Commutative quaternion algebra and DSP fundamental properties: Quaternion convolution and Fourier transform

Abstract

In recent years, quaternion algebra has been applied more and more in different applications. A primary motivation for investigating quaternion algebras is to extend the classical signal processing to quaternion signal processing theory. However, this algebra has non-commutative multiplication, leading to the uncertainty of such essential operations as the convolution and Fourier transform. As a result, it became difficult to process quaternion signals in the frequency domain. In this work, the arithmetic of quaternion numbers is analyzed using the Cayley-Dickson construction, but with an associative and commutative operation of multiplication. We call it the (2,2)-model of quaternions. This model's main properties of quaternions are described, including inverse, division, and conjugate operations. In the (2,2)-model, only two exponential functions with pure quaternion units exit, and therefore, only two types of the quaternion discrete Fourier transform (QDFT) can be defined. One of the crucial benefits of this model is that the unique definition of convolution will help reduce the complexity of quaternion convolution operation calculated using both types of QDFTs.