On the Distribution of Quadratic Expressions in Various Types of Random Vectors

Several approximations to the distribution of indefinite quadratic expressions in possibly singular Gaussian random vectors and ratios thereof are obtained in this dissertation. It is established that such quadratic expressions can be represented in their most general form as the difference of two positive definite quadratic forms plus a linear combination of Gaussian random variables. New advances on the distribution of quadratic expressions in elliptically contoured vectors, which are expressed as scalar mixtures of Gaussian vectors, are proposed as well. Certain distributional aspects of Hermitian quadratic expressions in complex Gaussian vectors are also investigated. Additionally, approximations to the distributions of quadratic forms in uniform, beta, exponential and gamma random variables as well as order statistics thereof are determined from their exact moments, for which explicit representations are derived. Closed form representations of the approximations to the density functions of the various types of quadratic expressions being considered herein are obtained by adjusting the base density functions associated with the quadratic forms appearing in the decompositions of the expressions by means of polynomials whose coefficients are determined from the moments of the target distributions. Quadratic forms being ubiquitous in Statistics, the proposed distributional results should prove eminently useful

Outage Probability and Outage-Based Robust Beamforming for MIMO Interference Channels with Imperfect Channel State Information

In this paper, the outage probability and outage-based beam design for multiple-input multiple-output (MIMO) interference channels are considered. First, closed-form expressions for the outage probability in MIMO interference channels are derived under the assumption of Gaussian-distributed channel state information (CSI) error, and the asymptotic behavior of the outage probability as a function of several system parameters is examined by using the Chernoff bound. It is shown that the outage probability decreases exponentially with respect to the quality of CSI measured by the inverse of the mean square error of CSI. Second, based on the derived outage probability expressions, an iterative beam design algorithm for maximizing the sum outage rate is proposed. Numerical results show that the proposed beam design algorithm yields better sum outage rate performance than conventional algorithms such as interference alignment developed under the assumption of perfect CSI.Comment: 41 pages, 14 figures. accepted to IEEE Transactions on Wireless Communication

On the distribution of indefinite quadratic forms in Gaussian random variables

In this work, we propose a transparent approach to evaluating the CDF of indefinite quadratic forms in Gaussian random variables and ratios of such forms. This quantity appears in the analysis of different receivers in communication systems and in various applications in signal processing. Instead of attempting to find the pdf of this quantity as is the case in many papers in literature, we focus on finding the CDF. The basic trick that we implement is to replace inequalities that appear in the CDF calculations with the unit step function and replace the latter with its Fourier transform. This produces a multi-dimensional integral that can be evaluated using complex integration. We show how our approach extends to nonzero mean Gaussian real/complex vectors and to the joint distribution of indefinite quadratic forms

Analysis of Gaussian Quadratic Forms with Application to Statistical Channel Modeling

Finalmente, en el contexto de modelado de canal, la metodología de análisis de variables propuesta permite obtener dos nuevas generalizaciones del conocido modelo de desvanecimiento kappa-mu shadowed. Estas dos nuevas distribuciones, nombradas Beckmann fluctuante y kappa-mu shadowed correlado, incluyen como casos particulares a la gran mayoría de distribuciones de desvanecimientos usadas en la literatura, abarcando desde los modelos clásicos de Rayleigh y Rice hasta otros más generales y complejos como el Beckmann y el kappa-mu. Para ambas distribuciones, se presenta su caracterización estadística de primer orden, i.e., función generadora de momentos (MGF), PDF y CDF; así como los estadísticos de segundo orden del modelo Beckmann fluctuante. Fecha de lectura de Tesis Doctoral: 24 Enero 2020En esta tesis se presenta una nueva aproximación a la distribución de de formas cuadráticas gaussianas (FCGs) no centrales tanto en variables reales como complejas. Para ello, se propone un nuevo método de análisis de variables aleatorias que, en lugar de centrarse en el estudio de la variable en cuestión, se basa en la caracterización estadística de una secuencia de variables aleatorias auxiliares convenientemente definida. Como consecuencia, las expresiones obtenidas, con independencia del grado de precisión adquirido, siempre representan una distribución válida, siendo ésta su principal ventaja. Aplicando este método, se obtienen simples expresiones recursivas para la función densidad de probabilidad (PDF) y la función de distribución (CDF) de las FCGs reales definidas positivas. En el caso de las formas complejas, esta nueva forma de análisis conduce a aproximaciones para los estadísticos de primer orden en términos de funciones elementales (exponenciales y potencias), siendo más convenientes para cálculos posteriores que otras soluciones disponibles en la literatura. La tratabilidad matemática se ejemplifica mediante el análisis de sistemas de combinación por razón máxima (MRC) sobre canales Rice correlados, proporcionando aproximaciones cerradas para la probabilidad de outage y la probabilidad de error de bit

Blind localization of radio emitters in wireless communications

The proliferation of wireless services is expected to increase the demand for radio spectrum in the foreseeable future. Given the limitations of the radio spectrum, it is evident that the current fixed frequency assignment policy fails to accommodate this increasing demand. Thus, the need for innovative technologies that can scale to accommodate future demands both in terms of spectrum efficiency and high reliable communication. Cognitive radio (CR) is one of the emerging technologies that offers a more flexible use of frequency bands allowing unlicensed users to exploit and use portions of the spectrum that are temporarily unused without causing any potential harmful interference to the incumbents. The most important functionality of a CR system is to observe the radio environment through various spectrum awareness techniques e.g., spectrum sensing or detection of spectral users in the spatio-temporal domain. In this research, we mainly focus on one of the key cognitive radio enabling techniques called localization, which provides crucial geo-location of the unknown radio transmitter in the surrounding environment. Knowledge of the user’s location can be very useful in enhancing the functionality of CRs and allows for better spectrum resource allocations in the spatial domain. For instance, the location-awareness feature can be harnessed to accomplish CR tasks such as spectrum sensing, dynamic channel allocation and interference management to enable cognitive radio operation and hence to maximize the spectral utilization. Additionally, geo-location can significantly expand the capabilities of many wireless communication applications ranging from physical layer security, geo-routing, energy efficiency, and a large set of emerging wireless sensor network and social networking applications. We devote the first part of this research to explore a broad range of existing cooperative localization techniques and through Monte-Carlo simulations analyze the performance of such techniques. We also propose two novel techniques that offer better localization performance with respect to the existing ones. The second and third parts of this research put forth a new analytical framework to characterize the performance of a particular low-complexity localization technique called weighted centroid localization (WCL), based on the statistical distribution of the ratio of two quadratic forms in normal variables. Specifically, we evaluate the performance of WCL in terms of the root mean square error (RMSE) and cumulative distribution function (CDF). The fourth part of this research focuses on studying the bias of the WCL and also provides solutions for bias correction. Throughout this research, we provide a case study analysis to evaluate the performance of the proposed approaches under changing channel and environment conditions. For the new theoretical framework, we compare analytical and Monte-Carlo simulation results of the performance metric of interest. A key contribution in our analysis is that we present not only the accurate performance in terms of the RMSE and CDF, but a new analytical framework that takes into consideration the finite nature of the network, overcoming the limitations of asymptotic results based on the central limit theorem. Remarkably, the numerical results unfold that the new analytical framework is able to predict the performance of WCL capturing all the essential aspects of propagation as well as the cognitive radio network spatial topology. Finally, we present conclusions gained from this research and possible future directions

Functional two-way analysis of variance and bootstrap methods for neural synchrony analysis

[Abstract] Background: Pairwise association between neurons is a key feature in understanding neural coding. Statistical neuroscience provides tools to estimate and assess these associations. In the mammalian brain, activating ascending pathways arise from neuronal nuclei located at the brainstem and at the basal forebrain that regulate the transition between sleep and awake neuronal firing modes in extensive regions of the cerebral cortex, including the primary visual cortex, where neurons are known to be selective for the orientation of a given stimulus. In this paper, the estimation of neural synchrony as a function of time is studied in data obtained from anesthetized cats. A functional data analysis of variance model is proposed. Bootstrap statistical tests are introduced in this context; they are useful tools for the study of differences in synchrony strength regarding 1) transition between different states (anesthesia and awake), and 2) affinity given by orientation selectivity. Results: An analysis of variance model for functional data is proposed for neural synchrony curves, estimated with a cross-correlation based method. Dependence arising from the experimental setting needs to be accounted for. Bootstrap tests allow the identification of differences between experimental conditions (modes of activity) and between pairs of neurons formed by cells with different affinities given by their preferred orientations. In our test case, interactions between experimental conditions and preferred orientations are not statistically significant. Conclusions: The results reflect the effect of different experimental conditions, as well as the affinity regarding orientation selectivity in neural synchrony and, therefore, in neural coding. A cross-correlation based method is proposed that works well under low firing activity. Functional data statistical tools produce results that are useful in this context. Dependence is shown to be necessary to account for, and bootstrap tests are an appropriate method with which to do so