Generalized Stokes Vectors and Generalized Power Spectra for Second-Order Stationary Vector-Processes

Abstract

In this paper, we present a generalization of the Stokes parameters and Stokes vectors for n-dimensional second-order stationary vector processes of the form ${\bf x}^T ( t ) = [ x_1 ( t ),x_2 ( t ), \cdots ,x_n ( t ) ]$. The Stokes parameters for $n = 2$ have been used for some time in describing the polarization characteristics of photons, but there has apparently been little attempt to generalize these parameters to arbitrary dimensions, with arbitrary bases. The Stokes parameters for $n = 2$ are found by expanding the density matrix in terms of the identity matrix plus the three Pauli spin matrices. The coefficients of this expansion are the Stokes parameters, and the ordered set of these coefficients is the Stokes vector. The basis set for $n = 2$ is trace-orthogonal and Hermitian, allowing a direct generalization to trace-orthogonal sets for $n > 2$. We show how these basis sets can be constructed from the outer products of complete sets of orthonormal vectors in unitary spaces. The Stokes vector sp...