Similarly to how complex numbers provide a possible framework for extending scalar signal processing techniques to 2-channel signals, the 4-dimensional hypercomplex algebra of quaternions can be used to represent signals with 3 or 4 components.
For a quaternion random vector to be suited for quaternion linear processing, it must be (second-order) proper.
We consider the likelihood ratio test (LRT) for propriety, and compute the exact distribution for statistics of Box type, which include this LRT. Various approximate distributions are compared. The Wishart distribution of a quaternion sample covariance matrix is derived from first principles.
Quaternions are isomorphic to an algebra of structured 4x4 real matrices.
This mapping is our main tool, and suggests considering more general real matrix problems as a way of investigating quaternion linear algorithms.
A quaternion vector autoregressive (VAR) time-series model is equivalent to a structured real VAR model. We show that generalised least squares (and Gaussian maximum likelihood) estimation of the parameters reduces to ordinary least squares, but only if the innovations are proper. A LRT is suggested to simultaneously test for quaternion structure in the regression coefficients and innovation covariance.
Matrix-valued wavelets (MVWs) are generalised (multi)wavelets for vector-valued signals. Quaternion wavelets are equivalent to structured MVWs.
Taking into account orthogonal similarity, all MVWs can be constructed from non-trivial MVWs. We show that there are no non-scalar non-trivial MVWs with short support [0,3]. Through symbolic computation we construct the families of shortest non-trivial 2x2 Daubechies MVWs and quaternion Daubechies wavelets.