Novel Adaptive Filtering Techniques for Multidimensional Signals

这一建议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。这将在四元数域中实现，受益于其除法代数，因此有望在此类信号的建模方面取得量子改进。将特别强调自适应信号处理问题的解决办法，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而得到提高。现有的算法不适用于具有非圆形(旋转相关)概率分布的大类过程，以及其分量表现出耦合和大的不平衡动力学的信号；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。所提出的研究将使三维和四维信号能够统一建模，以及更好地理解相关的非线性动力学和学习的几何学，并将作为一个框架来同时建模不同的数据源。这项工作的基本新颖性是我们最近提出的四元数最小均方(QLMS)算法，它充分利用了四元数代数，从而允许在现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，以设计一套新的自适应滤波和跟踪算法，基于标准和广义线性模型，将适合于处理四元数值信号的一般性。将进行全面的理论评估和实际测试，以证明所提出的方法的价值。所考虑的实际应用将是可再生能源中的短期风预测和智能环境中运动传感器的轨迹跟踪；在处理多尺度的大型和间歇性动力学(湍流、阵风、多个耦合的旋转轨迹)时，预计将获得特别的收益。这项研究提案由帝国理工学院与日本东京大学的国际领先研究小组合作，将找到解决这些问题的方案，并将为在处理不确定性、复杂性和多维数据性质的一些新兴领域取得进展打开新的可能性。

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.