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Redundancy of finite frames with a discussion on Gabor frames = /

Felix Ackermann

Redundancy of finite frames with a discussion on Gabor frames = /.

Rel. Bengt Patrik Martin Wahlberg. Politecnico di Torino, Corso di laurea magistrale in Ingegneria Matematica, 2022

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Frames represent an important theoretical concept for signal processing. A frame is a collection of vectors satisfying the frame inequality, i.e. a relaxed form of Parseval’s identity for which the vectors involved do not need to form an orthonormal basis. Frames are used in order to provide a redundant representation of a vector (signal) in terms of coefficients associated to the frame’s elements. The goal of this thesis is to quantify redundancy for frames. Therefore we present a combinatorial- and an analytical redundancy measure. Both yield information on the maximal number of spanning sets and the minimal number of linearly independent sets one can partition the frame into, though in general the two measures are not equivalent. For the combinatorial measure we show a proof of the Rado-Horn theorem and two results basing upon it. We then examine a well-known example: the Fourier frame. Instead the analytical measure is characterized in terms of properties of the redundancy function. It is known that these two measures coincide for an equal norm Parseval frame, but when relaxing equal norm, this does not hold anymore. Exploiting the properties of the redundancy function, we show that the two redundancy measures still coincide if we consider Parseval frames with some additional property. Finally, we examine Gabor frames. The frame elements are modulated translates of a window function. We characterize the frame both in general terms and with respect to its redundancy.

Relators: Bengt Patrik Martin Wahlberg
Academic year: 2021/22
Publication type: Electronic
Number of Pages: 55
Corso di laurea: Corso di laurea magistrale in Ingegneria Matematica
Classe di laurea: New organization > Master science > LM-44 - MATHEMATICAL MODELLING FOR ENGINEERING
Aziende collaboratrici: UNSPECIFIED
URI: http://webthesis.biblio.polito.it/id/eprint/23095
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