The ''covariance'' of complex random variables and processes, when defined consistently with the corresponding notion for real random variables, is shown to be determined by the usual (complex) covariance together with a quantity called the pseudo-covariance. A characterization of uncorrelatedness and wide-sense stationarity in terms of covariance and pseudo-covariance is given. Complex random variables and processes with a vanishing pseudo-covariance are called proper. It is shown that properness is preserved under affine transformations and that the complex-multivariate Gaussian density assumes a natural form only for proper random variables. The maximum-entropy theorem is generalized to the complex-multivariate case. The differential entropy of a complex random vector with a fixed correlation matrix is shown to be maximum, if and only if the random vector is proper, Gaussian and zero-mean. The notion of circular stationarity is introduced. For the class of proper complex random processes, a discrete Fourier transform correspondence is derived relating circular stationarity in the time domain to uncorrelatedness in the frequency domain. As an application of the theory, the capacity of a discrete-time channel with complex inputs, proper complex additive white Gaussian noise, and a finite complex unit-sample response is determined. This derivation is considerably simpler than an earlier deirivation for the real discrete-time Gaussian channel with intersymbol interference, whose capacity is obtained as a by-product of the results for the complex channel.
