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

HARDWARE/SOFTWARE CO-DESIGN: A SHORT COURSE FOR UNBELIEVERS

Hardware-Software Co-Design skills are increasingly needed to implement complex distributed embedded systems and systems-on-a-chip for applications in Telecommunications and related fields. Telecommunications and network engineers and computer scientists frequently lack the electronic engineering background to be able to design and implement such systems. Our short course, presented as an option within two different specialist postgraduate degree programmes, and also available direct to industry, has successfully demonstrated that foundation hardware-software co-design skills can be acquired in 2 days by students with limited previous hardware background