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000151505 001__ 151505 000151505 005__ 20190509132331.0 000151505 0247_ $$2doi$$a10.5075/epfl-thesis-4871 000151505 02470 $$2urn$$aurn:nbn:ch:bel-epfl-thesis4871-1 000151505 02471 $$2nebis$$a6153916 000151505 037__ $$aTHESIS 000151505 041__ $$aeng 000151505 088__ $$a4871 000151505 245__ $$bA Multidimensional Fourier Approach$$aSignal Processing in Space and Time 000151505 269__ $$a2010 000151505 260__ $$bEPFL$$c2010$$aLausanne 000151505 300__ $$a174 000151505 336__ $$aTheses 000151505 520__ $$aSound waves propagate through space and time by transference of energy between the particles in the medium, which vibrate according to the oscillation patterns of the waves. These vibrations can be captured by a microphone and translated into a digital signal, representing the amplitude of the sound pressure as a function of time. The signal obtained by the microphone characterizes the time-domain behavior of the acoustic wave field, but has no information related to the spatial domain. The spatial information can be obtained by measuring the vibrations with an array of microphones distributed at multiple locations in space. This allows the amplitude of the sound pressure to be represented not only as a function of time but also as a function of space. The use of microphone arrays creates a new class of signals that is somewhat unfamiliar to Fourier analysis. Current paradigms try to circumvent the problem by treating the microphone signals as multiple "cooperating" signals, and applying the Fourier analysis to each signal individually. Conceptually, however, this is not faithful to the mathematics of the wave equation, which expresses the acoustic wave field as a single function of space and time, and not as multiple functions of time. The goal of this thesis is to provide a formulation of Fourier theory that treats the wave field as a single function of space and time, and allows it to be processed as a multidimensional signal using the theory of digital signal processing (DSP). We base this on a physical principle known as the Huygens principle, which essentially says that the wave field can be sampled at the surface of a given region in space and subsequently reconstructed in the same region, using only the samples obtained at the surface. To translate this into DSP language, we show that the Huygens principle can be expressed as a linear system that is both space- and time-invariant, and can be formulated as a convolution operation. If the input signal is transformed into the spatio-temporal Fourier domain, the system can also be analyzed according to its frequency response. In the first half of the thesis, we derive theoretical results that express the 4-D Fourier transform of the wave field as a function of the parameters of the scene, such as the number of sources and their locations, the source signals, and the geometry of the microphone array. We also show that the wave field can be effectively analyzed on a small scale using what we call the space/time-frequency representation space, consisting of a Gabor representation across the spatio-temporal manifold defined by the microphone array. These results are obtained by treating the signals as continuous functions of space and time. The second half of the thesis is dedicated to processing the wave field in discrete space and time, using Nyquist sampling theory and multidimensional filter banks theory. In particular, we show examples of orthogonal filter banks that effectively represent the wave field in terms of its elementary components while satisfying the requirements of critical sampling and perfect reconstruction of the input. We discuss the architecture of such filter banks, and demonstrate their applicability in the context of real applications, such as spatial filtering and wave field coding. 000151505 6531_ $$aacoustic wave fields, array signal processing 000151505 6531_ $$amultidimensional Fourier analysis 000151505 6531_ $$aspace/time-frequency analysis 000151505 6531_ $$amultidimensional filter banks 000151505 6531_ $$adirectional filter banks 000151505 6531_ $$aspatial filtering 000151505 6531_ $$awave field coding 000151505 6531_ $$achamps d'ondes acoustiques 000151505 6531_ $$atraitement des signaux réseaux 000151505 6531_ $$aanalyse de Fourier multidimensionnelle 000151505 6531_ $$aanalyse espace/temps-fréquence 000151505 6531_ $$abancs de filtres multidimensionnels 000151505 6531_ $$abancs de filtres directionnels 000151505 6531_ $$afiltrage spatial 000151505 6531_ $$acodification du champ d'ondes 000151505 700__ $$aPinto, Francisco Pereira Correia 000151505 720_2 $$aVetterli, Martin$$edir.$$g107537$$0240184 000151505 8564_ $$uhttps://infoscience.epfl.ch/record/151505/files/EPFL_TH4871.pdf$$zTexte intégral / Full text$$s15546232$$yTexte intégral / Full text 000151505 909C0 $$xU10434$$0252056$$pLCAV 000151505 909CO $$pthesis-bn2018$$pDOI$$pIC$$ooai:infoscience.tind.io:151505$$qDOI2$$qGLOBAL_SET$$pthesis 000151505 918__ $$dEDIC2005-2015$$cISC$$aIC 000151505 919__ $$aLCAV1 000151505 920__ $$b2010 000151505 970__ $$a4871/THESES 000151505 973__ $$sPUBLISHED$$aEPFL 000151505 980__ $$aTHESIS