Magnetotelluric data, stable distributions and impropriety: an existential combination

Author Posting. © Author, 2014. This article is posted here by permission of The Royal Astronomical Society for personal use, not for redistribution. The definitive version was published in Geophysical Journal International 198 (2014): 622-636, doi: 10.1093/gji/ggu121.The robust statistical model of a Gaussian core contaminated by outlying data that underlies robust estimation of the magnetotelluric (MT) response function has been re-examined. The residuals from robust estimators are systematically long tailed compared to a distribution based on the Gaussian, and hence are inconsistent with the robust model. Instead, MT data are pervasively described by the alpha stable distribution family whose variance and sometimes mean are undefined. A maximum likelihood estimator (MLE) that exploits the stable nature of MT data is formulated, and its two-stage implementation in which stable parameters are first fit to the data and then the MT responses are solved for is described. The MLE is shown to be inherently robust, but differs from the conventional robust estimator because it is based on a model derived from the data, while robust estimators are ad hoc, being based on the robust model that is inconsistent with actual data. Propriety versus impropriety of the complex MT response was investigated, and a likelihood ratio test for propriety and its null distribution was established. The Cramér-Rao lower bounds for the covariance matrix of proper and improper MT responses were specified. The MLE was applied to exemplar long period and broad-band data sets from South Africa. Both are shown to be significantly stably distributed using the Kolmogorov–Smirnov goodness of fit and Ansari-Bradley non-parametric dispersion tests. Impropriety of the MT responses at both sites is pervasive, hence the improper Cramér-Rao bound was used to estimate the MLE covariance. The MLE is shown to be nearly unbiased and well described by a Gaussian distribution based on bootstrap simulation. The MLE was compared to a conventional robust estimator, establishing that the standard errors of the former are systematically smaller than for the latter and that the standardized differences between them exhibit excursions that are both too frequent and too large to be described by a Gaussian model. This is ascribed to pervasive bias of the robust estimator that is to some degree obscured by their systematically large confidence bounds. Finally, a series of topics for further investigation is proposed.This work was supported by NSF grant EAR0809074

Widely Linear State Space Filtering of Improper Complex Signals

Complex signals are the backbone of many modern applications, such as power systems, communication systems, biomedical sciences and military technologies. However, standard complex valued signal processing approaches are suited to only a subset of complex signals known as proper, and are inadequate of the generality of complex signals, as they do not fully exploit the available information. This is mainly due to the inherent blindness of the algorithms to the complete second order statistics of the signals, or due to under-modelling of the underlying system. The aim of this thesis is to provide enhanced complex valued, state space based, signal processing solutions for the generality of complex signals and systems. This is achieved based on the recent advances in the so called augmented complex statistics and widely linear modelling, which have brought to light the limitations of conventional statistical complex signal processing approaches. Exploiting these developments, we propose a class of widely linear adaptive state space estimation techniques, which provide a unified framework and enhanced performance for the generality of complex signals, compared with conventional approaches. These include the linear and nonlinear Kalman and particle filters, whereby it is shown that catering for the complete second order information and system models leads to significant performance gains. The proposed techniques are also extended to the case of cooperative distributed estimation, where nodes in a network collaborate locally to estimate signals, under a framework that caters for general complex signals, as well as the cross-correlations between observation noises, unlike earlier solutions. The analysis of the algorithms are supported by numerous case studies, including frequency estimation in three phase power systems, DIFAR sonobuoy underwater target tracking, and real-world wind modeling and prediction.Open Acces

Adaptive filtering algorithms for quaternion-valued signals

Advances in sensor technology have made possible the recoding of three and four-dimensional signals which afford a better representation of our actual three-dimensional world than the ``flat view'' one and two-dimensional approaches. Although it is straightforward to model such signals as real-valued vectors, many applications require unambiguous modeling of orientation and rotation, where the division algebra of quaternions provides crucial advantages over real-valued vector approaches. The focus of this thesis is on the use of recent advances in quaternion-valued signal processing, such as the quaternion augmented statistics, widely-linear modeling, and the HR-calculus, in order to develop practical adaptive signal processing algorithms in the quaternion domain which deal with the notion of phase and frequency in a compact and physically meaningful way. To this end, first a real-time tracker of quaternion impropriety is developed, which allows for choosing between strictly linear and widely-linear quaternion-valued signal processing algorithms in real-time, in order to reduce computational complexity where appropriate. This is followed by the strictly linear and widely-linear quaternion least mean phase algorithms that are developed for phase-only estimation in the quaternion domain, which is accompanied by both quantitative performance assessment and physical interpretation of operations. Next, the practical application of state space modeling of three-phase power signals in smart grid management and control systems is considered, and a robust complex-valued state space model for frequency estimation in three-phase systems is presented. Its advantages over other available estimators are demonstrated both in an analytical sense and through simulations. The concept is then expanded to the quaternion setting in order to make possible the simultaneous estimation of the system frequency and its voltage phasors. Furthermore, a distributed quaternion Kalman filtering algorithm is developed for frequency estimation over power distribution networks and collaborative target tracking. Finally, statistics of stable quaternion-valued random variables, that include quaternion-valued Gaussian random variables as a special case, is investigated in order to develop a framework for the modeling and processing of heavy-tailed quaternion-valued signals.Open Acces

Complex-Valued Random Vectors and Channels: Entropy, Divergence, and Capacity

Recent research has demonstrated significant achievable performance gains by exploiting circularity/non-circularity or propeness/improperness of complex-valued signals. In this paper, we investigate the influence of these properties on important information theoretic quantities such as entropy, divergence, and capacity. We prove two maximum entropy theorems that strengthen previously known results. The proof of the former theorem is based on the so-called circular analog of a given complex-valued random vector. Its introduction is supported by a characterization theorem that employs a minimum Kullback-Leibler divergence criterion. In the proof of latter theorem, on the other hand, results about the second-order structure of complex-valued random vectors are exploited. Furthermore, we address the capacity of multiple-input multiple-output (MIMO) channels. Regardless of the specific distribution of the channel parameters (noise vector and channel matrix, if modeled as random), we show that the capacity-achieving input vector is circular for a broad range of MIMO channels (including coherent and noncoherent scenarios). Finally, we investigate the situation of an improper and Gaussian distributed noise vector. We compute both capacity and capacity-achieving input vector and show that improperness increases capacity, provided that the complementary covariance matrix is exploited. Otherwise, a capacity loss occurs, for which we derive an explicit expression.Comment: 33 pages, 1 figure, slightly modified version of first paper revision submitted to IEEE Trans. Inf. Theory on October 31, 201

Maximally improper signaling in underlay MIMO cognitive radio networks

Improper Gaussian signaling is a well-known technique that has been shown to improve performance in different multi-user scenarios. In this paper, we analyze the benefit of improper signaling in underlay cognitive radio when users are equipped with multiple antennas. Specifically, we assume that the primary user is protected by the so-called interference temperature constraint, which guarantees a prescribed rate requirement. In this setting, we study how the maximum tolerable interference power changes when the interference is additionally constrained to be maximally improper (strictly noncircular, or rectilinear). We observe that the correlation structure of a maximally improper interference is an additional degree of freedom that can be exploited to improve the SU performance. Because of that, we propose two different protection strategies for the PU where this structure is either constrained or unconstrained, and derive the interference temperature threshold for both cases. We then focus on the secondary user and provide designs of the transmission parameters under the proposed protection strategies.The work of C. Lameiro and P. J. Schreier was supported by the German Research Foundation (DFG) under Grants SCHR 1384/6-1 and LA 4107/1-1. The work of I. Santamaría was supported by the Ministerio de Economía y Competitividad and AEI/FEDER funds of the UE, Spain, under Project CARMEN (TEC2016-75067-C4-4-R)

Rate region boundary of the SISO Z-interference channel with improper signaling

This paper provides a complete characterization of the boundary of an achievable rate region, called the Pareto boundary, of the single-antenna Z interference channel (Z-IC), when interference is treated as noise and users transmit complex Gaussian signals that are allowed to be improper. By considering the augmented complex formulation, we derive a necessary and sufficient condition for improper signaling to be optimal. This condition is stated as a threshold on the interference channel coefficient, which is a function of the interfered user rate and which allows insightful interpretations into the behavior of the achievable rates in terms of the circularity coefficient (i.e., degree of impropriety). Furthermore, the optimal circularity coefficient is provided in closed form. The simplicity of the obtained characterization permits interesting insights into when and how improper signaling outperforms proper signaling in the single-antenna Z-IC. We also provide an in-depth discussion on the optimal strategies and the properties of the Pareto boundary.The work of C. Lameiro and P. J. Schreier was supported by the German Research Foundation (DFG) under grant SCHR 1384/6-1. The work of I. Santamaría was supported by the Ministerio de Economía y Competitividad (MINECO), Spain, under projects RACHEL (TEC2013-47141-C4-3-R) and CARMEN (TEC2016-75067-C4-4-R)

Distributed adaptive signal processing for frequency estimation

It is widely recognised that future smart grids will heavily rely upon intelligent communication and signal processing as enabling technologies for their operation. Traditional tools for power system analysis, which have been built from a circuit theory perspective, are a good match for balanced system conditions. However, the unprecedented changes that are imposed by smart grid requirements, are pushing the limits of these old paradigms. To this end, we provide new signal processing perspectives to address some fundamental operations in power systems such as frequency estimation, regulation and fault detection. Firstly, motivated by our finding that any excursion from nominal power system conditions results in a degree of non-circularity in the measured variables, we cast the frequency estimation problem into a distributed estimation framework for noncircular complex random variables. Next, we derive the required next generation widely linear, frequency estimators which incorporate the so-called augmented data statistics and cater for the noncircularity and a widely linear nature of system functions. Uniquely, we also show that by virtue of augmented complex statistics, it is possible to treat frequency tracking and fault detection in a unified way. To address the ever shortening time-scales in future frequency regulation tasks, the developed distributed widely linear frequency estimators are equipped with the ability to compensate for the fewer available temporal voltage data by exploiting spatial diversity in wide area measurements. This contribution is further supported by new physically meaningful theoretical results on the statistical behavior of distributed adaptive filters. Our approach avoids the current restrictive assumptions routinely employed to simplify the analysis by making use of the collaborative learning strategies of distributed agents. The efficacy of the proposed distributed frequency estimators over standard strictly linear and stand-alone algorithms is illustrated in case studies over synthetic and real-world three-phase measurements. An overarching theme in this thesis is the elucidation of underlying commonalities between different methodologies employed in classical power engineering and signal processing. By revisiting fundamental power system ideas within the framework of augmented complex statistics, we provide a physically meaningful signal processing perspective of three-phase transforms and reveal their intimate connections with spatial discrete Fourier transform (DFT), optimal dimensionality reduction and frequency demodulation techniques. Moreover, under the widely linear framework, we also show that the two most widely used frequency estimators in the power grid are in fact special cases of frequency demodulation techniques. Finally, revisiting classic estimation problems in power engineering through the lens of non-circular complex estimation has made it possible to develop a new self-stabilising adaptive three-phase transformation which enables algorithms designed for balanced operating conditions to be straightforwardly implemented in a variety of real-world unbalanced operating conditions. This thesis therefore aims to help bridge the gap between signal processing and power communities by providing power system designers with advanced estimation algorithms and modern physically meaningful interpretations of key power engineering paradigms in order to match the dynamic and decentralised nature of the smart grid.Open Acces