Statistical distribution of complex random variables
In probability theory, the family of complex normal distributions, denoted
or
, 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
, and the relation matrix
. The standard complex normal is the univariate distribution with
,
, and
.
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:
and
.[2] This case is used extensively in signal processing, where it is sometimes referred to as just complex normal in the literature.
Definitions
Complex standard normal random variable
The standard complex normal random variable or standard complex Gaussian random variable is a complex random variable
whose real and imaginary parts are independent normally distributed random variables with mean zero and variance
.[3]: p. 494 [4]: pp. 501 Formally,
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd89b43ab0dd367fee11d712e350561c6d4264b8) | | (Eq.1) |
where
denotes that
is a standard complex normal random variable.
Complex normal random variable
Suppose
and
are real random variables such that
is a 2-dimensional normal random vector. Then the complex random variable
is called complex normal random variable or complex Gaussian random variable.[3]: p. 500
![{\displaystyle Z{\text{ complex normal random variable}}\quad \iff \quad (\Re (Z),\Im (Z))^{\mathrm {T} }{\text{ real normal random vector}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b815cafecf46dff47a56568979011aa92a151ab) | | (Eq.2) |
Complex standard normal random vector
A n-dimensional complex random vector
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
is a standard complex normal random vector is denoted
.
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32f9b19c30ca4edf99a19bf0290a9f60b4f7062a) | | (Eq.3) |
Complex normal random vector
If
and
are random vectors in
such that
is a normal random vector with
components. Then we say that the complex random vector
![{\displaystyle \mathbf {Z} =\mathbf {X} +i\mathbf {Y} \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73f7ad623585b9b12593652216785266cbf021bc)
is a complex normal random vector or a complex Gaussian 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54e6c5b7a324fae9ef70bb25f4eb3994b1d91991) | | (Eq.4) |
Mean, covariance, and relation
The complex Gaussian distribution can be described with 3 parameters:[5]
![{\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} }],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8186b266f6fa5b1c962c01bcc77a666d4ae579b)
where
denotes matrix transpose of
, and
denotes conjugate transpose.[3]: p. 504 [4]: pp. 500
Here the location parameter
is a n-dimensional complex vector; the covariance matrix
is Hermitian and non-negative definite; and, the relation matrix or pseudo-covariance matrix
is symmetric. The complex normal random vector
can now be denoted as
![{\displaystyle \mathbf {Z} \ \sim \ {\mathcal {CN}}(\mu ,\ \Gamma ,\ C).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec4fc0e80f3f11ab9cfad3d6767bdd41c4c1d954)
Moreover, matrices
![{\displaystyle \Gamma }](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19)
and
![{\displaystyle C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029)
are such that the matrix
![{\displaystyle P={\overline {\Gamma }}-{C}^{\mathrm {H} }\Gamma ^{-1}C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/105898b6cf444e1c5fec32d774b8a511697d605b)
is also non-negative definite where
denotes the complex conjugate of
.[5]
Relationships between covariance matrices
As for any complex random vector, the matrices
and
can be related to the covariance matrices of
and
via expressions
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7816194d319a6f156db69f13e9f2add16b9937fe)
and conversely
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebf86d3c93f2ebba884038416a3b999a39c17448)
Density function
The probability density function for complex normal distribution can be computed as
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ad0753ed7ddcae2d4deb9704bb6f3adfe23c4c5)
where
and
.
Characteristic function
The characteristic function of complex normal distribution is given by[5]
![{\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 \}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4960acae4bf65d255cae6480ec8149a59ee8b3da)
where the argument
is an n-dimensional complex vector.
Properties
- If
is a complex normal n-vector,
an m×n matrix, and
a constant m-vector, then the linear transform
will be distributed also complex-normally:
![{\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} })}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29500b1607c031f5161c0d34823cf1b200fcd758)
- If
is a complex normal n-vector, then
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4752c39ed434c1a5f987accdda45de63384153f)
- Central limit theorem. If
are independent and identically distributed complex random variables, then
![{\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),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a8e03845ead334b995c22f8fb6a63887477b36d)
- where
and
.
Circularly-symmetric central case
Definition
A complex random vector
is called circularly symmetric if for every deterministic
the distribution of
equals the distribution of
.[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
.
The circularly-symmetric (central) complex normal distribution corresponds to the case of zero mean and zero relation matrix, i.e.
and
.[3]: p. 507 [7] This is usually denoted
![{\displaystyle \mathbf {Z} \sim {\mathcal {CN}}(0,\,\Gamma )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e583264e8106c05c8fa13976d541ae1b9ba7f530)
Distribution of real and imaginary parts
If
is circularly-symmetric (central) complex normal, then the vector
is multivariate normal with covariance structure
![{\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 )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9886cae94e4561b3e7518b34ad8857b117831b75)
where
.
Probability density function
For nonsingular covariance matrix
, its distribution can also be simplified as[3]: p. 508
.
Therefore, if the non-zero mean
and covariance matrix
are unknown, a suitable log likelihood function for a single observation vector
would be
![{\displaystyle \ln(L(\mu ,\Gamma ))=-\ln(\det(\Gamma ))-{\overline {(z-\mu )}}'\Gamma ^{-1}(z-\mu )-n\ln(\pi ).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f5975072d2d6d3b5f25dda66c4ae29e39fdf21d)
The standard complex normal (defined in Eq.1) corresponds to the distribution of a scalar random variable with
,
and
. Thus, the standard complex normal distribution has density
![{\displaystyle f_{Z}(z)={\tfrac {1}{\pi }}e^{-{\overline {z}}z}={\tfrac {1}{\pi }}e^{-|z|^{2}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ca4d8b6b4c075a1365695f75e7e18e1877265fe)
Properties
The above expression demonstrates why the case
,
is called “circularly-symmetric”. The density function depends only on the magnitude of
but not on its argument. As such, the magnitude
of a standard complex normal random variable will have the Rayleigh distribution and the squared magnitude
will have the exponential distribution, whereas the argument will be distributed uniformly on
.
If
are independent and identically distributed n-dimensional circular complex normal random vectors with
, then the random squared norm
![{\displaystyle Q=\sum _{j=1}^{k}\mathbf {Z} _{j}^{\mathrm {H} }\mathbf {Z} _{j}=\sum _{j=1}^{k}\|\mathbf {Z} _{j}\|^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec0cb98d24e5b82a982d2ec37879b14903b5f63b)
has the generalized chi-squared distribution and the random matrix
![{\displaystyle W=\sum _{j=1}^{k}\mathbf {Z} _{j}\mathbf {Z} _{j}^{\mathrm {H} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f0a4ffa01bf823a44a450dd7134c349c121e1e5)
has the complex Wishart distribution with
degrees of freedom. This distribution can be described by density function
![{\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)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca0eb156ff8063e09d4501a04f047962c87d0d67)
where
, and
is a
nonnegative-definite matrix.
See also
References
- ^ Goodman, N.R. (1963). "Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction)". The Annals of Mathematical Statistics. 34 (1): 152–177. doi:10.1214/aoms/1177704250. JSTOR 2991290.
- ^ bookchapter, Gallager.R, pg9.
- ^ a b c d e f Lapidoth, A. (2009). A Foundation in Digital Communication. Cambridge University Press. ISBN 9780521193955.
- ^ a b c d Tse, David (2005). Fundamentals of Wireless Communication. Cambridge University Press. ISBN 9781139444668.
- ^ a b c Picinbono, Bernard (1996). "Second-order complex random vectors and normal distributions". IEEE Transactions on Signal Processing. 44 (10): 2637–2640. Bibcode:1996ITSP...44.2637P. doi:10.1109/78.539051.
- ^ Daniel Wollschlaeger. "The Hoyt Distribution (Documentation for R package 'shotGroups' version 0.6.2)".[permanent dead link]
- ^ bookchapter, Gallager.R
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