Quaternion digital signal processing: a hypercomplex approach to information processing

During the recent years, signal processing research started investigating hypercomplex numbers and their usefulness in the modeling of systems. Together, digital signal processing groups started developing multidimensional algorithms in the hypercomplex domains. At present, many proposals include quaternion, octonion and Lie algebra solutions. However, a canonical definition of discrete-time hypercircuit has not been written, yet. This paper has the aim to strengthen the concept of discrete-time hypercircuit theory. In addition, we intend to give a definition of quaternion convolution and Z-Transform in an established form. Besides this, filter design examples and simulations from the adaptive filtering context are supplied.