几何代数自适应滤波器
Geometric-Algebra Adaptive Filters
摘要：
本文提出了一类新的自适应滤波器，即几何代数自适应滤波器（GAAF）。它们是通过从几何代数（GA）的角度制定基本的最小化问题（确定性费用函数）生成的，几何代数是一种非常适合描述几何变换的综合数学语言。此外，与标准自适应滤波理论不同，几何微积分（GA到微分学的扩展）允许应用相同的推导技术，而不管数据的类型（子代数）如何，即，实数，复数，四元数等依赖于这些特性（除其他外），一个确定性的二次费用函数构成，从该GAAF的设计，提供了一个通用的常规自适应滤波器的子代数的GA。从获得的更新规则，它是如何恢复以下最小均方（LMS）自适应滤波器的变种：实条目LMS，复杂的LMS，和四元数LMS。均方分析和模拟中的系统辨识的情况下提供，显示出非常好的协议，为不同程度的测量噪声。
英文摘要：
This paper presents a new class of adaptive filters, namely Geometric-Algebra Adaptive Filters (GAAFs). They are generated by formulating the underlying minimization problem (a deterministic cost function) from the perspective of Geometric Algebra (GA), a comprehensive mathematical language well-suited for the description of geometric transformations. Also, differently from standard adaptive-filtering theory, Geometric Calculus (the extension of GA to differential calculus) allows for applying the same derivation techniques regardless of the type (subalgebra) of the data, i.e., real, complex numbers, quaternions, etc. Relying on those characteristics (among others), a deterministic quadratic cost function is posed, from which the GAAFs are devised, providing a generalization of regular adaptive filters to subalgebras of GA. From the obtained update rule, it is shown how to recover the following least-mean squares (LMS) adaptive filter variants: real-entries LMS, complex LMS, and quaternions LMS. Mean-square analysis and simulations in a system identification scenario are provided, showing very good agreement for different levels of measurement noise.
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