Novel Adaptive Filtering Techniques for Multidimensional Signals

This proposal seeks to develop a rigorous theoretical and computational framework for statistical signal processing of three- and four-dimensional real world signals. This will be achieved in the quaternion domain, benefiting from its division algebra, and thus promising a quantum improvement in the modelling of such signals. Particular emphasis will be on solutions for adaptive signal processing problems, whose accuracy will be enhanced through the use of quaternion statistics and the associated special forms of correlation- and eigen-structures. Current algorithms are less than adequate for the very large class of processes with noncircular (rotation dependent) probability distributions, and for signals whose components exhibit coupling and large unbalanced dynamics; these are common in array signal processing, wind modelling, motion tracking, and chaos engineering.The proposed research will enable unified modelling of three- and four-dimensional signals, together with better understanding of the associated nonlinear dynamics and geometry of learning, and will also serve as a framework for simultaneous modelling of heterogeneous data sources. The fundamental novelty of this work is our recently proposed quaternion least mean square (QLMS) algorithm, which makes full use of quaternion algebra, and thus allows for additional degrees of freedom and enhanced accuracy in the modelling of real world phenomena. This will also serve as a framework to design a suite of novel adaptive filtering and tracking algorithms, based on both standard and widely linear models, which will be suitable to deal with the generality of quaternion valued signals. Comprehensive theoretical evaluation and practical testing will be performed in order to prove the worthwhileness of the proposed approach. Practical applications considered will be short term wind forecasting in renewable energy and trajectory tracking from motion sensors in smart environments; particular gains are expected when dealing with large and intermittent dynamics at multiple scales (turbulence, gusts, multiple coupled rotation trajectories).This research proposal, based at Imperial College and in collaboration with an internationally leading research group from University of Tokyo Japan, will find solutions to these problems and will also open new possibilities for advances in a number of emerging areas dealing with uncertainty, complexity and multidimensional data natures.

This research proposes to introduce next generation solutions for adaptive filtering and tracking of multidimensional real world signals. Due to its fundamental nature, immediate benefits will be to the academic research and education communities. In the longer term, this research is likely to offer quantum improvement in a number of emerging practical applications. The output of this research will greatly enhance the understanding, and hence efficiency and reliability, of real time multidimensional adaptive filtering and tracking, especially for critical cases of intermittent and heterogeneous data sources, and thereby provide a significant increase in the performance and robustness. This will be achieved at a reduced costs, and will be of considerable value to UK electronics and other industries working in the field. The work in this proposal will also enable statistical modelling companies to gain competetive advantage in terms of speed, accuracy and ease of use in numerous applications based on real time modelling of three- and four-dimensional signals. These processes are common in a number of emerging applications, including renewable energy, robotics, and seismics, yet the existing algorithms are less than adequate for the very large class of signals which exhibit noncircular probability distributions and whose components have unbalanced dynamics. The two application areas considered within this proposal are wind prediction for renewable energy and motion trajectory tracking for robotics and biomedicine. Both areas are of strategic importance; they attract multibillion pound investments, and have direct impact on the imporant issues of green energy, quality of life, and wellbeing. This research will also develop highly skilled researchers, both through the workplan of this project and through related MEng, MSc,and group projects. This is likely to attract more interest and further research in this area, and will strenghten the position of the UK in statistical signal processing. These skills may be of considerable value to the industries working in this area; through our dissemination plan we have ensured that both the academic circles and relevant industries are aware of this work.