Price's Theorem for Quaternion Variables

Qiankun Diao, Dongpo Xu, Shuning Sun, Danilo P. Mandic

Abstract: Price's theorem in statistical signal processing relates the expectation of a nonlinear function of normally distributed random variables to their covariances. However, such a key theorem is yet to be established and explored in quaternion statistics. To this end, we introduce Price's theorem for quaternion variables using the generalized Hamilton-real (GHR) calculus. This is achieved by first employing the chain rule of GHR calculus to derive two crucial quaternion matrix derivatives of functions with respect to the product of quaternion matrix variables. Next, we leverage quaternion second-order statistics to establish the relationship between the derivative of a function with respect to the augmented quaternion covariance matrix and its real counterpart. Based on the above results and Price's theorem in real signal processing, we finally propose a novel formulation of Price's theorem for quaternion random variables. This finding not only enriches the theory of quaternion statistical signals processing but also extends its applicability.