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Title: Statistical signal processing of noncircular quaternion random variables
Author: Xiang, Min
ISNI:       0000 0004 7657 7404
Awarding Body: Imperial College London
Current Institution: Imperial College London
Date of Award: 2018
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The focus of this thesis is on the development of theory and methods of statistical signal processing for noncircular quaternion random variables. Although it is straightforward to model three- and four-dimensional signals as real-valued trivariate and quadrivariate vectors, the division algebra of quaternions provides several crucial advantages. These include nonexistence of the gimbal lock associated with vector algebras and generic extensions of real and complex learning algorithms. However, traditional quaternion techniques are based on the assumption of circular (rotation-invariantly distributed) quaternion variables, a very limiting assumption for real-world applications. Only recently, advances in augmented quaternion statistics have revealed the suboptimality of such an assumption and have introduced the widely linear processing methodlogy, which is suitable for second-order noncircular signals. Abstract First, a simultaneous diagonalisation of the covariance matrix and the three complementary covariance matrices of noncircular quaternion variables is developed. This forms a basis for the proposed quaternion approximate uncorrelating transform (QAUT), which can be used to decorrelate noncircular quaternion variables, and offers both computational and practical advantages in many applications. Abstract Using the QAUT, the performance of existing minimum mean square error estimation techniques for noncircular quaternion signals is thoroughly explored. The performance advantage of widely linear estimation over strictly linear estimation is theoretically quantified, and the convergence performance of a class of quaternion adaptive filters is established in the mean and mean square sense. Abstract Several novel adaptive estimation techniques for noncircular quaternion signals are proposed. A novel quaternion least mean magnitude-phase adaptive filter is shown to outperform the quaternion least mean square filter in several practical scenarios. Computationally efficient algorithms are also introduced for the quaternion estimation framework, through a special decomposition of the mean square error. Finally, a class of quaternion multiple model adaptive estimation algorithms are proposed to effectively deal with model uncertainty in quaternion-valued state estimation.
Supervisor: Mandic, Danilo Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral