Generating Augmented Quaternion Random Variable With Generalized Gaussian Distribution

Abstract

There is a need to generate quaternion valued processes with a given distribution ranging from super-Gaussian, Gaussian to sub-Gaussian with different degrees of properness for numerous practical applications, such as modeling the signal source, testing nonlinear adaptive filter models, watermarking, generating $\mathbb {Q}$ -proper and $\mathbb {Q}$ -improper processes, the synthesis of color images, and testing the algorithms of step detection. Therefore, in this paper, the alternative quaternion generalized Gaussian probability density function is derived for an augmented quaternion valued variable, and the procedure for generating the augmented quaternion valued random variables is defined for this distribution. Additionally, the 3-D generalized Gaussian distribution probability density function parameterized by the shape parameter $p$ and the covariance matrix $C$ is revised.