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The interest in wireless communications among consumers has exploded since the introduction of the "3G" cell phone standards. One reason for their success is the increasingly higher data rates achievable through the networks. A further increase in data rates is possible through the use of multiple antennas at either or both sides of the wireless links. Precoding and receive filtering using matrices obtained from a singular value decomposition (SVD) of the channel matrix is a transmission strategy for achieving the channel capacity of a deterministic narrowband multiple-input multiple-output (MIMO) communications channel. When signalling over wideband channels using orthogonal frequency-division multiplexing (OFDM), an SVD must be performed for every sub-carrier. As the number of sub-carriers of this traditional approach grow large, so does the computational load. It is therefore interesting to study alternate means for obtaining the decomposition. A wideband MIMO channel can be modeled as a matrix filter with a finite impulse response, represented by a polynomial matrix. This thesis is concerned with investigating algorithms which decompose the polynomial channel matrix directly. The resulting decomposition factors can then be used to obtain the sub-carrier based precoding and receive filtering matrices. Existing approximative polynomial matrix QR and singular value decomposition algorithms were modified, and studied in terms of decomposition quality and computational complexity. The decomposition algorithms were shown to give decompositions of good quality, but if the goal is to obtain precoding and receive filtering matrices, the computational load is prohibitive for channels with long impulse responses. Two algorithms for performing exact rational decompositions (QRD/SVD) of polynomial matrices were proposed and analyzed. Although they for simple cases resulted in excellent decompositions, issues with numerical stability of a spectral factorization step renders the algorithms in their current form purposeless. For a MIMO channel with exponentially decaying power-delay profile, the sum rates achieved by employing the filters given from the approximative polynomial SVD algorithm were compared to the channel capacity. It was shown that if the symbol streams were decoded independently, as done in the traditional approach, the sum rates were sensitive to errors in the decomposition. A receiver with a spatially joint detector achieved sum rates close to the channel capacity, but with such a receiver the low complexity detector set-up of the traditional approach is lost. Summarizing, this thesis has shown that a wideband MIMO channel can be diagonalized in space and frequency using OFDM in conjunction with an approximative polynomial SVD algorithm. In order to reach sum rates close to the capacity of a simple channel, the computational load becomes restraining compared to the traditional approach, for channels with long impulse responses.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Métodos estocásticos oferecem uma poderosa ferramenta para a execução da compressão de dados e decomposições de matrizes. O método estocástico para decomposição de matrizes estudado utiliza amostragem aleatória para identificar um subespaço que captura a imagem de uma matriz de forma aproximada, preservando uma parte de sua informação essencial. Estas aproximações compactam a informação possibilitando a resolução de problemas práticos de maneira eficiente. Nesta dissertação é calculada uma decomposição em valores singulares (SVD) utilizando técnicas estocásticas. Esta SVD aleatória é empregada na tarefa de reconhecimento de faces. O reconhecimento de faces funciona de forma a projetar imagens de faces sobre um espaço de características que melhor descreve a variação de imagens de faces conhecidas. Estas características significantes são conhecidas como autofaces, pois são os autovetores de uma matriz associada a um conjunto de faces. Essa projeção caracteriza aproximadamente a face de um indivíduo por uma soma ponderada das autofaces características. Assim, a tarefa de reconhecimento de uma nova face consiste em comparar os pesos de sua projeção com os pesos da projeção de indivíduos conhecidos. A análise de componentes principais (PCA) é um método muito utilizado para determinar as autofaces características, este fornece as autofaces que representam maior variabilidade de informação de um conjunto de faces. Nesta dissertação verificamos a qualidade das autofaces obtidas pela SVD aleatória (que são os vetores singulares à esquerda de uma matriz contendo as imagens) por comparação de similaridade com as autofaces obtidas pela PCA. Para tanto, foram utilizados dois bancos de imagens, com tamanhos diferentes, e aplicadas diversas amostragens aleatórias sobre a matriz contendo as imagens. / Stochastic methods offer a powerful tool for performing data compression and decomposition of matrices. These methods use random sampling to identify a subspace that captures the range of a matrix in an approximate way, preserving a part of its essential information. These approaches compress the information enabling the resolution of practical problems efficiently. This work computes a singular value decomposition (SVD) of a matrix using stochastic techniques. This random SVD is employed in the task of face recognition. The face recognition is based on the projection of images of faces on a feature space that best describes the variation of known image faces. These features are known as eigenfaces because they are the eigenvectors of a matrix constructed from a set of faces. This projection characterizes an individual face by a weighted sum of eigenfaces. The task of recognizing a new face is to compare the weights of its projection with the projection of the weights of known individuals. The principal components analysis (PCA) is a widely used method for determining the eigenfaces. This provides the greatest variability eigenfaces representing information from a set of faces. In this dissertation we discuss the quality of eigenfaces obtained by a random SVD (which are the left singular vectors of a matrix containing the images) by comparing the similarity with eigenfaces obtained by PCA. We use two databases of images, with different sizes and various random sampling applied on the matrix containing the images.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Métodos estocásticos oferecem uma poderosa ferramenta para a execução da compressão de dados e decomposições de matrizes. O método estocástico para decomposição de matrizes estudado utiliza amostragem aleatória para identificar um subespaço que captura a imagem de uma matriz de forma aproximada, preservando uma parte de sua informação essencial. Estas aproximações compactam a informação possibilitando a resolução de problemas práticos de maneira eficiente. Nesta dissertação é calculada uma decomposição em valores singulares (SVD) utilizando técnicas estocásticas. Esta SVD aleatória é empregada na tarefa de reconhecimento de faces. O reconhecimento de faces funciona de forma a projetar imagens de faces sobre um espaço de características que melhor descreve a variação de imagens de faces conhecidas. Estas características significantes são conhecidas como autofaces, pois são os autovetores de uma matriz associada a um conjunto de faces. Essa projeção caracteriza aproximadamente a face de um indivíduo por uma soma ponderada das autofaces características. Assim, a tarefa de reconhecimento de uma nova face consiste em comparar os pesos de sua projeção com os pesos da projeção de indivíduos conhecidos. A análise de componentes principais (PCA) é um método muito utilizado para determinar as autofaces características, este fornece as autofaces que representam maior variabilidade de informação de um conjunto de faces. Nesta dissertação verificamos a qualidade das autofaces obtidas pela SVD aleatória (que são os vetores singulares à esquerda de uma matriz contendo as imagens) por comparação de similaridade com as autofaces obtidas pela PCA. Para tanto, foram utilizados dois bancos de imagens, com tamanhos diferentes, e aplicadas diversas amostragens aleatórias sobre a matriz contendo as imagens. / Stochastic methods offer a powerful tool for performing data compression and decomposition of matrices. These methods use random sampling to identify a subspace that captures the range of a matrix in an approximate way, preserving a part of its essential information. These approaches compress the information enabling the resolution of practical problems efficiently. This work computes a singular value decomposition (SVD) of a matrix using stochastic techniques. This random SVD is employed in the task of face recognition. The face recognition is based on the projection of images of faces on a feature space that best describes the variation of known image faces. These features are known as eigenfaces because they are the eigenvectors of a matrix constructed from a set of faces. This projection characterizes an individual face by a weighted sum of eigenfaces. The task of recognizing a new face is to compare the weights of its projection with the projection of the weights of known individuals. The principal components analysis (PCA) is a widely used method for determining the eigenfaces. This provides the greatest variability eigenfaces representing information from a set of faces. In this dissertation we discuss the quality of eigenfaces obtained by a random SVD (which are the left singular vectors of a matrix containing the images) by comparing the similarity with eigenfaces obtained by PCA. We use two databases of images, with different sizes and various random sampling applied on the matrix containing the images.