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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 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.
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 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.