 As for any complex random vector, the matrices formula_1 and formula_2 can be related to the covariance matrices of formula_42 and formula_43 via expressions
and conversely

The probability density function for complex normal distribution can be computed as

where formula_47 and formula_48.The characteristic function of complex normal distribution is given by
where the argument formula_50 is a "n"-dimensional complex vector.A complex random vector formula_62 is called circularly symmetric if for every deterministic formula_63 the distribution of formula_64 equals the distribution of formula_62..
Gaussian complex random vectors that are circularly symmetric are of particular interest because they are fully specified by the covariance matrix formula_1.