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CLC number: TN911.7; O29

On-line Access: 2016-02-02

Received: 2015-10-15

Revision Accepted: 2016-01-06

Crosschecked: 2016-01-14

Cited: 4

Clicked: 11148

Citations:  Bibtex RefMan EndNote GB/T7714


Wei Liu


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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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Meng-di Jiang, Yi Li, Wei Liu. Properties of a general quaternion-valued gradient operator and its applications to signal processing[J]. Frontiers of Information Technology & Electronic Engineering, 2016, 17(2): 83-95.

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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.




Darkslateblue:Affiliate; Royal Blue:Author; Turquoise:Article


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