected using the cross-correlation information. . . . . . . . . . . . 75

3.28 II dataset - 45000 pulses, λ = 1550 nm - Wiener filtered signals am-plitude histogram (dots) and fit (line). 0γ peak is neglected using thecross-correlation information. . . . . . . . . . . . . . . . . . . . . . . 75

3.29 III dataset - 50000 pulses, λ = 1310 nm - Noisy signal amplitudeshistogram (dots) and fit (line). The 0γ peak has been isolated andneglected using the cross-correlation information. . . . . . . . . . . . 76

3.30 III dataset - 50000 pulses, λ = 1310 nm - Wiener filtered signalsamplitude histogram (dots) and fit (line). 0γ peak is neglected usingthe cross-correlation information. . . . . . . . . . . . . . . . . . . . . 76

3.31 TES polarization circuit. . . . . . . . . . . . . . . . . . . . . . . . . . 773.32 One photon average signal. . . . . . . . . . . . . . . . . . . . . . . . . 783.33 Linear trend of the TES amplitude response of the first 7 detected

photons, the non-linear region starts form 8γ. The inset shows theestimated error in the 8γ energy evaluation. . . . . . . . . . . . . . . 79

3.34 Linear trend of the TES amplitude response of the first 7 detectedphotons. Normalized to the unit amplitude and energy values as afunction of the number of carried photons. The only mean values arereported. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.1 LHC overview and the 4 experiments. . . . . . . . . . . . . . . . . . . 824.2 ALICE spectrometer schematic reported in [31]. . . . . . . . . . . . . 844.3 An overview of the ALICE TPC front end electronics reported in [31]. 84

160

LIST OF FIGURES

4.4 A simplified block diagram of the PreAmplifier-ShAper (PASA) signalprocessing chain reported in [31]. . . . . . . . . . . . . . . . . . . . . 85

4.5 Layout of the PASA chip presented in [31]. . . . . . . . . . . . . . . . 854.6 ALTRO chip block diagram reported in [31]. . . . . . . . . . . . . . . 864.7 ALTRO chip layout presented in [31]. . . . . . . . . . . . . . . . . . . 874.8 Zero Suppression scheme reported in [31]. . . . . . . . . . . . . . . . . 874.9 Baseline Correction II block operation principle presented in [31]. . . 884.10 Ideal Γ4 function. The amplitude scale is normalized, while both

scales are expressed in arbitrary units. . . . . . . . . . . . . . . . . . 914.11 Detailed example of ideal Γ4 sampling. The first 3 DGT samples with

amplitude higher then 0 (a.u.) are s1, s2 and s3. . . . . . . . . . . . . 924.12 Representative curve using three (higher than 0) consecutive samples

of the ideal Γ4 pulse. T is the sampling period. . . . . . . . . . . . . 934.13 Amplitude error (percentage of the maximum value) as a function

of the shift (from 0 to the value of the sampling period T ). Theprocedure has been applied to the first 3 samples with amplitudehigher then 0, no segmentation has been considered. . . . . . . . . . . 93

4.14 Example of sampling period subdivision: on the first 3 samples wehave 4 segmentations. The first is in green colour, the second in cian,the third in magenta and the fourth in yellow. . . . . . . . . . . . . . 95

4.15 Amplitude error (percentage of the maximum value) when 4 planes(segments) are used on the first 3 samples. The red asterisk indicatesthe maximum error: 0.01 %, target 0.10 %. . . . . . . . . . . . . . . . 95

4.16 Amplitude error (percentage of the maximum value) using the first5 samples with no segmentation. The red asterisk indicates themaximum error: 0.05 %, target 0.10 %. . . . . . . . . . . . . . . . . . 96

4.17 Shift error (percentage of the maximum value) using the first 4 sam-ples with 2 hyperplanes (segments). The red asterisk indicates themaximum error: 0.05 %, the target was 0.10 %. . . . . . . . . . . . . 96

4.18 Example of random shift on 2 different noisy built signals over anensemble of 1000 waveforms. It is expressed in terms of sub-samples.The relative difference is of 0.4 T. . . . . . . . . . . . . . . . . . . . . 98

4.19 Example of the zero-suppression procedure. All noisy samples underthreshold are set to 0 a.u. The inset shows a detail of the suppression. 98

4.20 Right: difference between desired and estimated peak time. Themaximum error is of one sub-sample (one twentieth of T ). Left:distribution of the aforementioned difference. . . . . . . . . . . . . . . 100

4.21 Analog re-built signal and superimposed ideal Γ4. Example of a goodmatch according to a Pearson’s chi-square test with confidence levelof 5%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.22 Analog re-built signal and DGT reference for the filtering procedure. . 101

161

LIST OF FIGURES

4.23 Amplitude percent. error between noisy simulated and ideal signalsusing the first 3 coefficients and no period subdivision. Mean abs.error: 0.137 %. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.24 Amplitude percent. error between noisy simulated and ideal signalsusing the first 4 coefficients and 4 period subdivisions. Mean abs.error: 0.078 %. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.25 Reconstructed trajectories of particles from first proton-proton colli-sions at 7 TeV detected by ALICE TPC. . . . . . . . . . . . . . . . . 105

4.26 Not uniform time-slot filling. Period of 20 sub-samples, 1 sub-sampleis equal to 5 ns. Result obtained with the CM technique. . . . . . . . 106

4.27 Reference signal obtained averaging all realigned values that fill thesame time-slot. Period of 20 sub-samples. Result obtained with theCM technique. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.28 Not uniform time-slot filling even in the case of 10 sub-samples period,1 sub-sample is equal to 10 ns. Result obtained with the CM technique.107

4.29 Reference signal obtained averaging all realigned values that fill thesame time-slot in the case of 10 sub-samples period. Result obtainedwith the CM technique. . . . . . . . . . . . . . . . . . . . . . . . . . 108

4.30 Example of a fitted ALICE TPC noisy signal. The curve used forthe fit is a Γ4(A,t0,τ), see Sect. 4.3.1, good match according to aPearson’s chi-square test with confidence level of 5%. . . . . . . . . . 109

4.31 Reference signal obtained averaging all realigned values with the fit-ting method that fill the same time-slot in the case of 20 sub-samplesperiod. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.32 Reference signal obtained averaging all realigned values with the fit-ting method that fill the same time-slot in the case of 10 sub-samplesperiod. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.33 Analog candidate for the Buzuloiu’s procedure obtained fitting aver-aged real data. Example of a good match according to a Pearson’schi-square test with confidence level of 5%. . . . . . . . . . . . . . . . 111

4.34 Real TPC filtered data with Buzuloiu’s procedure working on the first3 samples and no period segmentation. Red dots represent the max-imum DGT amplitude value, the blue asterisks the analog maximumestimates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.35 Real TPC filtered data with Buzuloiu’s procedure working on the first4 samples and 2 period segmentations. Red dots represent the maxi-mum DGT amplitude values, the blue asterisks the analog maximumestimates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.36 Example of a comparison between two different fitting models: a Γ4

and a LC(Γ4,G). There are reported the results of the χ2 test togetherwith the residual estimation on the third sample for one fitting model. 113

162

LIST OF FIGURES

5.1 Typical optical setup of single point OCT. . . . . . . . . . . . . . . . 116

5.2 Example of OCT application to art conservation. . . . . . . . . . . . 117

5.3 Example of OCT application to study of biological tissues: a fingertip.117

5.4 Image of human eye anatomy. . . . . . . . . . . . . . . . . . . . . . . 118

5.5 Retina: 3D OCT detail of optic nerve connection over retinal internalplane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.6 Cornea: 3D OCT acquisition. . . . . . . . . . . . . . . . . . . . . . . 119

5.7 Example of OCT corneal medical examination, courtesy of Dr. M.Frisani. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.8 OCT corneal tomogram and our region of interest (ROI) on thecorneal axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.9 Comparison between histological reference representation of corneallayers (left) and our OCT ROI (right) to be processed. . . . . . . . . 122

5.10 3D representation of one single gray-scale tomogram, the pixel inten-sity is used as third dimension and ranges form 0 to 255. . . . . . . . 122

5.11 ROI width calculation: each pixel row must contain the same corneallayer to produce a consistent treatment. lagMAX represents the halfof the maximum number of columns that can fill the ROI. . . . . . . 123

5.12 Detailed example of ROI subdivision in three slices one next to eachother. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.13 Pixel intensity profile on a slice 20 pixel wide, centered on the cornealaxis. In green the intensity of the only axial column is represented,while in red and blue, the averaged intensities in pixel and subpixelelaborations respectively. . . . . . . . . . . . . . . . . . . . . . . . . . 124

5.14 Detail of the pixel analysis. In the box, thickness estimates arereported both in pixel and subpixel scale. . . . . . . . . . . . . . . . . 125

5.15 Pixel intensity Gaussian distribution in the Stromal region of a pixelrow with medium point centred on the corneal axis and 90 pixelwide, being lagMAX equal to 45 pix. This result is in according to aPearson’s chi-square test with confidence level of 5%. . . . . . . . . . 126

5.16 Pictorial representation of SNR evaluation on the central column(green) of an OCT tomogram. Dashed purple lines show the ex-tremal values of amplitude (A) and 95% noise band (4σ). The bluesolid line is the linear trend. The noise has been considered in theonly Stromal region. . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5.17 Pixel intensity Gaussian distribution in the Stromal region of a pixelcolumn centred on the corneal axis. This result is in according toa Pearson’s chi-square test with confidence level of 5%. There hasbeen subtracted the linear trend to perform the analysis, therefore,negative pixel intensities correspond to values lower than the trend. . 128

163

LIST OF FIGURES

5.18 Pixel intensity Gaussian distribution in the Stromal region of a pixelaveraged column (lag = 15 pix) centred on the corneal axis. Thisresult is in according to a Pearson’s chi-square test with confidencelevel of 5%. As in Fig. 5.17, negative pixel intensities correspond tovalues lower than the trend. . . . . . . . . . . . . . . . . . . . . . . . 128

5.19 Gaussian distribution of SNR estimations evaluated on the cornealaxis pixel column. Sample of 52 patients, right eyes, third acquisi-tion. This result is in according to a Pearson’s chi-square test withconfidence level of 5%. . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.20 Gaussian distribution of SNR estimations evaluated on the averagedpixel column. The lag has been chosen equal to 15 pix, thereforethe average has been performed on 30 pixel columns centred on thecorneal axis. Sample of 52 patients, right eyes, third acquisition. Thisresult is in according to a Pearson’s chi-square test with confidencelevel of 5%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.21 Example of analysis applied to marginal corneal regions. Blue linesrepresent the directions on which layer thicknesses have been estimat-ed. Pink dots are positioned on layer boundaries per each considereddirection. The inset presents a detail of this procedure. . . . . . . . . 130

5.22 Detailed representation of thickness profile estimation on a patientbefore (red) and after (blue) an orthokeratology treatment, courtesyof Dr. M. Frisani. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

A.1 Sign-LMS filter structure Simulink schematics, the green squaresrepresent the sign evaluation of the estimated error e[n] and of thedata x[n]. The red circles are the filter input and the stepsize, whilethe blue circle is the filter output. . . . . . . . . . . . . . . . . . . . . 135

C.1 PSD comparison among a noisy signal carrying one photon (blue),pure noise (green) and the signal obtained as the average of 10000pulses carrying one γ (red). . . . . . . . . . . . . . . . . . . . . . . . 141

D.1 Superimposition of noisy and Wiener filtered fitted histograms. Thereis present a spread in the mean peak values. Wiener attenuates 1γ,2γ while amplifies 4γ,5γ peak amplitudes. . . . . . . . . . . . . . . . 144

D.2 Amplitude trend for noisy and Wiener filtered waveforms. Points rep-resent mean amplitude values of a specific subset of carried photons,bars are the corresponding errors calculated on the mean values. . . . 144

164

LIST OF FIGURES

D.3 Superimposition of noisy and Wiener filtered fitted histograms ona sample of 10000 noisy simulated pulses. The SNR has been main-tained fixed to 3 dB. All Wiener mean peak amplitudes are attenuatedwith respect to noisy ones. . . . . . . . . . . . . . . . . . . . . . . . . 146

D.4 Superimposition of noisy and Wiener filtered fitted histograms ona sample of 10000 noisy simulated pulses. The SNR has been main-tained fixed to 15 dB. All Wiener mean peak amplitudes are amplifiedwith respect to noisy ones. . . . . . . . . . . . . . . . . . . . . . . . . 146

D.5 Amplitude trend for noisy signals (blue), Wiener filtered at SNR15 dB (red) and Wiener filtered at SNR 3 dB (green) waveforms.Points represent mean amplitude values of a specific subset of carriedphotons. Error bars are not appreciable at this scale. . . . . . . . . . 147

D.6 Estimated amplification factor fA of the Wiener transfer function ex-pressed as a function of the SNR. fA has been calculated as the ampli-tude difference between 1γ noisy and Wiener filtered peak normalizedto 1γ noisy peak. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

E.1 Simulated 2D normalized to unit Γ4 pulse to be analysed. Red dotsrepresent the first DGT sample to apply the procedure and evaluatethe coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

E.2 Amplitude error (percentage of the maximum value) as a functionof the shift along x and y time directions (from 0 to the value ofthe sampling period T). The procedure has been applied to the first3x3 samples with amplitude higher then 0, no segmentation has beenconsidered. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

E.3 Amplitude error (percentage of the maximum value) in the same con-ditions of the previous figure but with two segmentations on both xand y time axes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

F.1 Simulated 2-dimensional surface reconstruction of corneal layers. Onthe top, the Epithelium-Bowman surface is appreciable, while on thebottom, the Stroma-Endothelium one. . . . . . . . . . . . . . . . . . 150

165

List of tables

2.1 SNR improvement [dB], our choice in bold typeface. . . . . . . . . . . 25

2.2 PD [% error], our choice in bold typeface. . . . . . . . . . . . . . . . 25

2.3 LMS performances vs µ, our choice in bold typeface. . . . . . . . . . 31

2.4 Performance of Matlab-simulted Butterworth and LMS filters. . . . 41

2.5 Performance of FPGA-optimized Butterworth and LMS filtering struc-ture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.6 Compared performance for different structures of Butterworth filterimplemented on a Xilinx Virtex 4 ML402. . . . . . . . . . . . . . . . 45

2.7 Compared performance for different structures of LMS filter imple-mented on a Xilinx Virtex 4 ML402. . . . . . . . . . . . . . . . . . . 45

2.8 FPGA consumptions on different boards: Xilinx Virtex 4 XC4VSX3-10 (V4 ) and Virtex 5 X5VSX35T-3 (V5 ). . . . . . . . . . . . . . . . 46

3.1 SNR evaluation on noisy and filtered signals as a function of thenumber of carried photons. . . . . . . . . . . . . . . . . . . . . . . . . 71

3.2 Time jitter estimates on noisy and filtered signals as a function of thenumber of carried photons. . . . . . . . . . . . . . . . . . . . . . . . . 72

4.1 Coefficients for the filtering procedure calculated using the reportednumber of first samples (3-4) and period segmentations. . . . . . . . . 103

4.2 Coefficients for the filtering procedure calculated using the first 5samples and reported period segmentations. . . . . . . . . . . . . . . 103

4.3 Amplitude mean error estimations expressed in percentage. Best andworst results in bold typeface. . . . . . . . . . . . . . . . . . . . . . . 103

4.4 Range of max. amplitude, TPC region, start time and length valuesused to select signals candidate for the filtering procedure. (∗) InnerRead-Out Chamber, see [31]. . . . . . . . . . . . . . . . . . . . . . . . 106

4.5 Mean values and RMS of the residuals estimated on the first 5 sampleswith a Γ4 and a LC(Γ4,G) models. . . . . . . . . . . . . . . . . . . . . 113

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LIST OF TABLES

5.1 Thickness estimation of the four main corneal layers obtained aver-aging the evaluations performed on more than 150 images per eye. . . 125

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Digital Signal Processing applied to Physical Signals - CERNcds.cern.ch/record/1341287/files/CERN-THESIS-2011-008.pdf · Digital Signal Processing applied to Physical Signals Diego