 For nonsingular covariance matrix formula_1,its distribution can also be simplified as

Therefore, if the non-zero mean formula_30 and covariance matrix formula_1 are unknown, a suitable log likelihood function for a single observation vector formula_79 would be

The standard complex normal (defined in )corresponds to the distribution of a scalar random variable with formula_3, formula_5 and formula_4.Thus, the standard complex normal distribution has density

The above expression demonstrates why the case formula_5, formula_3 is called “circularly-symmetric”.The density function depends only on the magnitude of formula_79 but not on its argument.