Location information is of extreme importance in every walk of life ranging from commercial applications such as location based advertising and location aware next generation communication networks such as the 5G networks to security based applications like threat localization and E-911 calling. In indoor and dense urban environments plagued by multipath effects there is usually a Non Line of Sight (NLOS) scenario preventing GPS based localization. Wireless localization using sensor networks provides a cost effective and accurate solution to the wireless source localization problem. Certain sensor geometries show significantly poor performance even in low noise scenarios when triangulation based localization methods are used. This brings the need for the design of an optimum sensor placement scheme for better performance in the source localization process. The optimum sensor placement is the one that optimizes the underlying Fisher Information Matrix(FIM) . This thesis will present a class of canonical optimum sensor placements that produce the optimum FIM for N-dimensional source localization N greater than or equal to 2 for a case where the source location has a radially symmetric probability density function within a N-dimensional sphere and the sensors are all on or outside the surface of a concentric outer N-dimensional sphere. While the canonical solution that we designed for the 2D problem represents optimum spherical codes, the study of 3 or higher dimensional design provides great insights into the design of measurement matrices with equal norm columns that have the smallest possible condition number. Such matrices are of importance in compressed sensing based applications. This thesis also presents an optimum sensing matrix design for energy efficient source localization in 2D. Specifically, the results relate to the worst case scenario when the minimum number of sensors are active in the sensor network. We also propose a distributed control law that guides the motion of the sensors on the circumference of the outer circle so that achieve the optimum sensor placement with minimum communication overhead. The design of equal norm column sensing matrices has a variety of other applications apart from the optimum sensor placement for N-dimensional source localization. One such application is fourier analysis in Magnetic Resonance Imaging (MRI). Depending on the method used to acquire the MR image, one can choose an appropriate transform domain that transforms the MR image into a sparse image that is compressible. Some such transform domains include Wavelet Transform and Fourier Transform. The inherent sparsity of the MR images in an appropriately chosen transform domain, motivates one of the objectives of this thesis which is to provide a method for designing a compressive sensing measurement matrix by choosing a subset of rows from the Discrete Fourier Transform (DFT) matrix. This thesis uses the spark of the matrix as the design criterion. The spark of a matrix is defined as the smallest number of linearly dependent columns of the matrix. The objective is to select a subset of rows from the DFT matrix in order to achieve maximum spark. The design procedure leads us to an interest study of coprime conditions on the row indices chosen with the size of the DFT matrix.

The exploration of oil and natural gas in offshore fields has motivated advanced researches about the environmental forces in the oceans. The waves, in particular, have been measured using different techniques, as meteorological buoys, with recent works proposing motion-based estimations procedures using the vessel, or a floating facility, in analogy with the buoys, as a wave sensor. Even though this approach has a number of benefits, the vessels, as dynamic systems, have a cut-off frequency that degrades the estimation of high-frequency waves, which are important for non-linear drift effects predictions. In order to solve this problem, it is proposed the incorporation of wave-probes - gauges used to measure the wave elevation in a point - installed on the hull of the vessel, based on literature suggestions and simple analytical arguments, using the Bayesian statistics as the standing point of a more complete estimation algorithm. In order to incorporate the measurements of the wave-probes, an extended linear model is proposed, showing that only corrections for the vertical motions of the vessel are necessary. The ideal installation positions of the wave-probes are defined using as base the utility Bayesian optimal design of experiments, which is shown to guarantee an upper bound for other optimal criteria, with the \'Elbow Criterion\" defining the optimal number of sensors to be employed. Based on the previous solutions, other proposals are made: a heuristic to solve the optimal sensor placement problem and an optimal prior exploring the probabilistic nature of the algorithm. Finally, all the proposals are tested numerically and experimentally, with a vessel model in a towing tank, concluding that the addition of the wave-probes is able to improve not only the estimation of high-frequency waves, but also the estimation over a large range of frequencies. For unimodal seas with intermediate draft, the addition of just one wave-probe reaches approximately a 37%-55% improvement in the energy parameter estimations - HS and TP; the addition of two or more probes reaches approximately a 62%-65% improvement in the same parameters estimations; the addition of four probes achieved the best cost benefit for mean direction estimation; and the addition of six probes is shown to be the recommendation for the best high-order directional estimation in the entire range of the spectrum. / A prospecção de óleo e gás natural em campos offshore tem motivado pesquisas avançadas sobre as forças ambientais em oceanos. As ondas, em particular, têm sido medidas através de diferentes técnicas, como boias meteorológicas, com trabalhos recentes propondo técnicas baseadas em movimento para que os navios, em analogia com as boias, possam ser usados como sensores de onda. Apesar desse método ter uma série de vantagens, os navios, como sistemas dinâmicos, têm uma frequência de corte que dificulta a estimação de ondas de altas frequências, que são importantes para a previsão de efeitos de deriva não-lineares. Para resolver esse problema, sugere-se a adição de wave-probes instalados no costado da embarcação, usando como justificativas sugestões da literatura e simples argumentos analíticos, com estatística Bayesiana como fundamentação para um algoritmo de estimação mais completo. Para que as medidas dos wave-probes possam ser incorporadas, um modelo linear estendido é proposto, mostrando que apenas correções para os movimentos verticais do navio são necessárias. A posição ideal de instalação dos wave-probes é definida usando como base o projeto ótimo de experimentos Bayesianos por utilidade, mostrando que o mesmo garante o limite superior de outros critérios de optimalidade, com o \"critério cotovelo\" definindo o número ótimo de sensores a serem usados. Com base nas soluções anteriores, outras propostas são feitas: uma heurística para resolver o problema de posicionamento ótimo dos sensores e uma priori ótima, explorando a natureza probabilística do algoritmo. Ao final, todas as propostas são testadas numericamente e experimentalmente, utilizando um modelo em escala em um tanque de provas, concluindo que a adição de wave-probes é capaz de melhorar não só a estimação de ondas em alta-frequência, mas também a estimação em uma ampla gama de frequências. Para mares unimodais, com calado intermediário, a adição de apenas um sensor alcançou uma melhoria de aproximadamente 37-55% na estimação dos parâmetros relacionados à energia - HS e TP; a adição de dois ou mais sensores alcançou melhorias de 62-65% na estimação de tais parâmetros; a adição de quatro sensores alcançou o melhor custo benefício para estimação da direção média; e a adição de seis sensores se mostrou ideal para estimação de ordem elevada do espectro direcional de energia.