Forschung

FramMulAc: General Frame Multiplier Theory

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)