Complex normal distribution

Γ ∈ C n × n {displaystyle Gamma in mathbb {C} ^{n imes n}} — covariance matrix (positive semi-definite matrix)In probability theory, the family of complex normal distributions 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} . Z ∼ C N ( 0 , 1 ) ⟺ ℜ ( Z ) ⊥ ⊥ ℑ ( Z )  and  ℜ ( Z ) ∼ N ( 0 , 1 / 2 )  and  ℑ ( Z ) ∼ N ( 0 , 1 / 2 ) {displaystyle Zsim {mathcal {CN}}(0,1)quad iff quad Re (Z)perp !!!perp Im (Z){ ext{ and }}Re (Z)sim {mathcal {N}}(0,1/2){ ext{ and }}Im (Z)sim {mathcal {N}}(0,1/2)}     (Eq.1) Z  complex normal random variable ⟺ ( ℜ ( Z ) , ℑ ( Z ) ) T  real normal random vector {displaystyle Z{ ext{ complex normal random variable}}quad iff quad (Re (Z),Im (Z))^{mathrm {T} }{ ext{ real normal random vector}}}     (Eq.2) Z ∼ C N ( 0 , I n ) ⟺ ( Z 1 , … , Z n )  independent  and for  1 ≤ i ≤ n : Z i ∼ C N ( 0 , 1 ) {displaystyle mathbf {Z} sim {mathcal {CN}}(0,{oldsymbol {I}}_{n})quad iff (Z_{1},ldots ,Z_{n}){ ext{ independent}}{ ext{ and for }}1leq ileq n:Z_{i}sim {mathcal {CN}}(0,1)}     (Eq.3) Z  complex normal random vector ⟺ ( ℜ ( Z 1 ) , … , ℜ ( Z n ) , ℑ ( Z 1 ) , … , ℑ ( Z n ) ) T  real normal random vector {displaystyle mathbf {Z} { ext{ complex normal random vector}}quad iff quad (Re (Z_{1}),ldots ,Re (Z_{n}),Im (Z_{1}),ldots ,Im (Z_{n}))^{mathrm {T} }{ ext{ real normal random vector}}}     (Eq.4) In probability theory, the family of complex normal distributions 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 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} . Circular symmetric complex normal random variables are used extensively in signal processing, and are sometimes referred to as just complex normal in signal processing literature. The standard complex normal random variable or standard complex Gaussian random variable is a complex random variable Z {displaystyle Z} whose real and imaginary parts are independent normally distributed random variables with mean zero and variance 1 / 2 {displaystyle 1/2} .:p. 494:pp. 501 Formally, where Z ∼ C N ( 0 , 1 ) {displaystyle Zsim {mathcal {CN}}(0,1)} denotes that Z {displaystyle Z} is a standard complex normal random variable. Suppose X {displaystyle X} and Y {displaystyle Y} are real random variables such that ( X , Y ) T {displaystyle (X,Y)^{mathrm {T} }} is a 2-dimensional normal random vector. Then the complex random variable Z = X + i Y {displaystyle Z=X+iY} is called complex normal random variable or complex Gaussian random variable.:p. 500 A n-dimensional complex random vector Z = ( Z 1 , … , Z n ) T {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.:p. 502:pp. 501That Z {displaystyle mathbf {Z} } is a standard complex normal random vector is denoted Z ∼ C N ( 0 , I n ) {displaystyle mathbf {Z} sim {mathcal {CN}}(0,{oldsymbol {I}}_{n})} . If X = ( X 1 , … , X n ) T {displaystyle mathbf {X} =(X_{1},ldots ,X_{n})^{mathrm {T} }} and Y = ( Y 1 , … , Y n ) T {displaystyle mathbf {Y} =(Y_{1},ldots ,Y_{n})^{mathrm {T} }} are random vectors in R n {displaystyle mathbb {R} ^{n}} such that [ X , Y ] {displaystyle } is a normal random vector with 2 n {displaystyle 2n} components. Then we say that the complex random vector has the is a complex normal random vector or a complex Gaussian random vector. The symbol N C {displaystyle {mathcal {N}}_{mathcal {C}}} is also used for the complex normal distribution.