Institut für Schallforschung der Österreichischen Akademie der Wissenschaften: A-1010 Wien; Reichsratsrasse 17. Tel. +43 1/4277-29500; Fax +43 1/4277-9296;

Institut für Schallforschung der Österreichischen Akademie der Wissenschaften: A-1010 Wien; Reichsratsrasse 17. Tel. +43 1/4277-29500; Fax +43 1/4277-9296; email: [email protected]; http://www.kfs.oeaw.ac.at

Frames and Frame Multiplier

Peter Balazs

Acoustics Research Institute, Austrian Academy of Science, Vienna- http://www.kfs.oeaw.ac.at- [email protected]

Groupe de Traitement du Signal, LATP, CMI, Marseille - http://www.cmi.univ-mrs.fr/%7Emolla/GTS/index.htm- [email protected]

in cooperation with:

NuHAG, Mathematics Institute, University Vienna - http://www.univie.ac.at/NuHAG/- [email protected]


SHORT TIME FOURIER TRANSFORMATION

Short Time Fourier transformation, STFT


INTERPRETATION OF THE PHASE VOCODER (STFT)

Fourier-transform

Windowed signal

mathematically equivalent

Filterbank

A channel of the filterbank

Gabor projection

Shifted atoms


overlap addoscillator bankprojection on dual atoms

SYNTHESIS

REDUNDANCYSTFT

PERCEPTUAL REDUNDANCY

masking

Basis or frame

SYNTHESIS AND REDUNDANCY

Gabor frame :


Discrete Representation

Sampling



Modification of Time Frequency or Time Scale Parameters

e.g. Denoising, masking, edge detection ......



Coding /Decoding e.g. in MP3



Coding / Decoding or Encryption / Decryption



Represent a „real world analog thing“ by discrete coefficents


Orthonormal Bases in Hilbert Spaces

Hilbert space Definition, examples

Bases & ONBsComplete, basis, ONS, ONB

Plancherel:

Properties of an ONB

Examples:

SVD, PCA, ICA


Problems with bases

Error - robustness

Properties of the elements

Perturbation of basis elements

Flexibility:

Uniqueness <-> redundancy

Gabor: Balian-Low

Wavelet


Frames

Definition

Properties, frame operator S

Dual frame

Canonical

Tight frame, canonical tight

Finite setting, Algorithms for S-1


Results for frames

Images of ONBs


Results for frames

Representation with frames


Gabor frames

Definition

Existence, Wiener space, Walnut theorem

Duality

Balian Low


Gabor multiplier


Masking Filter

Best representation for masking threshold resp. auditory filters?

Gabor Multiplier for irregular grid (bark)

Time Frequency localization operator (continuous)

Wavelet Multiplier

Multiple Gabor Extension Multiplier

(Or something completely different:) Auditory Filter Multiplier ?

Gamma-tone Filter Multiplier?

Spectrum and relevance threshold


Frame multiplier

Definition


REFERENCES:

[1] O. Christensen, An Introduction To Frames And Riesz Bases , Birkhäuser Boston (2003)

[2] M. Dörfler, Gabor Analysis for a Class of Signals called Music, PhD thesis Vienna (2002)

[3] H.G. Feichtinger, T. Stromer, Gabor Analysis and Algorithms, Theory and Applications, Birkhäuser Boston (1998)

[4] H. G. Feichtinger, T. Strohmer, Advances in Gabor Analysis , Birkhäuser Bosten (2003)

[5] K. Gröchenig, Foundations of Time-Frequency Analysis, Birkhäuser Boston (2001)

Documents

Institut für Schallforschung der Österreichischen Akademie der Wissenschaften: A-1010 Wien; Reichsratsrasse 17. Tel. +43 1/4277-29500; Fax +43 1/4277-9296;