In probability theory, the family of complex normal distributions, denoted C N {\displaystyle {\mathcal {CN}}} or N C {\displaystyle {\mathcal {N}}_{\mathcal {C}}} , characterizes complex random variables whose real and imaginary parts are jointly normal. The complex normal family has three parameters: location parameter μ, covariance matrix Γ {\displaystyle \Gamma } , and the relation matrix C {\displaystyle C} . The standard complex normal is the univariate distribution with μ = 0 {\displaystyle \mu =0} , Γ = 1 {\displaystyle \Gamma =1} , and C = 0 {\displaystyle C=0} .
An important subclass of complex normal family is called the circularly-symmetric (central) complex normal and corresponds to the case of zero relation matrix and zero mean: μ = 0 {\displaystyle \mu =0} and C = 0 {\displaystyle C=0} . This case is used extensively in signal processing, where it is sometimes referred to as just complex normal in the literature.