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



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 quaternionvalued 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.


Quaternion Gradient operator Signal processing Least mean square (LMS) algorithm Nonlinear adaptive filtering Adaptive beamforming 

CLC number

TN911.7 O29 


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Copyright information

© Journal of Zhejiang University Science Editorial Office and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Electronic and Electrical EngineeringUniversity of SheffieldSheffieldUK
  2. 2.School of Mathematics and StatisticsUniversity of SheffieldSheffieldUK

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