There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

In this book, Andrew Harvey sets out to provide a unified and comprehensive theory of structural time series models. Unlike the traditional ARIMA models, structural time series models consist explicitly of unobserved components, such as trends and seasonals, which have a direct interpretation. As a result the model selection methodology associated with structural models is much closer to econometric methodology. The link with econometrics is made even closer by the natural way in which the models can be extended to include explanatory variables and to cope with multivariate time series. From the technical point of view, state space models and the Kalman filter play a key role in the statistical treatment of structural time series models. The book includes a detailed treatment of the Kalman filter. This technique was originally developed in control engineering, but is becoming increasingly important in fields such as economics and operations research. This book is concerned primarily with modelling economic and social time series, and with addressing the special problems which the treatment of such series poses. The properties of the models and the methodological techniques used to select them are illustrated with various applications. These range from the modellling of trends and cycles in US macroeconomic time series to to an evaluation of the effects of seat belt legislation in the UK.

Forecasting, Structural Time Series Models and the Kalman Filter

On robust estimation of the location parameter

Filtering and smoothing methods are used to produce an accurate estimate of the state of a time-varying system based on multiple observational inputs (data). Interest in these methods has exploded in recent years, with numerous applications emerging in fields such as navigation, aerospace engineering, telecommunications, and medicine. This compact, informal introduction for graduate students and advanced undergraduates presents the current state-of-the-art filtering and smoothing methods in a unified Bayesian framework. Readers learn what non-linear Kalman filters and particle filters are, how they are related, and their relative advantages and disadvantages. They also discover how state-of-the-art Bayesian parameter estimation methods can be combined with state-of-the-art filtering and smoothing algorithms. The book’s practical and algorithmic approach assumes only modest mathematical prerequisites. Examples include MATLAB computations, and the numerous end-of-chapter exercises include computational assignments. MATLAB/GNU Octave source code is available for download at www.cambridge.org/sarkka, promoting hands-on work with the methods.

贝叶斯滤波与平滑 (Bayesian filtering and smoothing)

Most numerical integration techniques consist of approximating the integrand by a polynomial in a region or regions and then integrating the polynomial exactly. Often a complicated integrand can be factored into a non-negative ''weight'' function and another function better approximated by a polynomial, thus $\int_{a}^{b} g(t)dt = \int_{a}^{b} \omega (t)f(t)dt \approx \sum_{i=1}^{N} w_i f(t_i)$. Hopefully, the quadrature rule ${\{w_j, t_j\}}_{j=1}^{N}$ corresponding to the weight function $\omega$(t) is available in tabulated form, but more likely it is not. We present here two algorithms for generating the Gaussian quadrature rule defined by the weight function when: a) the three term recurrence relation is known for the orthogonal polynomials generated by $\omega$(t), and b) the moments of the weight function are known or can be calculated.

Calculation of Gauss quadrature rules.

In the last decade, the research on and the technology for outdoor tracking have seen an explosion of advances. It is expected that in the near future, we will witness similar trends for indoor scenarios where people spend more than 70% of their lives. The rationale for this is that there is a need for reliable and high-definition real-time tracking systems that have the ability to operate in indoor environments, thus complementing those based on satellite technologies, such as the Global Positioning System (GPS). The indoor environments are very challenging, and as a result, a large variety of technologies have been proposed for coping with them, but no legacy solution has emerged. This paper presents a survey on indoor wireless tracking of mobile nodes from a signal processing perspective. It can be argued that the indoor tracking problem is more challenging than the problem on indoor localization. The reason is simple: From a set of measurements, one has to estimate not one location but a series of correlated locations of a mobile node. The paper illustrates the theory, the main tools, and the most promising technologies for indoor tracking. New directions of research are also discussed.

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Indoor Tracking: Theory, Methods, and Technologies

In this letter, we obtain the maximum likelihood estimator of position in the framework of global navigation satellite systems. This theoretical result is the basis of a completely different approach to the positioning problem, in contrast to the conventional two-step position estimation, consisting of estimating the synchronization parameters of the in-view satellites and then performing a position estimation with that information. To the authors' knowledge, this is a novel approach that copes with signal fading, and it mitigates multipath and jamming interferences. Besides, the concept of position-based synchronization is introduced, which states that synchronization parameters can be recovered from a user position estimation. We provide computer simulation results showing the robustness of the proposed approach in fading multipath channels. The root mean square error performance of the proposed algorithm is compared to those achieved with state-of-the-art synchronization techniques. A sequential Monte Carlo-based method is used to deal with the multivariate optimization problem resulting from the maximum likelihood solution in an iterative way

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Maximum Likelihood Estimation of Position in GNSS

Multipath is known to be one of the most dominant sources of accuracy degradation in satellite-based navigation systems. Multipath may cause biased position estimates that could jeopardize high-precision applications. This paper considers the problem of tracking the time-variant synchronization parameters of both the line-of-sight signal (LOSS) and its multipath replicas. In particular, the proposed algorithm tracks time-delays, amplitudes, phases and proposes a procedure to extract Doppler shifts from complex amplitudes. However, the interest is focused on LOSS time-delay estimates, since those provide the means to compute user's position. The undertaken Bayesian approach is implemented by a particle filter. The selection of the importance density function, from which particles are generated, is performed using a Gaussian approximation of the posterior function. This selection provides a particle generating function close to the optimal, which yields to an efficient usage of particles. The complex-linear part of the model, i.e., complex amplitudes, is tackled by a Rao-Blackwellization procedure that implements a complex Kalman filter for each generated particle, thus reducing the computational load. Computer simulation results are compared to other Bayesian filtering alternatives (namely, the extended Kalman filter, the unscented Kalman filter and the sequential importance resampling algorithms) and the posterior Cramer-Rao bound.

A Bayesian Approach to Multipath Mitigation in GNSS Receivers

One of the main vulnerabilities of GNSS receivers is their exposure to intentional or unintentional jamming signals, which could even cause service unavailability. Several alternatives to counteract these effects were proposed in the literature, being the most promising those based on multiple antenna architectures. This is specially the case for high-grade receivers used in applications requiring reliability and robustness. This article provides an overview of the possible receiver architectures encompassing antenna arrays and the associated signal processing techniques. Emphasis is also put on the most typical implementation issues found when dealing with such technology. A thorough survey is complemented with a set of experiments, including real data processing by a working prototype, which exemplifies the above ideas.

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Robust GNSS Receivers by Array Signal Processing: Theory and Implementation

This special issue addresses various jammers and their effect on different processing stages and overall Global Navigation Satellite System (GNSS) receiver performance, and presents countermeasures and solutions to combat interference.

Pau Closas

Papers

Indoor Tracking: Theory, Methods, and Technologies

Maximum Likelihood Estimation of Position in GNSS

A Bayesian Approach to Multipath Mitigation in GNSS Receivers

Robust GNSS Receivers by Array Signal Processing: Theory and Implementation

Vulnerabilities, threats, and authentication in satellite-based navigation systems [scanning the issue]