In probability theory, the family of complex normal distributions, denoted ${\displaystyle {\mathcal {CN}}}$ or ${\displaystyle {\mathcal {N}}_{\mathcal {C}}}$, characterizes complex random variables whose real and imaginary parts are jointly normal. The complex normal family has three parameters: location parameter μ, covariance matrix ${\displaystyle \Gamma }$, and the relation matrix ${\displaystyle C}$. The standard complex normal is the univariate distribution with ${\displaystyle \mu =0}$, ${\displaystyle \Gamma =1}$, and ${\displaystyle C=0}$.

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: ${\displaystyle \mu =0}$ and ${\displaystyle C=0}$. This case is used extensively in signal processing, where it is sometimes referred to as just complex normal in the literature.

The standard complex normal random variable or standard complex Gaussian random variable is a complex random variable ${\displaystyle Z}$ whose real and imaginary parts are independent normally distributed random variables with mean zero and variance ${\displaystyle 1/2}$. Formally,

where ${\displaystyle Z\sim {\mathcal {CN}}(0,1)}$ denotes that ${\displaystyle Z}$ is a standard complex normal random variable.

Suppose ${\displaystyle X}$ and ${\displaystyle Y}$ are real random variables such that ${\displaystyle (X,Y)^{\mathrm {T} }}$ is a 2-dimensional normal random vector. Then the complex random variable ${\displaystyle Z=X+iY}$ is called complex normal random variable or complex Gaussian random variable.

A n-dimensional complex random vector ${\displaystyle \mathbf {Z} =(Z_{1},\ldots ,Z_{n})^{\mathrm {T} }}$ is a complex standard normal random vector or complex standard Gaussian random vector if its components are independent and all of them are standard complex normal random variables as defined above. That ${\displaystyle \mathbf {Z} }$ is a standard complex normal random vector is denoted ${\displaystyle \mathbf {Z} \sim {\mathcal {CN}}(0,{\boldsymbol {I}}_{n})}$.

If ${\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{n})^{\mathrm {T} }}$ and ${\displaystyle \mathbf {Y} =(Y_{1},\ldots ,Y_{n})^{\mathrm {T} }}$ are random vectors in ${\displaystyle \mathbb {R} ^{n}}$ such that ${\displaystyle [\mathbf {X} ,\mathbf {Y} ]}$ is a normal random vector with ${\displaystyle 2n}$ components. Then we say that the complex random vector

is a complex normal random vector or a complex Gaussian random vector.