Massive MIMO systems are shown to be a promising technology for next generations of wireless communication networks. The realization of the attractive merits promised by massive MIMO systems requires advanced linear precoding and receiving techniques in order to mitigate the interference in downlink and uplink transmissions. This work considers the precoder and receiver design in massive MIMO systems. We first consider the design of the linear precoder and receiver that maximize the minimum signal-to-interference-plus-noise ratio (SINR) subject to a given power constraint. The analysis is carried out under the asymptotic regime in which the number of the BS antennas and that of the users grow large with a bounded ratio. This allows us to leverage tools from random matrix theory in order to approximate the parameters of the optimal linear precoder and receiver by their deterministic approximations. Such a result is of valuable practical interest, as it provides a handier way to implement the optimal precoder and receiver. To reduce further the complexity, we propose to apply the truncated polynomial expansion (TPE) concept on a per-user basis to approximate the inverse of large matrices that appear on the expressions of 4 the optimal linear transceivers. Using tools from random matrix theory, we determine deterministic approximations of the SINR and the transmit power in the asymptotic regime. Then, the optimal per-user weight coefficients that solve the max-min SINR problem are derived. The simulation results show that the proposed precoder and receiver provide very close to optimal performance while reducing significantly the computational complexity. As a second part of this work, the TPE technique in a per-user basis is applied to the optimal linear precoding that minimizes the transmit power while satisfying a set of target SINR constraints. Due to the emerging research field of green cellular networks, such a problem is receiving increasing interest nowadays. Closed form expressions of the optimal parameters of the proposed low complexity precoding for power minimization are derived. Numerical results show that the proposed power minimization precoding approximates well the performance of the optimal linear precoding while being more practical for implementation.

Linear estimation is a fundamental branch of signal processing that deals with estimating the values of parameters from a corrupted measured data. Throughout the years, several optimization criteria have been used to achieve this task. The most astonishing attempt among theses is the linear least-squares. Although this criterion enjoyed a wide popularity in many areas due to its attractive properties, it appeared to suffer from some shortcomings. Alternative optimization criteria, as a result, have been proposed. These new criteria allowed, in one way or another, the incorporation of further prior information to the desired problem. Among theses alternative criteria is the regularized least-squares (RLS). In this thesis, we propose two new algorithms to find the regularization parameter for linear least-squares problems. In the constrained perturbation regularization algorithm (COPRA) for random matrices and COPRA for linear discrete ill-posed problems, an artificial perturbation matrix with a bounded norm is forced into the model matrix. This perturbation is introduced to enhance the singular value structure of the matrix. As a result, the new modified model is expected to provide a better stabilize substantial solution when used to estimate the original signal through minimizing the worst-case residual error function. Unlike many other regularization algorithms that go in search of minimizing the estimated data error, the two new proposed algorithms are developed mainly to select the artifcial perturbation bound and the regularization parameter in a way that approximately minimizes the mean-squared error (MSE) between the original signal and its estimate under various conditions. The first proposed COPRA method is developed mainly to estimate the regularization parameter when the measurement matrix is complex Gaussian, with centered unit variance (standard), and independent and identically distributed (i.i.d.) entries. Furthermore, the second proposed COPRA method deals with discrete ill-posed problems when the singular values of the linear transformation matrix are decaying very fast to a significantly small value. For the both proposed algorithms, the regularization parameter is obtained as a solution of a non-linear characteristic equation. We provide a details study for the general properties of these functions and address the existence and uniqueness of the root. To demonstrate the performance of the derivations, the first proposed COPRA method is applied to estimate different signals with various characteristics, while the second proposed COPRA method is applied to a large set of different real-world discrete ill-posed problems. Simulation results demonstrate that the two proposed methods outperform a set of benchmark regularization algorithms in most cases. In addition, the algorithms are also shown to have the lowest run time.

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Testing the equality between two high-dimensional covariance matrices is challenging. As the most efficient way to measure evidential discrepancies in observed data, the likelihood ratio test is expected to be powerful when the null hypothesis is violated. However, when the data dimensionality becomes large and potentially exceeds the sample size by a substantial margin, likelihood ratio based approaches face practical and theoretical challenges. To solve this problem, this study proposes a method by which we first randomly project the original high-dimensional data into lower-dimensional space, and then apply the corrected likelihood ratio tests developed with random matrix theory. We show that testing with a single random projection is consistent under the null hypothesis. Through evaluating the power function, which is challenging in this context, we provide evidence that the test with a single random projection based on a random projection matrix with reasonable column sizes is more powerful when the two covariance matrices are unequal but component-wise discrepancy could be small -- a weak and dense signal setting. To more efficiently utilize this data information, we propose combined tests from multiple random projections from the class of meta-analyses. We establish the foundation of the combined tests from our theoretical analysis that the p-values from multiple random projections are asymptotically independent in the high-dimensional covariance matrices testing problem. Then, we show that combined tests from multiple random projections are consistent under the null hypothesis. In addition, our theory presents the merit of certain meta-analysis approaches over testing with a single random projection. Numerical evaluation of the power function of the combined tests from multiple random projections is also provided based on numerical evaluation of power function of testing with a single random projection. Extensive simulations and two real genetic data analyses confirm the merits and potential applications of our test. / Statistics

[ES] Durante las primeras décadas del siglo pasado se estudiaron las inversas generalizadas que hoy en día se conocen como inversas generalizadas clásicas. Entre ellas cabe mencionar la inversa de Moore-Penrose (1955) y la inversa de Drazin (1958). Mientras que la inversa de Moore-Penrose se definió originalmente para matrices complejas rectangulares, la inversa de Drazin fue tratada, en un primer momento, únicamente para matrices cuadradas. Más tarde, en 1980, Cline y Greville realizaron la extensión del caso cuadrado al caso rectangular, mediante la consideración de una matriz de ponderación rectangular. Diferentes propiedades, caracterizaciones y aplicaciones fueron obtenidas para estos tipos de inversas generalizadas hasta finales del siglo pasado. En la última década, han aparecido nuevas nociones de inversas generalizadas. La primera de ellas fue la inversa core, introducida en el año 2010 por los autores Baksalary y Trenkler. La misma tuvo una amplia repercusión en la comunidad matemática debido a la sencillez de su definición, a su aplicación en la resolución de algunos sistemas lineales con restricciones que surgen en la teoría de redes eléctricas y también por su conexión con la inversa de Bott- Duffin. Muchos trabajos de investigación han surgido a partir de la inversa core, incluyendo sus extensiones a conjuntos más generales como el álgebra de operadores lineales acotados sobre espacios de Hilbert y/o al ámbito de anillos abstractos. El objetivo principal de esta tesis doctoral es definir y estudiar en profundidad una nueva inversa generalizada para matrices rectangulares, llamada inversa inversa de grupo débil ponderada, la cual extiende al caso rectangular la inversa de grupo débil recientemente definida (para el caso cuadrado) por Wang y Chen. También se considera un amplio estudio de la inversa core-EP ponderada definida por Ferreyra, Levis y Thome en el año 2018, y que extiende al caso rectangular inversa core-EP introducida por Manjunatha-Prasad y Mohana en el año 2014. Para ambas inversas generalizadas se obtienen nuevas propiedades, representaciones, caracterizaciones como así también su relación con otras inversas conocidas en la literatura. Además, se presentan dos algoritmos que permiten realizar un cálculo efectivo de las mismas. / [CA] Durant les primeres dècades del segle passat es van estudiar les inverses generalitzades que hui dia es coneixen com a inverses generalitzades clàssiques. Entre elles cal esmentar la inversa de Moore-Penrose (1955) i la inversa de Drazin (1958). Mentre que la inversa de Moore-Penrose es va definir originalment per a matrius complexes rectangulars, la inversa de Drazin va ser tractada, en un primer moment, únicament per a matrius quadrades. Més tard, en 1980, Cline i Greville van realitzar l'extensió del cas quadrat al cas rectangular, mitjançant la consideració d'una matriu de ponderació rectangular. Diferents propietats, caracteritzacions i aplicacions van ser obtingudes per a aquests tipus d'inverses generalitzades fins a finals del segle passat. En l'última dècada, han aparegut noves nocions d'inverses generalitzades. La primera d'elles va ser la inversa core, introduïda l'any 2010 pels autors Baksalary i Trenkler. La mateixa va tindre una àmplia repercussió en la comunitat matemàtica a causa de la senzillesa de la seua definició, a la seua aplicació en la resolució d'alguns sistemes lineals amb restriccions que sorgeixen en la teoria de xarxes elèctriques i també per la seua connexió amb la inversa de Bott-Duffinn. Molts treballs de recerca han sorgit a partir de la inversa core, incloent les seues extensions a conjunts més generals com l'àlgebra d'operadors lineals delimitats sobre espais de Hilbert i/o a l'àmbit d'anells abstractes. L'objectiu principal d'aquesta tesi doctoral és definir i estudiar en profunditat una nova inversa generalitzada per a matrius rectangulars, anomenada inversa inversa de grup feble ponderada, la qual estén al cas rectangular la inversa de grup feble recentment definida (per al cas quadrat) per Wang i Chen. Tamb é es considera un ampli estudi de la inversa core-EP ponderada definida per Ferreyra, Levis i Thome l'any 2018, i que estén al cas rectangular inversa core-EP introduïda per Manjunatha-Prasad i Mohana l'any 2014. Per a totes dues inverses generalitzades s'obtenen noves propietats, representacions, caracteritzacions com així també la seua relació amb altres inverses conegudes en la literatura. A més, es presenten dos algorismes que permeten realitzar un càlcul efectiu d'aquestes. / [EN] Generalized inverses, known today as Classical Generalized Inverses, were studied during the first decades of the last century. Two important classical generalized inverses are the Moore-Penrose inverse (1955) and the Drazin inverse (1958). The Moore-Penrose inverse was originally defined for complex rectangular matrices. In turn, the Drazin inverse was studied, at first, only for square matrices. It was in 1980 when Cline and Greville extended the case of square matrices to the case of rectangular matrices by considering a weight rectangular matrix. Throughout the entire past century there appeared difierent properties, characterizations and applications of these types of generalized inverses. This last decade gave rise to new notions of generalized inverses. The first of these new notions is known as the core inverse. Core inverses were introduced in 2010 by Baksalary and Trenkler. Their work had a wide repercussion in the mathematical community due to the simplicity of its denition and its application in the solution of some linear systems with restrictions. The core inverse further gain in interest due to their connection to the Bott-Duffin inverse. There is a large body of work on the core inverse, including extensions to more general sets if such as the algebra of bounded linear operators on Hilbert spaces and/or abstract rings. The main goal of this thesis is to define and study in depth a new generalized inverse for rectangular matrices. This new inverse is called weighted weak group inverse (or weighted WG inverse). Weighted WG inverses extend weak group inverse, recently defined for the square case by Wang and Chen, to the rectangular case. We also consider an extensive study of the weighted core-EP inverse. The latter type of inverse was dened by Ferreyra, Levis, and Thome in 2018. This inverse extends the core-EP inverse introduced by Manjunatha- Prasad and Mohana in 2014 to the rectangular case. This thesis presents new properties, representations, characterizations, as well as their relation with other inverses known in the literature are obtained, for weighted WG inverses and weighted core-EP inverse. In addition, the thesis presents two algorithms that allow for an efiective computation weighted WG inverses and weighted core-EP inverse. / Orquera, V. (2022). La inversa core-EP y la inversa de grupo débil para matrices rectangulares [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/185227 / TESIS

No / Opportunistic secondary usage of underutilised radio spectrum is currently of great interest and the use of TV White Spaces (TVWS) has been considered for Long Term Evolution (LTE) broadband services. However, wireless microphones operating in TV bands pose a challenge to TVWS opportunistic access. Efficient and proactive spectrum sensing could prevent harmful interference between collocated devices, but existing spectrum sensing schemes such as energy detection and schemes based on Random Matrix Theory (RMT) have performance limitations. We propose a new blind spectrum sensing scheme with higher performance based on RMT supported by a new formula for the estimation of noise variance. The performance of the proposed scheme has been evaluated through extensive simulations on wireless microphone signals. The proposed scheme has also been compared to energy detection schemes, and shows higher performance in terms of the probability of false alarm (Pfa) and probability of detection (Pd).

In this Bachelor thesis project, the Lindblad master equation is derived, both as the most general way of modeling interaction with an environment that lacks memory, and through microscopic derivations focused on assumptions about the way the system interacts with its environment (weak-coupling, Born-Markov and rotating wave approximations). It is then applied to a two-level system (qubit).

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