About QLMS Derivations

International audienceIn this letter, a review of the quaternionic least mean squares (QLMS) algorithm is proposed. Three versions coming from three derivation ways exist: the original QLMS based on component wise gradients, the HR-QLMS based on a quaternion gradient operator and iQLMS based on an involutions-gradient. Noting and investigating the differences between the three QLMS formulations, we show that the original QLMS suffers from a mistake in the derivation calculus. Thus, we propose to derive rigorously the criterion following the first way, giving the correct version of QLMS. A comparison with the other QLMS versions validates these results on simulated data

Employing data fusion & diversity in the applications of adaptive signal processing

The paradigm of adaptive signal processing is a simple yet powerful method for the class of system identification problems. The classical approaches consider standard one-dimensional signals whereby the model can be formulated by flat-view matrix/vector framework. Nevertheless, the rapidly increasing availability of large-scale multisensor/multinode measurement technology has render no longer sufficient the traditional way of representing the data. To this end, the author, who from this point onward shall be referred to as `we', `us', and `our' to signify the author myself and other supporting contributors i.e. my supervisor, my colleagues and other overseas academics specializing in the specific pieces of research endeavor throughout this thesis, has applied the adaptive filtering framework to problems that employ the techniques of data diversity and fusion which includes quaternions, tensors and graphs. At the first glance, all these structures share one common important feature: invertible isomorphism. In other words, they are algebraically one-to-one related in real vector space. Furthermore, it is our continual course of research that affords a segue of all these three data types. Firstly, we proposed novel quaternion-valued adaptive algorithms named the n-moment widely linear quaternion least mean squares (WL-QLMS) and c-moment WL-LMS. Both are as fast as the recursive-least-squares method but more numerically robust thanks to the lack of matrix inversion. Secondly, the adaptive filtering method is applied to a more complex task: the online tensor dictionary learning named online multilinear dictionary learning (OMDL). The OMDL is partly inspired by the derivation of the c-moment WL-LMS due to its parsimonious formulae. In addition, the sequential higher-order compressed sensing (HO-CS) is also developed to couple with the OMDL to maximally utilize the learned dictionary for the best possible compression. Lastly, we consider graph random processes which actually are multivariate random processes with spatiotemporal (or vertex-time) relationship. Similar to tensor dictionary, one of the main challenges in graph signal processing is sparsity constraint in the graph topology, a challenging issue for online methods. We introduced a novel splitting gradient projection into this adaptive graph filtering to successfully achieve sparse topology. Extensive experiments were conducted to support the analysis of all the algorithms proposed in this thesis, as well as pointing out potentials, limitations and as-yet-unaddressed issues in these research endeavor.Open Acces

Adaptive signal processing algorithms for noncircular complex data

The complex domain provides a natural processing framework for a large class of signals encountered in communications, radar, biomedical engineering and renewable energy. Statistical signal processing in C has traditionally been viewed as a straightforward extension of the corresponding algorithms in the real domain R, however, recent developments in augmented complex statistics show that, in general, this leads to under-modelling. This direct treatment of complex-valued signals has led to advances in so called widely linear modelling and the introduction of a generalised framework for the differentiability of both analytic and non-analytic complex and quaternion functions. In this thesis, supervised and blind complex adaptive algorithms capable of processing the generality of complex and quaternion signals (both circular and noncircular) in both noise-free and noisy environments are developed; their usefulness in real-world applications is demonstrated through case studies. The focus of this thesis is on the use of augmented statistics and widely linear modelling. The standard complex least mean square (CLMS) algorithm is extended to perform optimally for the generality of complex-valued signals, and is shown to outperform the CLMS algorithm. Next, extraction of latent complex-valued signals from large mixtures is addressed. This is achieved by developing several classes of complex blind source extraction algorithms based on fundamental signal properties such as smoothness, predictability and degree of Gaussianity, with the analysis of the existence and uniqueness of the solutions also provided. These algorithms are shown to facilitate real-time applications, such as those in brain computer interfacing (BCI). Due to their modified cost functions and the widely linear mixing model, this class of algorithms perform well in both noise-free and noisy environments. Next, based on a widely linear quaternion model, the FastICA algorithm is extended to the quaternion domain to provide separation of the generality of quaternion signals. The enhanced performances of the widely linear algorithms are illustrated in renewable energy and biomedical applications, in particular, for the prediction of wind profiles and extraction of artifacts from EEG recordings

Quaternionic Sparse Approximation

ISBN 978-0-8176-4267-9International audienceIn this paper, we introduce a new processing procedure for quaternionic signals through consideration of the well-known orthogonal matching pursuit (OMP), which provides sparse approximation. We present a quaternionic extension, the quaternionic OMP, that can be used to process a right-multiplication linear combination of quaternionic signals. As validation, this quaternionic OMP is applied to simulated data. Deconvolution is carried out and presented here with a new spikegram that is designed for visualization of quaternionic coefficients, and finally this is compared to multivariate OMP

Single-state weighted particle filter with application to Earth Observation missions

To push the boundaries of autonomy in space, the spacecraft must rely on its own sensors to achieve positioning and environmental perception. In this context, the key problem of autonomous navigation is the nonlinear state estimation of the spacecraft in a dynamic 3D environment. In this paper, we propose a new approach based on a single-state sub-partitioning of the state vector and a partial updating of the vector of weights according to the specific information provided by each sensor. In this way, we avoid to lose information in the resampling phase thanks to a parallelization approach. The proposed method has been applied to an Earth observation mission and the efficacy of the proposed approach is demonstrated with a numerical example using a high-fidelity orbital simulator