Towards Robust Spectrum Sensing in Cognitive Radio Networks

This thesis focuses on multi-antenna assisted energy based spectrum sensing. The studies leading to this thesis have been motivated by some practical issues with energy based detection. These include the noise uncertainty problem at the secondary receiver, the presence of multiple active primary users in cognitive cellular networks, the existence of unknown noise correlations and detection in the low signal-to-noise ratio regime. In this thesis, the aim is to incorporating these practical concerns into the design of spectrum sensing algorithms. To this end, we propose the use of various detectors that are suitable for different scenarios. We consider detectors derived from decision-theoretical criteriors as well as heuristic detectors. We analyze the performance of the proposed detectors by deriving their false alarm probability, detection probability and receiver operating characteristic. The main contribution of this thesis consists of the derived closed-form performance metrics. These results are obtained by utilizing tools from multivariate analysis, moment based approximations, Mellin transforms, and random matrix theory. Numerical results show that the proposed detectors have indeed resolved the concerns raised by the above practical issues. Some detectors could meet the needs of one of the practical challenges, while others are shown to be robust when several practical issues are taken into account. The use of detectors constructed with decision-theoretical considerations over the heuristically proposed ones is justified as well

Compact smallest eigenvalue expressions in Wishart-Laguerre ensembles with or without fixed-trace

The degree of entanglement of random pure states in bipartite quantum systems can be estimated from the distribution of the extreme Schmidt eigenvalues. For a bipartition of size M\geq N, these are distributed according to a Wishart-Laguerre ensemble (WL) of random matrices of size N x M, with a fixed-trace constraint. We first compute the distribution and moments of the smallest eigenvalue in the fixed trace orthogonal WL ensemble for arbitrary M\geq N. Our method is based on a Laplace inversion of the recursive results for the corresponding orthogonal WL ensemble by Edelman. Explicit examples are given for fixed N and M, generalizing and simplifying earlier results. In the microscopic large-N limit with M-N fixed, the orthogonal and unitary WL distributions exhibit universality after a suitable rescaling and are therefore independent of the constraint. We prove that very recent results given in terms of hypergeometric functions of matrix argument are equivalent to more explicit expressions in terms of a Pfaffian or determinant of Bessel functions. While the latter were mostly known from the random matrix literature on the QCD Dirac operator spectrum, we also derive some new results in the orthogonal symmetry class.Comment: 25 pag., 4 fig - minor changes, typos fixed. To appear in JSTA

Exact distributions of finite random matrices and their applications to spectrum sensing

The exact and simple distributions of finite random matrix theory (FRMT) are critically important for cognitive radio networks (CRNs). In this paper, we unify some existing distributions of the FRMT with the proposed coefficient matrices (vectors) and represent the distributions with the coefficient-based formulations. A coefficient reuse mechanism is studied, i.e., the same coefficient matrices (vectors) can be exploited to formulate different distributions. For instance, the same coefficient matrices can be used by the largest eigenvalue (LE) and the scaled largest eigenvalue (SLE); the same coefficient vectors can be used by the smallest eigenvalue (SE) and the Demmel condition number (DCN). A new and simple cumulative distribution function (CDF) of the DCN is also deduced. In particular, the dimension boundary between the infinite random matrix theory (IRMT) and the FRMT is initially defined. The dimension boundary provides a theoretical way to divide random matrices into infinite random matrices and finite random matrices. The FRMT-based spectrum sensing (SS) schemes are studied for CRNs. The SLE-based scheme can be considered as an asymptotically-optimal SS scheme when the dimension K is larger than two. Moreover, the standard condition number (SCN)-based scheme achieves the same sensing performance as the SLE-based scheme for dual covariance matrix [Formula: see text]. The simulation results verify that the coefficient-based distributions can fit the empirical results very well, and the FRMT-based schemes outperform the IRMT-based schemes and the conventional SS schemes

From phenomenological modelling of anomalous diffusion through continuous-time random walks and fractional calculus to correlation analysis of complex systems

This document contains more than one topic, but they are all connected in ei- ther physical analogy, analytic/numerical resemblance or because one is a building block of another. The topics are anomalous diffusion, modelling of stylised facts based on an empirical random walker diffusion model and null-hypothesis tests in time series data-analysis reusing the same diffusion model. Inbetween these topics are interrupted by an introduction of new methods for fast production of random numbers and matrices of certain types. This interruption constitutes the entire chapter on random numbers that is purely algorithmic and was inspired by the need of fast random numbers of special types. The sequence of chapters is chrono- logically meaningful in the sense that fast random numbers are needed in the first topic dealing with continuous-time random walks (CTRWs) and their connection to fractional diffusion. The contents of the last four chapters were indeed produced in this sequence, but with some temporal overlap. While the fast Monte Carlo solution of the time and space fractional diffusion equation is a nice application that sped-up hugely with our new method we were also interested in CTRWs as a model for certain stylised facts. Without knowing economists [80] reinvented what physicists had subconsciously used for decades already. It is the so called stylised fact for which another word can be empirical truth. A simple example: The diffusion equation gives a probability at a certain time to find a certain diffusive particle in some position or indicates concentration of a dye. It is debatable if probability is physical reality. Most importantly, it does not describe the physical system completely. Instead, the equation describes only a certain expectation value of interest, where it does not matter if it is of grains, prices or people which diffuse away. Reality is coded and “averaged” in the diffusion constant. Interpreting a CTRW as an abstract microscopic particle motion model it can solve the time and space fractional diffusion equation. This type of diffusion equation mimics some types of anomalous diffusion, a name usually given to effects that cannot be explained by classic stochastic models. In particular not by the classic diffusion equation. It was recognised only recently, ca. in the mid 1990s, that the random walk model used here is the abstract particle based counterpart for the macroscopic time- and space-fractional diffusion equation, just like the “classic” random walk with regular jumps ±∆x solves the classic diffusion equation. Both equations can be solved in a Monte Carlo fashion with many realisations of walks. Interpreting the CTRW as a time series model it can serve as a possible null- hypothesis scenario in applications with measurements that behave similarly. It may be necessary to simulate many null-hypothesis realisations of the system to give a (probabilistic) answer to what the “outcome” is under the assumption that the particles, stocks, etc. are not correlated. Another topic is (random) correlation matrices. These are partly built on the previously introduced continuous-time random walks and are important in null- hypothesis testing, data analysis and filtering. The main ob jects encountered in dealing with these matrices are eigenvalues and eigenvectors. The latter are car- ried over to the following topic of mode analysis and application in clustering. The presented properties of correlation matrices of correlated measurements seem to be wasted in contemporary methods of clustering with (dis-)similarity measures from time series. Most applications of spectral clustering ignores information and is not able to distinguish between certain cases. The suggested procedure is sup- posed to identify and separate out clusters by using additional information coded in the eigenvectors. In addition, random matrix theory can also serve to analyse microarray data for the extraction of functional genetic groups and it also suggests an error model. Finally, the last topic on synchronisation analysis of electroen- cephalogram (EEG) data resurrects the eigenvalues and eigenvectors as well as the mode analysis, but this time of matrices made of synchronisation coefficients of neurological activity

Exact Demmel Condition Number Distribution of Complex Wishart Matrices via the Mellin Transform

In this letter we study the distribution of the Demmel condition number of complex Wishart matrices, which has various applications in the performance analysis of communication systems. New exact expressions are derived for the probability density function and cumulative distribution function, which are very simple and valid for any matrix dimensions. These results are obtained by taking advantage of properties of the Mellin transform for products of independent random variables

Multidimensional and temporal SAR data representation and processing based on binary partition trees

This thesis deals with the processing of different types of multidimensional SAR data for distinct applications. Instead of handling the original pixels of the image, which correspond to very local information and are strongly contaminated by speckle noise, a region-based and multiscale data abstraction is defined, the Binary Partition Tree (BPT). In this representation, each region stands for an homogeneous area of the data, grouping pixels with similar properties and making easier its interpretation and processing. The work presented in this thesis concerns the definition of the BPT structures for Polarimetric SAR (PolSAR) images and also for temporal series of SAR acquisitions. It covers the description of the corresponding data models and the algorithms for BPT construction and its exploitation. Particular attention has been paid to the speckle filtering application. The proposed technique has proven to achieve arbitrarily large regions over homogeneous areas while also preserving the spatial resolution and the small details of the original data. As a consequence, this approach has demonstrated an improvement in the performance of the target response estimation with respect to other speckle filtering techniques. Moreover, due to the flexibility and convenience of this representation, it has been employed for other applications as scene segmentation and classification. The processing of SAR time series has also been addressed, proposing different approaches for dealing with the temporal information of the data, resulting into distinct BPT abstractions. These representations have allowed the development of speckle filtering techniques in the spatial and temporal domains and also the improvement and the definition of additional methods for classification and temporal change detection and characterization

Enhanced Spectrum Sensing Techniques for Cognitive Radio Systems

Due to the rapid growth of new wireless communication services and applications, much attention has been directed to frequency spectrum resources. Considering the limited radio spectrum, supporting the demand for higher capacity and higher data rates is a challenging task that requires innovative technologies capable of providing new ways of exploiting the available radio spectrum. Cognitive radio (CR), which is among the core prominent technologies for the next generation of wireless communication systems, has received increasing attention and is considered a promising solution to the spectral crowding problem by introducing the notion of opportunistic spectrum usage. Spectrum sensing, which enables CRs to identify spectral holes, is a critical component in CR technology. Furthermore, improving the efficiency of the radio spectrum use through spectrum sensing and dynamic spectrum access (DSA) is one of the emerging trends. In this thesis, we focus on enhanced spectrum sensing techniques that provide performance gains with reduced computational complexity for realistic waveforms considering radio frequency (RF) impairments, such as noise uncertainty and power amplifier (PA) non-linearities. The first area of study is efficient energy detection (ED) methods for spectrum sensing under non-flat spectral characteristics, which deals with relatively simple methods for improving the detection performance. In realistic communication scenarios, the spectrum of the primary user (PU) is non-flat due to non-ideal frequency responses of the devices and frequency selective channel conditions. Weighting process with fast Fourier transform (FFT) and analysis filter bank (AFB) based multi-band sensing techniques are proposed for overcoming the challenge of non-flat characteristics. Furthermore, a sliding window based spectrum sensing approach is addressed to detect a re-appearing PU that is absent in one time and present in other time. Finally, the area under the receiver operating characteristics curve (AUC) is considered as a single-parameter performance metric and is derived for all the considered scenarios. The second area of study is reduced complexity energy and eigenvalue based spectrum sensing techniques utilizing frequency selectivity. More specifically, novel spectrum sensing techniques, which have relatively low computational complexity and are capable of providing accurate and robust performance in low signal-to-noise ratio (SNR) with noise uncertainty, as well as in the presence of frequency selectivity, are proposed. Closed-form expressions are derived for the corresponding probability of false alarm and probability of detection under frequency selectivity due the primary signal spectrum and/or the transmission channel. The offered results indicate that the proposed methods provide quite significant saving in complexity, e.g., 78% reduction in the studied example case, whereas their detection performance is improved both in the low SNR and under noise uncertainty. Finally, a new combined spectrum sensing and resource allocation approach for multicarrier radio systems is proposed. The main contribution of this study is the evaluation of the CR performance when using wideband spectrum sensing methods in combination with water-filling and power interference (PI) based resource allocation algorithms in realistic CR scenarios. Different waveforms, such as cyclic prefix based orthogonal frequency division multiplexing (CP-OFDM), enhanced orthogonal frequency division multiplexing (E-OFDM) and filter bank based multicarrier (FBMC), are considered with PA nonlinearity type RF impairments to see the effects of spectral leakage on the spectrum sensing and resource allocation performance. It is shown that AFB based spectrum sensing techniques and FBMC waveforms with excellent spectral containment properties have clearly better performance compared to the traditional FFT based spectrum sensing techniques with the CP-OFDM. Overall, the investigations in this thesis provide novel spectrum sensing techniques for overcoming the challenge of noise uncertainty with reduced computational complexity. The proposed methods are evaluated under realistic signal models