Mathematics and Signal Processing in Acoustics

  • The FWF project "Time-Frequency Implementation of HRTFs" has started.

    Principal Investigator: Damian Marelli

    Co-Applicants: Peter BalazsPiotr Majdak

  • Computer werden ständig schneller und die schnelle Entwicklung von Audio-Interface and Audio-Transmissions Technologien haben zu einem neuen Zeitalter von Audio-Systemen geführt, die mittels Surround Lautsprechern räumliche Schallerlebnisse reproduzieren können.

    Viele dieser Anwendungen benötigen eine genaue, effiziente und robuste Darstellung des Schalls in der Raum-Zeit-Frequenzebene. Das gemeinesame Projekt von ISF und IRCAM verbindet die mathematischen Konzepte, die am ARI verwendet und entwickelt werden mit der profunden Kenntnis in Signalverarbeitung in Echtzeit am IRCAM. Das Projekt versucht grundlegende Fragen in beiden Forschungsfeldern zu beantworten und hat als Ziel die Entwicklung von verbesserten Methoden für die oben erwähnten Anwendungen.

    Spezielle Fragen, die in diesem Projekt geklärt werden sollen, sind:

    • Kann mittels Wavelets und Frames eine effiziente Raum-Zeit-Frequenz Darstellung von Wellenfeldern gefunden werden, die robuster als derzeitig existente Methoden sind?
    • Ist es möglich, auf Frames basierenden Methoden an ein (sphärisches) Lautsprecher-, bzw. Mikrophonarray mit vorgegeben Anordnung von Lautsprechen, bzw. Mikrophonen anzupassen (z.B. das 64 Kanal Array am IRCAM)
    • Wie kann das akustische Feld auf einer Kugel mit Frames dargestellt werden, um bessere Raum-Zeit-Frequenz Darstellung des akustischen Felds an bestimmten Teilen der Kugel zu erhalten?
    • Ist es möglich, diese Raum-Zeit-Frequenz Darstellung in mehreren Auflösungen für Raumaufnahmen mittles sphärischen Mehrkanal-Mikrophonarray zu verwenden (z.B. um eine höhere räumliche Auflösung von frühen Raumreflexionen zu erreichen)?
  • S&T cooperation project 'Amadee' Austria-France 2013-14, "Frame Theory for Sound Processing and Acoustic Holophony", FR 16/2013

    Project Partner: The Institut de recherche et coordination acoustique/musique (IRCAM)

  • Objective:

    For many important applications - like virtual reality, communication, sound design, audio compression & coding, and hearing aids - a mathematical representation that matches or approximates the perception of the human auditory system is needed. For solving this critical and prominent problem, a trans-disciplinary approach is necessary. The goals of this project are:

    • design, development and evaluation of new representations of audio signals,
    • development of new tools based on the mathematic theory of time-frequency (or time-scale) representations (Gabor, wavelet or other),
    • development of the mathematic background for applications in audio perception and psychoacoustics,
    • and evaluation of these representations with auditory and psychoacoustic tests.

    Partners:

    • R. Kronland-Martinet, S. Ystad, T. Necciari, Modélisation, Synthèse et Contrôle des Signaux Sonores et Musicaux of the LMA / CRNS Marseille
    • S. Meunier, S. Savel, Acoustique perceptive et qualité de l’environnement sonore of the LMA / CRNS Marseille

    This is a partner project of the ANR project senSons.

  • Scientific and Technological Cooperation between Austria and Serbia (SRB 01/2018)

    Duration of the project: 01.07.2018 - 30.06.2020

     

    Project partners:

    Acoustics Research Institute, ÖAW (Austria)

    University of Vienna (Austria)

    University of Novi Sad (Republic of Serbia)

     

    Project website: http://nuhag.eu/anacres

  • Objective:

    Another project has investigated the basic properties of frame and Bessel multipliers. This project aims to generalize this concept so that it will work with Banach spaces also.

    Method:

    As the Gram matrix plays an important role in the investigation of multipliers, it is quite natural to look at the connection to localized frames and multipliers. The dependency of the operator class on the symbol class can be researched.

    The following statements will be investigated:

    • Theorem: If G is a localized frame and a is a bounded sequence, then the frame multiplier Ta is bounded on all associated Banach spaces (the associated co-orbit spaces).
    • Theorem: If G is a localized frame and a is a bounded sequence, such that the frame multiplier Ta is invertible on the Hilbert space H, then Ta is simultaneously invertible on the associated Banach spaces.

    The applications of these results to Gabor frames and Gabor multipliers will be further investigated.

    Application:

    Although Banach spaces are more general a concept than Hilbert spaces, Banach theory has found applications. For example, if any norm other than L2 (least square error) is used for approximation, Banach theory tools have to be applied.

    Partners:

    • K. Gröchenig, NuHAG, Faculty of Mathematics, University of Vienna

    Project-completion:

    This project ended on 28.02.2008 and is incorporated into the 'High Potential'-Project of the WWTF, MULAC.

  • Objective:

    From many previous applications, it is known that inverse problems often require a regularization that makes the inversion numerically stable. In this project, sequences that allow a bounded, injective analysis (that is not boundedly invertible) are investigated, .

    Method:

    Even for general sequences, analysis operator and synthesis operator can be defined. The first part of this project will investigate the most general results of these definitions. For example, it can be shown that the analysis operator is always a closed operator. Although it can be shown that the existence of another sequence that allows a perfect reconstruction fit can not be bounded, the question of how to construct such a "dual sequence" will be investigated.

    Application:

    Such sequences have already found applications in wavelet analysis, in which dual sequences were constructed algorithmically. Also, the original system investigated by Gabor with a redundancy of 1 satisfies this condition.

    Partners:

    • M. El-Gebeily, Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Saudi Arabia
    • J. P. Antoine, Unité de physique théorique et de physique mathématique – FYMA, Belgium
  • Objective:

    The applications involving signal processing algorithms (like adaptive or time variant filters) are numerous. If the STFT, the Short Time Fourier Transformation, is used in its sampled version, the Gabor transform, the use of Gabor multipliers creates a possibility to construct a time-variant filter. The Gabor transform is used to calculate time frequency coefficients, which are multiplied with a fixed time-frequency mask. Then the result is synthesized. If another way of calculating these coefficients is chosen or if another synthesis is used, many modifications can still be implemented as multipliers. For example, it seems quite natural to define wavelet multipliers. Therefore, for this case, it is quite natural to continue generalizing and look at multipliers with frames lacking any further structure.

    Method:

    Therefore, for Bessel sequences, the investigation of operators

    M = ∑ mk < f , ψk > φk

    where the analysis coefficients, < f , ψk >, are multiplied by a fixed symbol mk before resynthesis (with φk), is very natural and useful. These are the Bessel multipliers investigated in this project. The goal of this project is to set the mathematical basis to unify the approach to the Bessel multipliers for all possible analysis / synthesis sequences that form a Bessel sequence.

    Application:

    Bessel sequences and frames are used in many applications. They have the big advantage of allowing the possibility to interpret the analysis coefficients. This makes the formulation of a multiplier concept for other analysis / synthesis systems very profitable. One such system involves gamma tone filter banks, which are mainly used for analysis based on the auditory system.

    Publications:

    • Balazs, P. (2007), "Basic Definition and Properties of Bessel Multipliers", Journal of Mathematical Analysis and Applications, 325, 1: 571--585. doi:10.1016/j.jmaa.2006.02.012, preprint

    Project-completion:

    This project ended on 01.01.2007. Its completion allowed the sucessfull application for a 'High Potential'-Project of the WWTF, see MULAC.

  • Objective:

    Practical experience has shown that the concept of an orthonormal basis is not always useful. This led to the concept of frames. Models in physics and other application areas, including sound vibration analysis, are mostly continuous models. Many continuous model problems can be formulated as operator theory problems, as in differential or integral equations. An interesting class of operators is the Hilbert Schmidt class. This project aims to find the best approximation of any matrix by a frame multiplier, using the Hilbert Schmidt norm.

    Method:

    In finite dimensions, every sequence is a frame sequence, so the best approximation of any element can be found only via the frame operator using the dual frame for synthesis. Furthermore, the present best approximation algorithm involves the following steps: 1) The Hilbert-Schmidt inner product of the matrix and the projection operators involved is calculated in an efficient way; 2) Then the pseudo inverse of the Grame matrix is used to avoid the so-called calculation of the dual frames; The pseudo inverse is applied to the coefficients found above to find the lower symbol of the multiplier.

    Application:

    To find the best approximation of matrices via multipliers gives a way to find efficient algorithms to implement such operators. Any time-variant linear system can be modeled by a matrix. Time-invariant systems can be described as circulating matrices. Slowly-time-varying linear systems have a good chance at closely resembling Gabor multipliers. Other matrices can be well approximated by a "diagonalization" using other frames.

    Publications:

    • P. Balazs, "Hilbert-Schmidt Operators and Frames - Classification, Approximation by Multipliers and Algorithms" , International Journal of Wavelets, Multiresolution and Information Processing, (2007, accepted)  preprint, Codes and Pictures: here

    Project-completion:

    This project ended on 01.01.2009. Its completion allowed the sucessfull application for a 'High Potential'-Project of the WWTF, see MULAC

  • Biotop Beschreibung
    Workflow Biotop

    Einführung

    Die Lokalisierung von Schallquellen spielt eine wichtige Rolle im täglichem Leben. Die Form des menschlichen Kopfs, des Torsos und vor allem des Außenohrs (Pinna) bewirken einen Filtereffekt für einfallenden Schall und spielen daher eine wichtige Rolle bei der Ortung einer Schallquelle. Dieser Filtereffekt kann mittels der s.g. head related transfer functions (HRTFs, kopfbezogene Übertragungsfunktionen) beschrieben werden. Diese Filterfunktionen können mittels numerischer Methoden (zum Beispiel der Randelemente Methode, BEM) berechnet werden. In BIOTOP sollen diese Berechnungen durch Anwendung adaptiver Wavelet und Frame Methoden effizienter gemacht werden.

    Ziel

    Verglichen mit den herkömmlichen BEM Ansatzfunktionen haben Wavelets den Vorteil, besser an gegebene Schallverteilungen angepasst werden zu können. Als Verallgemeinerung von Wavelets sollen Frames dabei helfen, eine noch flexiblere Berechnungsmethode und damit eine noch bessere Anpassung an das gegebene Problem zu entwickeln. BIOTOP verbindet abstrakte mathematische Theorie mit numerischer und angewandter Mathematik. BIOTOP ist ein internationales DACH-Projekt (DFG-FWF-SFG) zwischen der Philipps-Universität Marburg (Stephan Dahlke), der Unicersität Basel (Helmut Harbrecht) und dem Institut für Schallforschung. Die gemeinsame Erfahrung dieser drei Forschungsgruppen soll helfen, neue numerische Strategien und Methoden zu entwickeln. Das Projekt wird vom FWF (Proj. Nummer: I-1018 N25) gefördert.

     

  • Basic Description:

    This project line has the goal of finding efficient algorithms for signal processing applications. To apply the results of signal processing, Gabor or wavelet theory, the algorithms must be formulated for finite dimensional vectors. These discrete results are motivated by the continuous setting, but also often provide some insight. Furthermore, the efficient implementation of algorithms becomes important. For the consistency of these algorithms, it is useful to incorporate them into a maintained software package.

    Subprojects:

    • Double Preconditioning for Gabor Frames: This project develops a way to find an analysis-synthesis system with perfect reconstruction in a numerically efficient way using double preconditioning.
    • Perfect Reconstruction Overlap Add Method (PROLA): The classic overlap-add synthesis method is systematically compared to a new method motivated by frame theory.
    • Numerics of Block Matrices: In some applications in acoustics, it is apparent that block matrices are a powerful tool to find numerically efficient algorithms.
    • Practical Time Frequency Analysis: This project evaluates the usefulness of a time-frequency toolbox for acoustic applications and STx.

    Partners:

    • H.G. Feichtinger et al., NuHAG, Faculty of Mathematics, University of Vienna
    • B. Torrésani, Groupe de Traitement du Signal, Laboratoire d'Analyse Topologie et Probabilités, LATP/ CMI, Université de Provence, Marseille
    • P. Soendergaard, Department of Mathematics, Technical University of Denmark
    • J. Walker, Department of Mathematics, University of Wisconsin-Eau Claire

    Publications:

    • P. Balazs, H.G. Feichtinger, M. Hampejs, G. Kracher; "Double Preconditioning for Gabor Frames”; IEEE Transactions on Signal Processing, Vol. 54, No.12, December 2006 (2006), preprint
    • P. Balazs, H.G. Feichtinger, M. Hampejs, G. Kracher; "Double Preconditioning for the Gabor Frame Operator”; Proceedings ICASSP '06, May 14-19, Toulouse, DVD (2006)
  • Objective:

    The Short-Time Fourier Transform (STFT), in its sampled version (the Gabor transform), is a well known, valuable tool for displaying the energy distribution of a signal over the time-frequency plane. The equivalence between Gabor analysis and certain filter banks is a well-known fact. The main task is how to find a Gabor analysis-synthesis system with perfect (or depending on the application, satisfactorily accurate) reconstruction in a numerically efficient way. This is done by using the dual Gabor frame, which implies the need to invert the Gabor frame operator.

    Method:

    This project incorporates an application of the general idea of preconditioning in the context of Gabor frames. While most (iterative) algorithms aim at a relatively costly exact numeric calculation of the inverse Gabor frame matrix, we will use a "cheap method" to find an approximation. The inexpensive method will be based on (double) preconditioning using diagonal and circulant preconditioners. As a result, good approximations of the true dual Gabor atom can be obtained at low computational costs.

    Application:

    For a number of applications, such as time stretching without changing the frequency content in audio processing or more complex modifications like psychoacoustical masking, the time domain signal needs to be reconstructed using the time-frequency domain coefficients.

    Partners:

    H. G. Feichtinger et al., NuHAG, Facultyof Mathematics, University of Vienna

    Publications:

    • P. Balazs, H.G. Feichtinger, M. Hampejs, G. Kracher; "Double Preconditioning for Gabor Frames"; IEEE Transactions on Signal Processing, Vol. 54, No.12, December 2006 (2006), preprint
    • P. Balazs, H.G. Feichtinger, M. Hampejs, G. Kracher; "Double Preconditioning for the Gabor Frame Operator"; Proceedings ICASSP '06, May 14-19, Toulouse, DVD (2006)
  • START project of P. Balazs.

    FLAME

     

    Diese Seite ist eine Projektbeschreibung und als solche in englischer Sprache verfasst.

    This international, multi-disciplinary and team-oriented project will expand the group Mathematics and Acoustical Signal Processing at the Acoustic Research Institute in cooperation with NuHAG Vienna (Hans G. Feichtinger, M. Dörfler, K. Gröchenig), Institute of TelecommunicationVienna (Franz Hlawatsch), LATP Marseille (Bruno Torrésani) LMA (Richard Kronland-Martinet). CAHR (Torsten Dau, Peter Soendergaard), the FYMA Louvain-la-Neuve (Jean-Pierre Antoine), AG Numerics (Stephan Dahlke), School of Electrical Engineering and Computer Science (Damian Marelli) as well as the BKA Wiesbaden (Timo Becker).

    Within the institute the groups Audiological Acoustics and Psychoacoutics, Computational Acoustics, Acoustic Phonetics and Software Development are involved in the project.

    This project is funded by the FWF as a START price . It is planned to run from May 2012 to April 2018.

     

    Workshops:

     

    Multipliers

     

    General description:

    We live in the age of information where the analysis, classification, and transmission of information is f essential importance. Signal processing tools and algorithms form the backbone of important technologieslike MP3, digital television, mobile phones and wireless networking. Many signal processing algorithms have been adapted for applications in audio and acoustics, also taking into account theproperties of the human auditory system.

    The mathematical concept of frames describes a theoretical background for signal processing. Frames are generalizations of orthonormal bases that give more freedom for the analysis and modificationof information - however, this concept is still not firmly rooted in applied research. The link between the mathematical frame theory, the signal processing algorithms, their implementations andfinally acoustical applications is a very promising, synergetic combination of research in different fields.

    Therefore the main goal of this multidisciplinary project is to

    -> Establish Frame Theory as Theoretical Backbone of Acoustical Modeling

    in particular in psychoacoustics, phonetic and computational acoustics as well as audio engineering.

    Overview

     

    For this auspicious connection of disciplines, FLAME will produce substantial impact on both the heory and applied research.

    The theory-based part of FLAME consists of the following topics:

    • T1 Frame Analysis and Reconstruction Beyond Classical Approaches
    • T2 Frame Multipliers, Extended
    • T3 Novel Frame Representation of Operators Motivated by Computational Acoustics

    The application-oriented part of FLAME consists of:

    • A1 Advanced Frame Methods for Perceptual Sparsity in the Time-Frequency Plane
    • A2 Advanced Frame Methods for the Analysis and Classification of Speech
    • A3 Advanced Frame Methods for Signal Enhancement and System Estimation

    Press information:

     

     

     

  • This project ended in September 2011.

    Media Coverage:


    Meetings:

    The final MulAc Meeting was in Vienna from 29th to 30th of August 2011.

    The ARI Mulac Frame Meeting was held on Tuesday, June 15th 2010at ARI.

    The MULAC Mid-term Meeting was held in Marseille from 12. to 13. April 2010. See the Registration-Webpage or the Program.

    The FYMA Mulac seminar was held in Louvain-la-Neuve in the 11th of March, 2010. (Talks by Jean-Pierre Antoine, Jean-Pierre Gazeau, Diana Stoeva and Peter Balazs.)

    The MULAC - Kick-Off Meeting took place at ARI in Vienna from September 23rd to 24th 2008.


    This international, multi-disciplinary and team-oriented project allowed P. Balazs to form a small group 'Mathematics and Acoustical Signal Processing’ at the Acoustic Research Institute in cooperation with NuHAG Vienna (Hans G. Feichtinger), LMA (Richard Kronland-Martinet) and LATP Marseille (Bruno Torrésani) as well as the FYMA Louvain-la-Neuve (Jean-Pierre Antoine).

    Within the institute the groups 'Audiological Acoustics' and 'Software Development' are involved.

    This project is funded by the WWTF . It will run for 3,5 years and post-docs will be employed for six years total, as well as master students for 36 months total.

    In December 2007 the Austrian Academy of Sciences was presenting 'mathematics in...' as the topic of the month . This included 'mathematics at the Acoustics Research Institute', which describes this project.

    General description:

    "Frame Multipliers” are a promising mathematical concept, which can be applied to retrieve desired information out of acoustic signals. P. Balazs introduced them by successfully generalizing existing time-variant filter approaches. This project aims to establish new results in the mathematical theory of frame multipliers, to integrate them in efficient digital signal processing algorithms and to make them available for use in 'real-world' acoustical applications. A multi-disciplinary and international cooperation has been established and will be extended in the project to create new significant impulses for the involved disciplines: mathematics, numerics, engineering, physics and cognitive sciences. Various acoustical applications like modelling of auditory perception, measurement of sound absorption coefficients and system identification of the head related transfer functions are included. The results of the project will allow their future integration into practical areas such as audio coding, noise abatement, sound quality design, virtual reality and hearing aids. 

    Media coverage:

  • This page provides resources for the research article:

    "Frame Theory for Signal Processing in Psychoacoustics"

    by Peter Balazs, Nicki Holighaus, Thibaud Necciari, and Diana Stoeva

    to appear in the book "Excursions in Harmonic Analysis" published by Springer.

    Abstract: This review chapter aims to strengthen the link between frame theory and signal processing tasks in psychoacoustics. On the one side, the basic concepts of frame theory are presented and some proofs are provided to explain those concepts in some detail. The goal is to reveal to hearing scientists how this mathematical theory could be relevant for their research. In particular, we focus on frame theory in a filter bank approach, which is probably the most relevant view-point for scientists in audio signal processing. On the other side, basic psychoacoustic concepts are presented to stimulate mathematicians to apply their knowledge in this field.

    The present ZIP archive features Matlab/Octave scripts that will allow to reproduce the results presented in Figures 7, 10, and 11 of the article.

    IMPORTANT NOTE: The Matlab/Octave toolbox Large Time-Frequency Analysis (LTFAT, version 1.2.0 and above) must be installed to run the codes. This toolbox is freely available at Sourceforge.

    If you encounter any issue with the files, please do not hesitate to contact the authors.

     

  • Objective:

    In recent years, frames in signal processing applications have received more and more attention. Models, data, and operators must be discretized in order to function numerically. As a result, applications and algorithms always work with finite dimensional data. In the finite dimensional case, frames are equivalent to a spanning system. If reconstruction is wanted, frames are the only feasible generalization of bases. In contrast to bases, frames lose their linear independency. This project aims to investigate the properties of frames in finite dimensional spaces.

    Method:

    In this project, we will implement algorithms to work with frames in finite dimensional spaces. We will look at a way to "switch" between different frames, i.e. find a way to bijectively map between their coefficient spaces and provide the corresponding algorithm. This will be done by using the Cross-Gram matrix of the two involved frames. This matrix is a canonical extension of the basic transformation matrix used for orthonormal bases (ONB). The properties of the Gram matrix use a frame and its dual. We will investigate a criterion for finite dimensional spaces using frames. In particular, a space is finite dimensional if and only if Σ||gk||2 < ∞.

    Application:

    Any analysis / synthesis system that allows perfect reconstruction is equivalent to a frame in its discrete version. This can be applied to Gabor, wavelet, or any other such system (e.g. a Gamma tone filter bank).

    Publications

  • Objective:

    Gabor multipliers are an efficient tool for time-variant filtering. They are used implicitly in many engineering applications in signal processing. For these operators, the result of a Gabor transform (the sampled version of the Short Time Fourier Transform) is multiplied by a fixed function (called the time-frequency mask or symbol). Then the result is synthesized.

    Other transforms beyond the Gabor transform, for example the wavelet transform, are more suitable for certain applications. The concept of multipliers can easily be extended to these transforms. More precisely, the concept of multipliers can be applied to general frames without any further structure. This results in the introduction of operators called frame multipliers, which will be investigated in detail in this project in order to precisely define their mathematical properties and optimize their use in applications.

    Method:

    The problem will be approached using modern frame theory, functional analysis, numeric tools, and linear algebra tools. Systematic numeric experiments will be conducted to observe the different properties of frame multipliers. This observations will support the analytical formulation and demonstration of these properties.

    The following topics will be investigated in the project:

    • Eigenvalues and eigenvectors of frame multipliers
    • Invertibility, injectivity, and surjectivity of frame multipliers
    • Reproducing kernel invariance
    • Generalization of multipliers to Banach frames and p-frames
    • Connection of frame multipliers to weighted frames
    • Discretization and implementation of frame multipliers
    • Best approximation of operators by frame multipliers and identification of frame multipliers

    Application:

    The applications of frame multipliers in signal processing are numerous and include any application requiring time-variant filtering. Some applications of frame multipliers will be investigated further in the following parallel projects:

    • Mathematical Modeling of Auditory Time-Frequency Masking Functions
    • Improvement of Head-Related Transfer Function Measurements
    • Advanced Method of Sound Absorption Measurements

    Publications:

    • P. Balazs, "Matrix Representation of Bounded Linear Operators By Bessel Sequences, Frames and Riesz Sequence", SampTA'09, 8th International Conference on Sampling and Applications, May 2009, Marseille, France 
    • P. Balazs, J.-P. Antoine, A. Grybos, "Weighted and Controlled Frames: Mutual relationship and first Numerical Properties", accepted for publication in International Journal of Wavelets, Multiresolution and Information Processing (2009), preprint
    • A. Rahimi, P. Balazs, "Multipliers for p-Bessel sequences in Banach spaces", submitted (2009)
    • D. Stoeva, P. Balazs, "Unconditional convergence and Invertibility of Multipliers", preprint (2009)
  • Scientific and Technological Cooperation with Macedonia 2016-18
    Project duration: 01.07.2016 – 30.06.2018

    The main aim of the project is to combine the research areas of Frame Theory and Generalized Asymptotic Analysis.

    Project partner institutions:
    Acoustics Research Institute (ARI), Austrian Academy of Sciences, Vienna, Austria
    Ss. Cyril and Methodius University, Skopje, The Former Yugoslav Republic of Macedonia

    Project members:
    Diana T. Stoeva (Project coordinator Austria), Peter Balazs, Nicki Holighaus, Zdenek Prusa
    Katerina Hadzi-Velkova Saneva (Project coordinator FYROM), Sanja Atanasova, Pavel Dimovski, Zoran Hadzi-Velkov, Bojan Prangoski, Biljana Stanoevska-Angelova, Daniel Velinov, Jasmina Veta Buralieva


    Project Workshops and Activities:

    1) Nov. 24-26, 2016, Ss. Cyril and Methodius University, Skopje

    Project Kickoff-workshop

    Program of the workshop

    2) Nov. 15-19, 2017, ARI, Vienna

    Research on project-related topics

    3) April 14-19, 2018, ARI, Vienna

    Research on project-related topics

    and

    ARI-Guest-Talk given at ARI on the 17th of April, 2018: Prof. Zoran Hadzi-Velkov, "The Emergence of Wireless Powered Communication Networks"

    4) May 25-30, Ss. Cyril and Methodius University, Skopje

    Research on project-related topics

    and

    Workshop "Women in mathematics in the Balkan region" (May 28 - May 29, Ss. Cyril and Methodius University, Skopje)

    5) June 14-18, Ss. Cyril and Methodius University, Skopje

    Research on project-related topics

    and

    Summer course "An Introduction to Frame Theory and the Large Time/Frequency Analysis Toolbox" (June 14-15), Lecturers: Diana Stoeva and Zdenek Prusa (from ARI)

    6) Mini-Symposium "Frame Theory and Asymptotic Analysis" organized at the European Women in Mathematics General Meeting 2018, Karl-Franzens-Universität Graz, Austria, 3-7 September 2018.

    Link to Conference website

    7) November 17-20, 2018, ARI, Vienna

    Work on project-related topics

     

     

     

  • Objective:

    Gabor multipliers are an efficient tool for time variant filtering used implicitly in many engineering applications in signal processing. For these operators, the result of a Gabor transform (the sampled version of the Short Time Fourier Transform) is multiplied by a fixed function, called the time-frequency mask or symbol. The result is then synthesized.

    While Gabor multipliers are widely and practically used, some of their theoretical properties are not well known. The goal of this project is to improve the mathematical knowledge about Gabor multipliers, in order to optimize their use in applications.

    Method:

    The problem will be approached using modern Gabor theory, harmonic analysis tools, and numeric tools. Formulation and demonstration of analytical statements will be conducted jointly with systematic numeric experiments to study the properties of Gabor multipliers.

    The following topics will be investigated in the project:

    • Eigenvalues and eigenvectors of Gabor multipliers and their localization
    • Invertibility and injectivity of Gabor multipliers
    • Reproducing kernel invariance
    • Connection of irregular Gabor multipliers and irregular frames of translates
    • Discretization and implementation of Gabor multipliers
    • Best approximation of operators by Gabor multipliers and identification of Gabor multipliers.

    Application:

    The applications of Gabor multipliers in signal processing are numerous, and include any application requiring time-variant filtering. Some applications of Gabor multipliers will be investigated further in the following parallel projects:

    • Mathematical Modeling of Auditory Time-Frequency Masking Functions
    • Improvement of Head-Related Transfer Function Measurements
    • Advanced Method of Sound Absorption Measurements

    Software:

    The implementation of a Gabor multiplier in the software system STx has already proceeded quite far, see Stx-Mulac.

    Publications:

    • Monika Dörfler and Bruno Torrésani, “Representation of operators in the time-frequency domain and generalized Gabor multipliers”, J. Fourier Anal. Appl., 2009 (in press)
    • Yohan Frutiger: "Multiplicateurs de Gabor pour les transformations sonores" (Gabor Multipliers for sound transformations) Master thesis under the supervision of R. Kronland-Martinet, June 2008 
    • F. Jaillet, P. Balazs, M. Dörfler and N. Engelputzeder, “On the Structure of the Phase around the Zeros of the Short-Time Fourier Transform”, NAG/DAGA 2009, International Conference on Acoustics, March 2009, Rotterdam, Nederland
    • F. Jaillet, P. Balazs and M. Dörfler, “Nonstationary Gabor Frames”, SampTA'09, 8th International Conference on Sampling and Applications, May 2009, Marseille, France
  • Basic Description:

    HASSIP is a Research Training Network funded by the European Commission within the Improving the Human Potential program. The aim of the HASSIP network is to develop research activities and systematic interactions in mathematical analysis and statistics that are directly connected to signal and image processing. Although the Acoustics Research Institute was not initially a partner of this network, P. Balazs became a fellow of this network through cooperation with the group NuHAG.

    Partners:

    • NuHAG, Faculty of Mathematics, University of Vienna
    • Groupe de Traitement du Signal, Laboratoire d'Analyse Topologie et Probabilités, LATP/ CMI, Université de Provence, Marseille
    • Modélisation, Synthèse et Contrôle des Signaux Sonores et Musicaux des LMA / CRNS Marseille
    • Unité de physique théorique et de physique mathématique – FYMA

    Subprojects:

    • Basic Properties of Bessel and Frame Multipliers: For Bessel sequences, the investigation of operators M = ∑ mk < f , ψk > is very natural and useful. The above M are Bessel multipliers. The goal of this project is to set the mathematical basis for this kind of operator.
    • Best Approximation of Matrices by Frame Multipliers: Finding the best approximation by multipliers of matrices that represent time-variant systems gives a way to find efficient algorithms to implement such operators. 

    Publications:

    • P. Balazs, "Hilbert-Schmidt Operators and Frames - Classification, Approximation by Multipliers and Algorithms" , International Journal of Wavelets, Multiresolution and Information Processing, Vol. 6, No. 2, pp. 315 - 330, March 2008, preprint, Codes and Pictures: here
    • P. Balazs, "Basic Definition and Properties of Bessel Multipliers", Journal of Mathematical Analysis and Applications, 325, 1: 571--585. (2007) doi:10.1016/j.jmaa.2006.02.012, preprint

    Project-completion:

    This project ended on 01.01.2009. Its completion allowed the sucessfull application for a 'High Potential'-Project of the WWTF, see MULAC.