1,122 research outputs found
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
The HR-Calculus: Enabling Information Processing with Quaternion Algebra
From their inception, quaternions and their division algebra have proven to
be advantageous in modelling rotation/orientation in three-dimensional spaces
and have seen use from the initial formulation of electromagnetic filed theory
through to forming the basis of quantum filed theory. Despite their impressive
versatility in modelling real-world phenomena, adaptive information processing
techniques specifically designed for quaternion-valued signals have only
recently come to the attention of the machine learning, signal processing, and
control communities. The most important development in this direction is
introduction of the HR-calculus, which provides the required mathematical
foundation for deriving adaptive information processing techniques directly in
the quaternion domain. In this article, the foundations of the HR-calculus are
revised and the required tools for deriving adaptive learning techniques
suitable for dealing with quaternion-valued signals, such as the gradient
operator, chain and product derivative rules, and Taylor series expansion are
presented. This serves to establish the most important applications of adaptive
information processing in the quaternion domain for both single-node and
multi-node formulations. The article is supported by Supplementary Material,
which will be referred to as SM
Beamforming and Direction of Arrival Estimation Based on Vector Sensor Arrays
Array signal processing is a technique linked closely to radar and sonar systems. In communication, the antenna array in these systems is applied to cancel the interference, suppress the background noise and track the target sources based on signals'parameters. Most of existing work ignores the polarisation status of the impinging signals and is mainly focused on their direction parameters. To have a better performance in array processing, polarized signals can be considered in array signal processing and their property can be exploited by employing various electromagnetic vector sensor arrays.
In this thesis, firstly, a full quaternion-valued model for polarized array processing is proposed based on the Capon beamformer. This new beamformer uses crossed-dipole array and considers the desired signal as quaternion-valued. Two scenarios are dealt with, where the beamformer works at a normal environment without data model errors or with model errors under the worst-case constraint. After that, an algorithm to solve the joint DOA and polarisation estimation problem is proposed. The algorithm applies the rank reduction method to use two 2-D searches instead of a 4-D search to estimate the joint parameters. Moreover, an analysis is given to introduce the difference using crossed-dipole sensor array and tripole sensor array, which indicates that linear crossed-dipole sensor array has an ambiguity problem in the estimation work and the linear tripole sensor array avoid this problem effectively. At last, we study the problem of DOA estimation for a mixture of single signal transmission (SST) signals and duel signal transmission (DST) signals. Two solutions are proposed: the first is a two-step method to estimate the parameters of SST and DST signals separately; the second one is a unified one-step method to estimate SST and DST signals together, without treating them separately in the estimation process
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
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Advanced navigation algorithms for precision landing
A detailed analysis of autonomous navigation algorithms to achieve autonomous
precision landing is presented. The problem of integrated attitude determination
and inertial navigation is solved. The theoretical results are applied and tested
in three different applications. Optimality conditions for constrained quaternion
estimation using the Kalman filter are derived.
It is common in spacecraft applications to separate the attitude determination
from the inertial navigation system. While this approach has worked in the
past, it inevitably degrades the navigation performance when the correlations between
the two systems are not correctly accounted for. It is shown how to optimally
include an attitude determination algorithm into the Kalman filter. When the conditions
to achieve optimality are not met, it is shown how to achieve sub-optimality
by properly accounting for the correlation.
The traditional approach to inertial navigation is to employ the inertial measurement
unit (IMU) outputs to propagate the estimated states forward in time,
rather then use them to update the state. A detailed covariance analysis of deadreckoning
Mars entry navigation is performed. The contribution of various sources
of IMU errors are explicitly accounted for and the filter performance is validated
through Monte Carlo analysis.
The drawback of dead-reckoning is that this approach prevents the inertial
measurements from reducing the uncertainty of the estimated states. While this
shortcoming can be compensated by the availability of other measurements, it becomes
crucial when the IMU is the only sensor to provide measurements. Such a
situation arises, for example, during Mars atmospheric entry. In the second application
of this work, IMU measurements from a NASA mission are processed in an
extended Kalman filter, and the results are compared to dead-reckoning. It is shown
that is possible to reduce the uncertainty of the inertial states by filtering the IMU.
The final application is lunar descent to landing navigation. In this example
the IMU is filtered and the algorithms to include an attitude estimate into the
Kalman filter are tested. The design performance is confirmed by Monte Carlo
analysis.Aerospace Engineering and Engineering Mechanic
Quaternion Matrices : Statistical Properties and Applications to Signal Processing and Wavelets
Similarly to how complex numbers provide a possible framework for extending scalar signal processing techniques to 2-channel signals, the 4-dimensional hypercomplex algebra of quaternions can be used to represent signals with 3 or 4 components.
For a quaternion random vector to be suited for quaternion linear processing, it must be (second-order) proper.
We consider the likelihood ratio test (LRT) for propriety, and compute the exact distribution for statistics of Box type, which include this LRT. Various approximate distributions are compared. The Wishart distribution of a quaternion sample covariance matrix is derived from first principles.
Quaternions are isomorphic to an algebra of structured 4x4 real matrices.
This mapping is our main tool, and suggests considering more general real matrix problems as a way of investigating quaternion linear algorithms.
A quaternion vector autoregressive (VAR) time-series model is equivalent to a structured real VAR model. We show that generalised least squares (and Gaussian maximum likelihood) estimation of the parameters reduces to ordinary least squares, but only if the innovations are proper. A LRT is suggested to simultaneously test for quaternion structure in the regression coefficients and innovation covariance.
Matrix-valued wavelets (MVWs) are generalised (multi)wavelets for vector-valued signals. Quaternion wavelets are equivalent to structured MVWs.
Taking into account orthogonal similarity, all MVWs can be constructed from non-trivial MVWs. We show that there are no non-scalar non-trivial MVWs with short support [0,3]. Through symbolic computation we construct the families of shortest non-trivial 2x2 Daubechies MVWs and quaternion Daubechies wavelets.Open Acces
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