Novel Adaptive Filtering Techniques for Multidimensional Signals

Lead Research Organisation: Imperial College London
Department Name: Electrical and Electronic Engineering


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.

Planned Impact

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.


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Cheong Took C (2011) Augmented second-order statistics of quaternion random signals in Signal Processing

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Cheong Took C (2012) On Approximate Diagonalization of Correlation Matrices in Widely Linear Signal Processing in IEEE Transactions on Signal Processing

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Cheong Took C (2010) A Quaternion Widely Linear Adaptive Filter in IEEE Transactions on Signal Processing

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Cheong Took C (2011) On the unitary diagonalisation of a special class of quaternion matrices in Applied Mathematics Letters

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Jahanchahi C (2013) A class of quaternion valued affine projection algorithms in Signal Processing

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Jahanchahi C (2014) A class of quaternion Kalman filters. in IEEE transactions on neural networks and learning systems

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Javidi S (2011) Fast independent component analysis algorithm for quaternion valued signals. in IEEE transactions on neural networks

Description The project has made a quantum step forward in statistical signal processing of quaternion random signals, by introducing several enabling tools for the the development of quaternion valued algorithms and their use in practical applications. These include

1) Augmented quaternion statistics for second order modelling of noncircular signals, necessary for the modelling of quadrivariate signals with different power levels (a standard in practice)

2) Rigorous and compact definition of quaternion gradient, necessary for the development of learning algorithms

3) A class of widely linear adaptive filtering algorithms in the quaternion domain, applied to wind modelling for renewable energy and 3D inertial body motion sensors

4) A new gradient for nonlinear functions of quaternion variables and a class of nonlinear quaternion valued algorithms and the foundations for quaternion neural networks.

5) Quaternion Takagi factorisation for the analysis of quaternion matrices that arise in augmented quaternion statistics and convergence analysis of adaptive filters

6) Widely linear quaternion state space estimation and quaternion Kalman filter

7) Foundations for blind source separation in the quaternion domain, including independent component analysis and blind source extraction

8) Real-time trackers of the degree of quaternion noncircularity

9) Accurate modelling of three-dimensional wind and joint modelling of wind and atmospheric parameters for renewable energy

10) Enhanced modelling in human centred applications, such as 3D inertial body sensors and multichannel electroencepahlogram
Exploitation Route The work in this proposal is algorithmic and has numerous practical applications, including:

- monitoring of the elderly in smart environments

- stress and fatigue monitoring

- color imaging

- inertial navigation and tracking

- wind and renewables, and smart grid We have successfully exploited the findings through

- the University Defence Research Centre (UDRC) at Imperial, by establishing close links with DSTL and applying the findings to 3D target tracking in air and underwater

- through our project partner, Prof Aihara from the University of Tokyo, we explored the applications in the renewable energy sector

- through internal collaboration at Imperial (Welcome Osteoarthritis centre) we have provided proof-of-concept for the application of quaternions in bodysensor networks

- The work has been disseminated through academic journals and conference papers, and though several tutorial and plenary talks by the PI

- The work has also served as an algorithmic background for our novel in-ear vital signs monitor device
Sectors Digital/Communication/Information Technologies (including Software)

Description The quaternion algorithms developed in this grant have been successfully tested within the University Defence Research Centre (UDRC) at imperial in three-dimensional tracking of moving targets. Real vector algebras are not suitable for the modelling of rotation and orientation, suffering from the gimbal lock phenomenon. This limits their use in e.g. 3D tracking of flying or underwater objects which have arbitrary trajectories. The quaternion valued models developed in this work are enabling technology for accurate tracking in these scenarios. Beneficiaries: Defence sector Contribution Method: Enabling algorithms
Sector Digital/Communication/Information Technologies (including Software)
Impact Types Economic

Description Collaboration with the DSTL 
Organisation Defence Science & Technology Laboratory (DSTL)
Country United Kingdom 
Sector Public 
PI Contribution Prof Mandic collaborated with DSTL to apply the outcomes of this research to defence needs - bearings only tracking of aircraft based on quaternion Kalman filters. We provided a proof of concept of using quaternions in three-dimensional target tracking, which is of interest to MoD.
Start Year 2010
Description Task Force on Smart Grid 
Form Of Engagement Activity A formal working group, expert panel or dialogue
Part Of Official Scheme? No
Primary Audience Participants in your research or patient groups
Results and Impact Dr Mandic became a member of the Task Force on Smart Grid within the IEEE Computational Intelligence Society . Awarding Body - IEEE Computational Intelligence Society, Name of Scheme - Task Force on Smart Grid, a member
Year(s) Of Engagement Activity 2010
Description This is a special issue in an international journal, Co-edited by me 
Form Of Engagement Activity A magazine, newsletter or online publication
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact We organized a special issue in Elsevier Signal Processing entitled "Hypercomplex Signal Processing", which was guest edited jointly with the colleagues from France, USA, and UK.
The issue has appeared in volume 136C in 2017
Year(s) Of Engagement Activity 2016,2017