### Basic Description:

Signal processing has entered into today's life on a broad range, from mobile phones, UMTS, xDSL, and digital television to scientific research such as psychoacoustic modeling, acoustic measurements, and hearing prosthesis. Such applications often use time-invariant filters by applying the Fourier transform to calculate the complex spectrum. The spectrum is then multiplied by a function, the so-called transfer function. Such an operator can therefore be called a Fourier multiplier. Real life signals are seldom found to be stationary. Quasi-stationarity and fast-time variance characterize the majority of speech signals, transients in music, or environmental sounds, and therefore imply the need for non-stationary system models. Considerable progress can be achieved by reaching beyond traditional Fourier techniques and improving current time-variant filter concepts through application of the basic mathematical concepts of frame multipliers.

Several transforms, such as the Gabor transform (the sampled version of the Short-Time Fourier Transformation), the wavelet transform, and the Bark, Mel, and Gamma tone filter banks are already in use in a large number of signal processing applications. Generalization of these techniques can be obtained via the mathematical frame theory. The advantage of introducing the frame theory consists particularly in the interpretability of filter and analysis coefficients in terms of frequency and time localization, as opposed to techniques based on orthonormal bases.

One possibility to construct time-variant filters exists through the use of Gabor multipliers. For these operators the result of a Gabor transform is multiplied by a given function, called the time-frequency mask or symbol, followed by re-synthesis. These operators are already used implicitly in engineering applications, and have been investigated as Gabor filters in the fields of mathematics and signal processing theory. If alternative transforms are used, the concept of multipliers can be extended appropriately. So, for example, the concept of wavelet multipliers could be investigated for a wavelet transform.

Different kinds of applications call for different frames. Multipliers can be generalized to the abstract level of frames without any further structure. This concept will be further investigated in this project. Its feasibility will be evaluated in acoustic applications using special cases of Gabor and wavelet systems.

The project goal is to study both the mathematical theory of frame multipliers and their application among selected problems in acoustics. The project is divided into the following subprojects:

### Theory of Multipliers:

- General Frame Multiplier Theory
- Analytic and Numeric Properties of Gabor Multipliers
- Analytic and Numeric Properties of Wavelet Multipliers

### Application of Multipliers:

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

### Partners:

- H.G. Feichtinger et al., NuHAG, Faculty of Mathematics, University of Vienna
- R. Kronland-Martinet et al., Modélisation, Synthèse et Contrôle des Signaux Sonores et Musicaux of the LMA / CNRS Marseille
- B. Torrésani et al., LATP Université de Provence / CNRS Marseille
- J.P. Antoine et al., FYMA Université Catholique de Louvain

### Publications:

- P. Balazs, J.-P. Antoine, A. Gryboś, "Weighted and Controlled Frames: Mutual relationship and first Numerical Properties", accepted for publication in International Journal of Wavelets, Multiresolution and Information Processing (2009), preprint
- 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
- 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)
- 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
- P. Balazs, B. Laback, G. Eckel, W. Deutsch, "Introducing Time-Frequency Sparsity by Removing Perceptually Irrelevant Components Using a Simple Model of Simultaneous Masking", IEEE Transactions on Audio, Speech and Language Processing (2009), in press
- B. Laback, P. Balazs, G. Toupin, T. Necciari, S. Savel, S. Meunier, S. Ystad and R. Kronland-Martinet, "Additivity of auditory masking using Gaussian-shaped tones", Acoustics'08, Paris, 29.06.-04.07.2008 (03.07.2008)
- B. Laback, P. Balazs, T. Necciari, S. Savel, S. Ystad, S. Meunier and R. Kronland-Martinet, "Additivity of auditory masking for Gaussian-shaped tone pulses", preprint
- Anaïk Olivero: "Expérimentation des multiplicateurs temps-échelle" (On the time-scale multipliers) Master thesis under the supervision of R. Kronland-Martinet and B. Torrésani, June 2008