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Frontiers of Information Technology & Electronic Engineering
ISSN 2095-9184 (print), ISSN 2095-9230 (online)
2016 Vol.17 No.2 P.83-95
Properties of a general quaternion-valued gradient operator and its applications to signal processing
Abstract: The gradients of a quaternion-valued function are often required for quaternionic signal processing algorithms. The HR gradient operator provides a viable framework and has found a number of applications. However, the applications so far have been limited to mainly real-valued quaternion functions and linear quaternion-valued functions. To generalize the operator to nonlinear quaternion functions, we define a restricted version of the HR operator, which comes in two versions, the left and the right ones. We then present a detailed analysis of the properties of the operators, including several different product rules and chain rules. Using the new rules, we derive explicit expressions for the derivatives of a class of regular nonlinear quaternion-valued functions, and prove that the restricted HR gradients are consistent with the gradients in the real domain. As an application, the derivation of the least mean square algorithm and a nonlinear adaptive algorithm is provided. Simulation results based on vector sensor arrays are presented as an example to demonstrate the effectiveness of the quaternion-valued signal model and the derived signal processing algorithm.
Key words: Quaternion, Gradient operator, Signal processing, Least mean square (LMS) algorithm, Nonlinear adaptive filtering, Adaptive beamforming
创新点:在信号处理中,虽然很多优化函数的值都是实数,但在进行优化时,尤其是在非线性信号处理中,经常会遇到对取值为四元数的四元数函数求梯度。不同于以往只适用于实数值四元数函数梯度的定义,本文第一次就一般四元数函数的梯度给出了一个自洽的定义,并对其特性进行了详细的研究和描述。基于以上研究,本文对四元数值的最小均方(LMS)自适应算法,以及一个有代表性的非线性自适应算法进行了推导,并以矢量传感器阵列波束形成为例进行了计算机模拟。
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DOI:
10.1631/FITEE.1500334
CLC number:
TN911.7; O29
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2024-08-27
Received:
2023-10-17
Revision Accepted:
2024-05-08
Crosschecked:
2016-01-14