Testing for Quaternion Propriety

Abstract

We consider the problem of testing whether a quaternion-valued Gaussian random vector is proper.The quaternion covariance matrix fully describes the second order properties of a quaternion random vector only if the distribution is proper. The exact distribution of the likelihood ratio test under the hypothesis of propriety is derived for general sample size, N, and vector dimensionality, p. As this is in terms of Meijer's G-function, various approximations are considered, including Box-type and saddlepoint approximations. We find in particular that a new approach matching the first three cumulants is easy to implement and extremely accurate.

A Journey From Improper Gaussian Signaling to Asymmetric Signaling

“…Rr 1 r 1 + Rr i r i = Rr j r j + Rr k r k Rr 1 r 1 + Rr j r j = Rr i r i + Rr k r k Rr 1 r 1 + Rr k r k = Rr i r i + Rr j r j Rr i r 1 − Rr 1 r i = Rr k r j − Rr j r k Rr 1 r i + Rr i r 1 = −Rr k r j − Rr j r k Rr i r 1 + Rr 1 r i = Rr k r j + Rr j r k Rr 1 r j + Rr j r 1 = Rr i r k + Rr k r i Rr j r 1 − Rr 1 r j = Rr i r k − Rr k r i Rr 1 r j + Rr j r 1 = −Rr i r k − Rr k r i Rr 1 r k + Rr k r 1 = −Rr i r j − Rr j r i Rr 1 r k + Rr k r 1 = Rr i r j + Rr j r i Rr 1 r k − Rr k r 1 = Rr i r j − Rr j r i 12 set of equations (as given in Table II) obtained by setting R qq (i) , R qq (j) and R qq (k) to zero [166].…”

“…Intuitively, a Q-proper quaternion is C α -proper for all pure unit quaternions α = i, j, and k. Additionally, R α and C α -properness are complementary and together they result in Q-properness. As a special case, the propriety of scalar quaternion q = r 1 + ir i + jr j + kr k , is equivalent to sphericity of v, i.e., it is called proper iff r 1 , r i , r j and r k are independent and identically distributed (i.i.d) [166].…”

The geometry of proper quaternion random variables

“…One can see from property 7 that a (µ 1 , µ 2 )-proper quaternion random variable is completly described by four parameters. It can be also understood as a pair of improper complex random variables (z 1 and z 2 ) which are correlated and pseudo-correlated 5 .…”

“…Several papers exploring the concept of properness for quaternion valued random vectors and signals, and their applications, followed soon after [3], [4]. Tests to check the properness of quaternion random vectors were proposed in [5], [6]. Quaternion properness has then found applications among which widely-linear processing [7], [8], detection [9], [10], ICA [11], [12], adaptive filtering [8], [13], study of random monogenic signal [14], quaternion VAR processes [15], Gaussian graphical models [16] and directionnality in random fields [17].…”

Semi-widely linear estimation of Cη-proper quaternion random signal vectors under Gaussian and stationary conditions

“…In fact, a variety of problems have been solved using the properness hypothesis, e.g., classification of polarized signals [6], detection [10,11], quaternion VAR modeling and estimation [12], etc. On the other hand, several methods are available to determine whether a quaternion random vector is Q-proper, C η -proper, or improper [13][14][15].…”