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2016, Volume 17, Issue 2

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Frontiers of Information Technology & Electronic Engineering >> 2016, Volume 17, Issue 2 doi: 10.1631/FITEE.1500334

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

Affiliation(s): 1Department of Electronic and Electrical Engineering, University of Sheffield, Sheffield S1 3JD, UK; 2School of Mathematics and Statistics, University of Sheffield, Sheffield S3 7RH, UK; less

Received: 2015-10-15 Accepted: 2016-02-02 Available online: 2016-02-02
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Abstract

The gradients of a -valued function are often required for ic algorithms. The HR provides a viable framework and has found a number of applications. However, the applications so far have been limited to mainly real-valued functions and linear -valued functions. To generalize the operator to nonlinear 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 -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 -valued signal model and the derived algorithm.

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