Novel Adaptive Filtering Techniques for Multidimensional Signals

本提案旨在为三维和四维真实世界信号的统计信号处理开发一个严格的理论和计算框架。这将在四元数领域实现，受益于它的除法代数，从而有望在这些信号的建模方面得到量子改进。将特别强调自适应信号处理问题的解决方案，其准确性将通过使用四元数统计和相关的特殊形式的相关和特征结构来提高。目前的算法不足以满足具有非圆（旋转依赖）概率分布的非常大的一类过程，以及其分量表现出耦合和大不平衡动态的信号；这些在阵列信号处理、风建模、运动跟踪和混沌工程中很常见。拟议的研究将使三维和四维信号的统一建模成为可能，同时更好地理解相关的非线性动力学和学习的几何，也将作为异构数据源同时建模的框架。这项工作的基本新颖之处在于我们最近提出的四元数最小均方（QLMS）算法，该算法充分利用了四元数代数，从而允许在真实世界现象的建模中增加自由度和提高精度。这也将作为一个框架来设计一套新的自适应滤波和跟踪算法，基于标准和广泛的线性模型，这将适用于处理四元数值信号的一般性。为了证明所提出的方法的价值，将进行全面的理论评估和实际测试。实际应用将考虑可再生能源的短期风力预测和智能环境中运动传感器的轨迹跟踪；在处理多尺度（湍流、阵风、多重耦合旋转轨迹）的大型间歇动力学时，预期会有特别的增益。这项研究计划由帝国理工学院与日本东京大学的一个国际领先的研究小组合作，将找到这些问题的解决方案，并将为处理不确定性、复杂性和多维数据性质的一些新兴领域的进步开辟新的可能性。

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.