Jiang, M-D., Li, Y. and Liu, W. orcid.org/0000-0003-2968-2888 (2016) Properties of a general quaternion-valued gradient operator and its applications to signal processing. Frontiers of Information Technology & Electronic Engineering, 17 (2). pp. 83-95. ISSN 2095-9184
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 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.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2016 Springer Verlag. This is an author produced version of a paper subsequently published in Frontiers of Information Technology & Electronic Engineering. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | Quaternion; Gradient operator; Signal processing; Least mean square (LMS) algorithm; Nonlinear adaptive filtering; Adaptive beamforming |
Dates: |
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Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Engineering (Sheffield) > Department of Electronic and Electrical Engineering (Sheffield) The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield) |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 17 May 2016 13:06 |
Last Modified: | 14 Apr 2017 03:45 |
Published Version: | http://dx.doi.org/10.1631/FITEE.1500334 |
Status: | Published |
Publisher: | Springer Verlag |
Refereed: | Yes |
Identification Number: | 10.1631/FITEE.1500334 |
Related URLs: | |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:97387 |