A widely linear quaternion-valued least-mean kurtosis (WL-QLMK) algorithm is introduced for adaptive filtering of quaternion-valued circular and noncircular signals. In the design, kurtosis-based cost function is first defined in the quaternion domain by integrating the widely linear model, and augmented statistics, and then minimized using the recently developed generalized Hamilton-real (GHR) calculus. In this way, the novel WL-QLMK algorithm is obtained for training quaternion-valued adaptive filter structures. Furthermore, its steady-state performance is theoretically analyzed to determine the bounds of the step size, which provides a theoretical justification for simulations. The simulation results over both benchmark system identification scenarios, and one-step-ahead predictions of real-world 4D pathological resting tremors show that the proposed WL-QLMK algorithm, by virtue of its newly defined cost function, significantly enhances the performance compared to the recently developed quaternion-valued algorithms, especially for noncircular signals.

中文翻译：

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宽线性四元数值最小峰态算法

广泛线性四元数值最小均值峰态 (WL-QLMK) 算法被引入用于四元数值圆形和非圆形信号的自适应滤波。在设计中，基于峰度的成本函数首先通过集成宽线性模型和增强统计在四元数域中定义，然后使用最近开发的广义哈密顿实数 (GHR) 微积分最小化。通过这种方式，获得了用于训练四元数值自适应滤波器结构的新颖 WL-QLMK 算法。此外，从理论上分析了其稳态性能以确定步长的界限，这为模拟提供了理论依据。两种基准系统识别场景的仿真结果，