Circularly-Symmetric Gaussian random vectors

A number of basic properties about circularly-symmetric Gaussian random vectors are stated and proved here. These properties are each probably well known to most researchers who work with Gaussian noise, but I have not found them stated together with simple proofs in the literature. They are usually viewed as too advanced or too detailed for elementary texts but are used (correctly or incorrectly) without discussion in more advanced work. These results should have appeared in Section 7.8.1 of my book, R. G. Gallager, “Principles of Digital Communication,” Cambridge Press, 2008 (PDC08), but I came to understand them only while preparing a solution manual when the book was in the final production stage. 1 Pseudo-covariance and an example Let Z = (Z1, Z2, . . . , Zn) T be a complex jointly-Gaussian random vector. That is, (Zk) and (Zk) for 1 ≤ k ≤ n comprise a set of 2n jointly-Gaussian (real) random variables (rv’s). For a large portion of the situations in which it is useful to view 2n jointly-Gaussian rv’s as a vector of n complex jointly-Gaussian rv’s, these vectors have an additional property called circular symmetry. By definition, Z is circularly symmetric if eZ has the same probability distribution as Z for all real φ. For n = 1, i.e., for the case where Z is a complex Gaussian random variable Z, circular symmetry holds if and only if (Z) and (Z) are statistically independent and identically distributed (iid) with zero mean, i.e., if and only if (Z) and (Z) are jointly Gaussian with equi-probability-density contours around 0. Since E[eZ ] = eE[Z ], any circularly-symmetric complex random vector must have E[Z ] = 0, i.e., must have zero mean. In a moment, we will see that a circularly-symmetric jointly-Gaussian complex random vector is completely determined by its covariance matrix, KZ = E[ZZ †], where Z † = Z T∗ is the complex conjugate of the transpose. A circularly-symmetric jointly-Gaussian complex random vector Z is denoted and referred to as Z ∼ CN (0,KZ ), where the C denotes that Z is both circularly symmetric and complex. Most communication engineers believe that vectors of Gaussian random variables (real or complex) are determined by their covariance matrix. For the real case, this is only