An expression is derived for the distribution of a mixture of real and complex normal variates. The asymptotic distribution of the resulting real-complex maximum-likelihood estimates is the real-complex normal distribution derived. The covariance matrix of this distribution is particularly important. It is the asymptotic covariance matrix for maximum-likelihood estimates and the Cramer-Rao lower bound on the variance of the real-complex estimates in general. From this covariance matrix, the variance of the reconstructed complex-valued exit wave then follows using the pertinent propagation formulas. The resulting expressions show the dependence of the variance on the free microscope parameters used for experimental design.

Price's (1958) theorem is derived for complex valued variates. The derivation differs from the existing derivation in two respects. First, the normal variates are not assumed to be circularly complex. Thus the result is more general. Second, the characteristic function of the complex variates is not used.

The multivariate complex normal distribution usually employed in the literature is a special case since certain restrictions have been imposed on the covariances of the real and imaginary parts of its variables. A more general distribution is proposed of which the usual distribution is shown to be a special case.>

Maximum entropy spectral analysis is a method for the estimation of power spectra with a higher resolution than can be obtained with conventional techniques. This is achieved by extrapolation of the autocorrelation function in such a way that the entropy of the corresponding probability density function is maximized in each step of the extrapolation. This correspondence also gives a simple interpretation of the method without entropy considerations.