Statistical distribution of complex random variables
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 .[ 1] 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} .[ 2] This case is used extensively in signal processing , where it is sometimes referred to as just complex normal in the literature.
Complex standard normal random variable [ edit ] 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} .[ 3] : p. 494 [ 4] : pp. 501 Formally,
Z ∼ C N ( 0 , 1 ) ⟺ ℜ ( Z ) ⊥ ⊥ ℑ ( Z ) and ℜ ( Z ) ∼ N ( 0 , 1 / 2 ) and ℑ ( Z ) ∼ N ( 0 , 1 / 2 ) {\displaystyle Z\sim {\mathcal {CN}}(0,1)\quad \iff \quad \Re (Z)\perp \!\!\!\perp \Im (Z){\text{ and }}\Re (Z)\sim {\mathcal {N}}(0,1/2){\text{ and }}\Im (Z)\sim {\mathcal {N}}(0,1/2)} (Eq.1 )
where Z ∼ C N ( 0 , 1 ) {\displaystyle Z\sim {\mathcal {CN}}(0,1)} denotes that Z {\displaystyle Z} is a standard complex normal random variable.
Complex normal random variable [ edit ] 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 .[ 3] : p. 500
Z complex normal random variable ⟺ ( ℜ ( Z ) , ℑ ( Z ) ) T real normal random vector {\displaystyle Z{\text{ complex normal random variable}}\quad \iff \quad (\Re (Z),\Im (Z))^{\mathrm {T} }{\text{ real normal random vector}}} (Eq.2 )
Complex standard normal random vector [ edit ] 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.[ 3] : p. 502 [ 4] : pp. 501 That 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,{\boldsymbol {I}}_{n})} .
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,{\boldsymbol {I}}_{n})\quad \iff (Z_{1},\ldots ,Z_{n}){\text{ independent}}{\text{ and for }}1\leq i\leq n:Z_{i}\sim {\mathcal {CN}}(0,1)} (Eq.3 )
Complex normal random vector [ edit ] 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 [\mathbf {X} ,\mathbf {Y} ]} is a normal random vector with 2 n {\displaystyle 2n} components. Then we say that the complex random vector
Z = X + i Y {\displaystyle \mathbf {Z} =\mathbf {X} +i\mathbf {Y} \,} is a complex normal random vector or a complex Gaussian random vector .
Z complex normal random vector ⟺ ( ℜ ( Z T ) , ℑ ( Z T ) ) T = ( ℜ ( Z 1 ) , … , ℜ ( Z n ) , ℑ ( Z 1 ) , … , ℑ ( Z n ) ) T real normal random vector {\displaystyle \mathbf {Z} {\text{ complex normal random vector}}\quad \iff \quad (\Re (\mathbf {Z} ^{\mathrm {T} }),\Im (\mathbf {Z} ^{\mathrm {T} }))^{\mathrm {T} }=(\Re (Z_{1}),\ldots ,\Re (Z_{n}),\Im (Z_{1}),\ldots ,\Im (Z_{n}))^{\mathrm {T} }{\text{ real normal random vector}}} (Eq.4 )
Mean, covariance, and relation[ edit ] The complex Gaussian distribution can be described with 3 parameters:[ 5]
μ = E [ Z ] , Γ = E [ ( Z − μ ) ( Z − μ ) H ] , C = E [ ( Z − μ ) ( Z − μ ) T ] , {\displaystyle \mu =\operatorname {E} [\mathbf {Z} ],\quad \Gamma =\operatorname {E} [(\mathbf {Z} -\mu )({\mathbf {Z} }-\mu )^{\mathrm {H} }],\quad C=\operatorname {E} [(\mathbf {Z} -\mu )(\mathbf {Z} -\mu )^{\mathrm {T} }],} where Z T {\displaystyle \mathbf {Z} ^{\mathrm {T} }} denotes matrix transpose of Z {\displaystyle \mathbf {Z} } , and Z H {\displaystyle \mathbf {Z} ^{\mathrm {H} }} denotes conjugate transpose .[ 3] : p. 504 [ 4] : pp. 500
Here the location parameter μ {\displaystyle \mu } is a n-dimensional complex vector; the covariance matrix Γ {\displaystyle \Gamma } is Hermitian and non-negative definite ; and, the relation matrix or pseudo-covariance matrix C {\displaystyle C} is symmetric . The complex normal random vector Z {\displaystyle \mathbf {Z} } can now be denoted as Z ∼ C N ( μ , Γ , C ) . {\displaystyle \mathbf {Z} \ \sim \ {\mathcal {CN}}(\mu ,\ \Gamma ,\ C).} Moreover, matrices Γ {\displaystyle \Gamma } and C {\displaystyle C} are such that the matrix
P = Γ ¯ − C H Γ − 1 C {\displaystyle P={\overline {\Gamma }}-{C}^{\mathrm {H} }\Gamma ^{-1}C} is also non-negative definite where Γ ¯ {\displaystyle {\overline {\Gamma }}} denotes the complex conjugate of Γ {\displaystyle \Gamma } .[ 5]
Relationships between covariance matrices [ edit ] As for any complex random vector, the matrices Γ {\displaystyle \Gamma } and C {\displaystyle C} can be related to the covariance matrices of X = ℜ ( Z ) {\displaystyle \mathbf {X} =\Re (\mathbf {Z} )} and Y = ℑ ( Z ) {\displaystyle \mathbf {Y} =\Im (\mathbf {Z} )} via expressions
V X X ≡ E [ ( X − μ X ) ( X − μ X ) T ] = 1 2 Re [ Γ + C ] , V X Y ≡ E [ ( X − μ X ) ( Y − μ Y ) T ] = 1 2 Im [ − Γ + C ] , V Y X ≡ E [ ( Y − μ Y ) ( X − μ X ) T ] = 1 2 Im [ Γ + C ] , V Y Y ≡ E [ ( Y − μ Y ) ( Y − μ Y ) T ] = 1 2 Re [ Γ − C ] , {\displaystyle {\begin{aligned}&V_{XX}\equiv \operatorname {E} [(\mathbf {X} -\mu _{X})(\mathbf {X} -\mu _{X})^{\mathrm {T} }]={\tfrac {1}{2}}\operatorname {Re} [\Gamma +C],\quad V_{XY}\equiv \operatorname {E} [(\mathbf {X} -\mu _{X})(\mathbf {Y} -\mu _{Y})^{\mathrm {T} }]={\tfrac {1}{2}}\operatorname {Im} [-\Gamma +C],\\&V_{YX}\equiv \operatorname {E} [(\mathbf {Y} -\mu _{Y})(\mathbf {X} -\mu _{X})^{\mathrm {T} }]={\tfrac {1}{2}}\operatorname {Im} [\Gamma +C],\quad \,V_{YY}\equiv \operatorname {E} [(\mathbf {Y} -\mu _{Y})(\mathbf {Y} -\mu _{Y})^{\mathrm {T} }]={\tfrac {1}{2}}\operatorname {Re} [\Gamma -C],\end{aligned}}} and conversely
Γ = V X X + V Y Y + i ( V Y X − V X Y ) , C = V X X − V Y Y + i ( V Y X + V X Y ) . {\displaystyle {\begin{aligned}&\Gamma =V_{XX}+V_{YY}+i(V_{YX}-V_{XY}),\\&C=V_{XX}-V_{YY}+i(V_{YX}+V_{XY}).\end{aligned}}} The probability density function for complex normal distribution can be computed as
f ( z ) = 1 π n det ( Γ ) det ( P ) exp { − 1 2 ( ( z ¯ − μ ¯ ) ⊺ , ( z − μ ) ⊺ ) ( Γ C C ¯ Γ ¯ ) − 1 ( z − μ z ¯ − μ ¯ ) } = det ( P − 1 ¯ − R ∗ P − 1 R ) det ( P − 1 ) π n e − ( z − μ ) ∗ P − 1 ¯ ( z − μ ) + Re ( ( z − μ ) ⊺ R ⊺ P − 1 ¯ ( z − μ ) ) , {\displaystyle {\begin{aligned}f(z)&={\frac {1}{\pi ^{n}{\sqrt {\det(\Gamma )\det(P)}}}}\,\exp \!\left\{-{\frac {1}{2}}{\begin{pmatrix}({\overline {z}}-{\overline {\mu }})^{\intercal },&(z-\mu )^{\intercal }\end{pmatrix}}{\begin{pmatrix}\Gamma &C\\{\overline {C}}&{\overline {\Gamma }}\end{pmatrix}}^{\!\!-1}\!{\begin{pmatrix}z-\mu \\{\overline {z}}-{\overline {\mu }}\end{pmatrix}}\right\}\\[8pt]&={\tfrac {\sqrt {\det \left({\overline {P^{-1}}}-R^{\ast }P^{-1}R\right)\det(P^{-1})}}{\pi ^{n}}}\,e^{-(z-\mu )^{\ast }{\overline {P^{-1}}}(z-\mu )+\operatorname {Re} \left((z-\mu )^{\intercal }R^{\intercal }{\overline {P^{-1}}}(z-\mu )\right)},\end{aligned}}} where R = C H Γ − 1 {\displaystyle R=C^{\mathrm {H} }\Gamma ^{-1}} and P = Γ ¯ − R C {\displaystyle P={\overline {\Gamma }}-RC} .
Characteristic function [ edit ] The characteristic function of complex normal distribution is given by[ 5]
φ ( w ) = exp { i Re ( w ¯ ′ μ ) − 1 4 ( w ¯ ′ Γ w + Re ( w ¯ ′ C w ¯ ) ) } , {\displaystyle \varphi (w)=\exp \!{\big \{}i\operatorname {Re} ({\overline {w}}'\mu )-{\tfrac {1}{4}}{\big (}{\overline {w}}'\Gamma w+\operatorname {Re} ({\overline {w}}'C{\overline {w}}){\big )}{\big \}},} where the argument w {\displaystyle w} is an n -dimensional complex vector.
If Z {\displaystyle \mathbf {Z} } is a complex normal n -vector, A {\displaystyle {\boldsymbol {A}}} an m×n matrix, and b {\displaystyle b} a constant m -vector, then the linear transform A Z + b {\displaystyle {\boldsymbol {A}}\mathbf {Z} +b} will be distributed also complex-normally: Z ∼ C N ( μ , Γ , C ) ⇒ A Z + b ∼ C N ( A μ + b , A Γ A H , A C A T ) {\displaystyle Z\ \sim \ {\mathcal {CN}}(\mu ,\,\Gamma ,\,C)\quad \Rightarrow \quad AZ+b\ \sim \ {\mathcal {CN}}(A\mu +b,\,A\Gamma A^{\mathrm {H} },\,ACA^{\mathrm {T} })} If Z {\displaystyle \mathbf {Z} } is a complex normal n -vector, then 2 [ ( Z − μ ) H P − 1 ¯ ( Z − μ ) − Re ( ( Z − μ ) T R T P − 1 ¯ ( Z − μ ) ) ] ∼ χ 2 ( 2 n ) {\displaystyle 2{\Big [}(\mathbf {Z} -\mu )^{\mathrm {H} }{\overline {P^{-1}}}(\mathbf {Z} -\mu )-\operatorname {Re} {\big (}(\mathbf {Z} -\mu )^{\mathrm {T} }R^{\mathrm {T} }{\overline {P^{-1}}}(\mathbf {Z} -\mu ){\big )}{\Big ]}\ \sim \ \chi ^{2}(2n)} Central limit theorem . If Z 1 , … , Z T {\displaystyle Z_{1},\ldots ,Z_{T}} are independent and identically distributed complex random variables, then T ( 1 T ∑ t = 1 T Z t − E [ Z t ] ) → d C N ( 0 , Γ , C ) , {\displaystyle {\sqrt {T}}{\Big (}{\tfrac {1}{T}}\textstyle \sum _{t=1}^{T}Z_{t}-\operatorname {E} [Z_{t}]{\Big )}\ {\xrightarrow {d}}\ {\mathcal {CN}}(0,\,\Gamma ,\,C),} where Γ = E [ Z Z H ] {\displaystyle \Gamma =\operatorname {E} [ZZ^{\mathrm {H} }]} and C = E [ Z Z T ] {\displaystyle C=\operatorname {E} [ZZ^{\mathrm {T} }]} . Circularly-symmetric central case [ edit ] A complex random vector Z {\displaystyle \mathbf {Z} } is called circularly symmetric if for every deterministic φ ∈ [ − π , π ) {\displaystyle \varphi \in [-\pi ,\pi )} the distribution of e i φ Z {\displaystyle e^{\mathrm {i} \varphi }\mathbf {Z} } equals the distribution of Z {\displaystyle \mathbf {Z} } .[ 4] : pp. 500–501
Central normal complex random vectors that are circularly symmetric are of particular interest because they are fully specified by the covariance matrix Γ {\displaystyle \Gamma } .
The circularly-symmetric (central) complex normal distribution corresponds to the case of zero mean and zero relation matrix, i.e. μ = 0 {\displaystyle \mu =0} and C = 0 {\displaystyle C=0} .[ 3] : p. 507 [ 7] This is usually denoted
Z ∼ C N ( 0 , Γ ) {\displaystyle \mathbf {Z} \sim {\mathcal {CN}}(0,\,\Gamma )} Distribution of real and imaginary parts [ edit ] If Z = X + i Y {\displaystyle \mathbf {Z} =\mathbf {X} +i\mathbf {Y} } is circularly-symmetric (central) complex normal, then the vector [ X , Y ] {\displaystyle [\mathbf {X} ,\mathbf {Y} ]} is multivariate normal with covariance structure
( X Y ) ∼ N ( [ 0 0 ] , 1 2 [ Re Γ − Im Γ Im Γ Re Γ ] ) {\displaystyle {\begin{pmatrix}\mathbf {X} \\\mathbf {Y} \end{pmatrix}}\ \sim \ {\mathcal {N}}{\Big (}{\begin{bmatrix}0\\0\end{bmatrix}},\ {\tfrac {1}{2}}{\begin{bmatrix}\operatorname {Re} \,\Gamma &-\operatorname {Im} \,\Gamma \\\operatorname {Im} \,\Gamma &\operatorname {Re} \,\Gamma \end{bmatrix}}{\Big )}} where Γ = E [ Z Z H ] {\displaystyle \Gamma =\operatorname {E} [\mathbf {Z} \mathbf {Z} ^{\mathrm {H} }]} .
Probability density function [ edit ] For nonsingular covariance matrix Γ {\displaystyle \Gamma } , its distribution can also be simplified as[ 3] : p. 508
f Z ( z ) = 1 π n det ( Γ ) e − ( z − μ ) H Γ − 1 ( z − μ ) {\displaystyle f_{\mathbf {Z} }(\mathbf {z} )={\tfrac {1}{\pi ^{n}\det(\Gamma )}}\,e^{-(\mathbf {z} -\mathbf {\mu } )^{\mathrm {H} }\Gamma ^{-1}(\mathbf {z} -\mathbf {\mu } )}} . Therefore, if the non-zero mean μ {\displaystyle \mu } and covariance matrix Γ {\displaystyle \Gamma } are unknown, a suitable log likelihood function for a single observation vector z {\displaystyle z} would be
ln ( L ( μ , Γ ) ) = − ln ( det ( Γ ) ) − ( z − μ ) ¯ ′ Γ − 1 ( z − μ ) − n ln ( π ) . {\displaystyle \ln(L(\mu ,\Gamma ))=-\ln(\det(\Gamma ))-{\overline {(z-\mu )}}'\Gamma ^{-1}(z-\mu )-n\ln(\pi ).} The standard complex normal (defined in Eq.1 ) corresponds to the distribution of a scalar random variable with μ = 0 {\displaystyle \mu =0} , C = 0 {\displaystyle C=0} and Γ = 1 {\displaystyle \Gamma =1} . Thus, the standard complex normal distribution has density
f Z ( z ) = 1 π e − z ¯ z = 1 π e − | z | 2 . {\displaystyle f_{Z}(z)={\tfrac {1}{\pi }}e^{-{\overline {z}}z}={\tfrac {1}{\pi }}e^{-|z|^{2}}.} The above expression demonstrates why the case C = 0 {\displaystyle C=0} , μ = 0 {\displaystyle \mu =0} is called “circularly-symmetric”. The density function depends only on the magnitude of z {\displaystyle z} but not on its argument . As such, the magnitude | z | {\displaystyle |z|} of a standard complex normal random variable will have the Rayleigh distribution and the squared magnitude | z | 2 {\displaystyle |z|^{2}} will have the exponential distribution , whereas the argument will be distributed uniformly on [ − π , π ] {\displaystyle [-\pi ,\pi ]} .
If { Z 1 , … , Z k } {\displaystyle \left\{\mathbf {Z} _{1},\ldots ,\mathbf {Z} _{k}\right\}} are independent and identically distributed n -dimensional circular complex normal random vectors with μ = 0 {\displaystyle \mu =0} , then the random squared norm
Q = ∑ j = 1 k Z j H Z j = ∑ j = 1 k ‖ Z j ‖ 2 {\displaystyle Q=\sum _{j=1}^{k}\mathbf {Z} _{j}^{\mathrm {H} }\mathbf {Z} _{j}=\sum _{j=1}^{k}\|\mathbf {Z} _{j}\|^{2}} has the generalized chi-squared distribution and the random matrix
W = ∑ j = 1 k Z j Z j H {\displaystyle W=\sum _{j=1}^{k}\mathbf {Z} _{j}\mathbf {Z} _{j}^{\mathrm {H} }} has the complex Wishart distribution with k {\displaystyle k} degrees of freedom. This distribution can be described by density function
f ( w ) = det ( Γ − 1 ) k det ( w ) k − n π n ( n − 1 ) / 2 ∏ j = 1 k ( k − j ) ! e − tr ( Γ − 1 w ) {\displaystyle f(w)={\frac {\det(\Gamma ^{-1})^{k}\det(w)^{k-n}}{\pi ^{n(n-1)/2}\prod _{j=1}^{k}(k-j)!}}\ e^{-\operatorname {tr} (\Gamma ^{-1}w)}} where k ≥ n {\displaystyle k\geq n} , and w {\displaystyle w} is a n × n {\displaystyle n\times n} nonnegative-definite matrix.
Discrete univariate
with finite support with infinite support
Continuous univariate
supported on a bounded interval supported on a semi-infinite interval supported on the whole real line with support whose type varies
Mixed univariate
Multivariate (joint) Directional Degenerate and singular Families