Quaternion adaptive filters have been widely used for processing 3D and 4D phenomena. Deficient length quaternion adaptive filters are explicitly or implicitly used in many practical applications where the length of system impulse response is large or unknown. However, their statistical behaviors are yet to be fully understood. As theoretical results on the class of “full length” quaternion least mean square (QLMS) algorithms do not necessarily apply to their deficient length versions, this article fills this void and analyses the mean and mean square convergence of the deficient length QLMS algorithms, both for the strictly linear and widely linear cases. Transient and steady-state performance is characterised by exploiting the augmented statistics of noncircular quaternion random vectors. A novel decorrelation technique in the quaternion domain is shown to allow for the development of intuitive closed-form solutions for correlated quaternion Gaussian inputs, thus unveiling the relationship between the algorithm behaviour and the noncircularity of quaternion input data. The analysis also provides a general framework whereby the strictly linear and semi-widely linear QLMS algorithms can be seen as “deficient-length” versions of the widely linear QLMS. Numerical simulations validate the accuracy of the theoretical results and support the behaviour of the considered algorithms.

中文翻译：

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缺陷长度四元数最小均方自适应滤波器的性能分析

四元数自适应滤波器已广泛用于处理 3D 和 4D 现象。在系统脉冲响应的长度很大或未知的许多实际应用中，显式或隐式地使用了缺陷长度四元数自适应滤波器。然而，他们的统计行为尚未完全了解。由于“全长”四元数最小均方 (QLMS) 算法类的理论结果不一定适用于它们的不足长度版本，本文填补了这一空白并分析了不足长度 QLMS 算法的均值和均方收敛性，两者都对于严格线性和宽线性的情况。瞬态和稳态性能的特点是利用非圆形四元数随机向量的增强统计。展示了四元数域中的一种新颖去相关技术，可以为相关四元数高斯输入开发直观的封闭形式解决方案，从而揭示算法行为与四元数输入数据的非圆形之间的关系。该分析还提供了一个通用框架，由此严格线性和半宽线性 QLMS 算法可以被视为宽线性 QLMS 的“长度不足”版本。数值模拟验证了理论结果的准确性并支持所考虑算法的行为。该分析还提供了一个通用框架，由此严格线性和半宽线性 QLMS 算法可以被视为宽线性 QLMS 的“长度不足”版本。数值模拟验证了理论结果的准确性并支持所考虑算法的行为。该分析还提供了一个通用框架，由此可以将严格线性和半宽线性 QLMS 算法视为宽线性 QLMS 的“长度不足”版本。数值模拟验证了理论结果的准确性并支持所考虑算法的行为。