JZUS - Journal of Zhejiang University SCIENCE

Frontiers of Information Technology & Electronic Engineering 2016 Vol.17 No.2 P.83-95

Properties of a general quaternion-valued gradient operator and its applications to signal processing

Author(s): Meng-di Jiang, Yi Li, Wei Liu
Affiliation(s): 1Department of Electronic and Electrical Engineering, University of Sheffield, Sheffield S1 3JD, UK; more
Corresponding email(s): w.liu@sheffield.ac.uk
Key Words: Quaternion, Gradient operator, Signal processing, Least mean square (LMS) algorithm, Nonlinear adaptive filtering, Adaptive beamforming

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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.

一般四元数函数梯度的定义、特性及在信号处理领域的应用

目的：随着四元数在信号处理各个领域越来越广泛的应用，基于四元数的信号处理理论也获得了快速发展。然而，制约其进一步应用的一个瓶颈就是对一般四元数函数的梯度的定义及特性还缺乏清晰并有说服力的描述。本文就试图对这一问题进行探索。
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。

关键词：四元数；梯度；信号处理；最小均方算法；非线性自适应滤波；波束形成

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一般四元数函数梯度的定义、特性及在信号处理领域的应用

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