View source for Complex normal distribution

{{Probability distribution
  | name = Complex normal
  | type = multivariate
  | pdf_image  = 
  | cdf_image  =
  | notation   = 
  | parameters = <math>\mathbf{\mu} \in \mathbb{C}^n</math> — [[location parameter|location]]<br/>
<math>\Gamma \in \mathbb{C}^{n \times n}</math> — [[covariance matrix]] ([[positive semi-definite matrix]])<br/>
<math>C \in \mathbb{C}^{n \times n}</math> — [[Complex random vector#Covariance matrix and pseudo-covariance matrix|relation matrix]] ([[complex symmetric matrix]])
  | support    = <math>\mathbb{C}^n</math>
  | pdf        = complicated, see text
  | mean       = <math>\mathbf{\mu}</math>
  | mode       = <math>\mathbf{\mu}</math>
  | variance   = <math>\Gamma</math>
  | cf         = <math>
    \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\}
  </math>
  }}
In [[probability theory]], the family of '''complex normal distributions''' characterizes [[complex random variable]]s whose real and imaginary parts are jointly [[Multivariate normal distribution|normal]].<ref>{{harvtxt|Goodman|1963}}</ref> The complex normal family has three parameters: ''location'' parameter ''μ'', ''covariance'' matrix <math>\Gamma</math>, and the ''relation'' matrix <math>C</math>. The '''standard complex normal''' is the univariate distribution with <math>\mu = 0</math>, <math>\Gamma=1</math>, and <math>C=0</math>.

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: <math> \mu = 0 </math>  and <math> C=0 </math>.<ref>[http://www.rle.mit.edu/rgallager/documents/CircSymGauss.pdf  ''bookchapter, Gallager.R''], pg9.</ref> 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.

==Definitions==

===Complex standard normal random variable===
The '''standard complex normal random variable''' or '''standard complex Gaussian random variable''' is a complex random variable <math>Z</math> whose real and imaginary parts are independent normally distributed random variables with mean zero and variance <math>1/2</math>.<ref name=Lapidoth>{{cite book | author=Lapidoth, A.| title=A Foundation in Digital Communication| publisher=Cambridge University Press | year=2009 | isbn=9780521193955}}</ref>{{rp|p. 494}}<ref name=TseViswanath>{{cite book |first=David |last=Tse |year=2005 |title=Fundamentals of Wireless Communication |publisher=Cambridge University Press|url=https://books.google.com/books?id=GdsLAQAAQBAJ&printsec=frontcover#v=snippet&q=%22random%20variable%22&f=false}}</ref>{{rp|pp. 501}} Formally,

{{Equation box 1
|indent =
|title=
|equation = {{NumBlk||<math>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)</math>|{{EquationRef|Eq.1}}}}
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|border colour = #0073CF
|background colour=#F5FFFA}}

where <math>Z \sim \mathcal{CN}(0,1)</math> denotes that <math>Z</math> is a standard complex normal random variable.

===Complex normal random variable===
Suppose <math>X</math> and <math>Y</math> are real random variables such that <math>(X,Y)^{\mathrm T}</math> is a 2-dimensional [[normal random vector]]. Then the complex random variable <math>Z=X+iY</math> is called '''complex normal random variable''' or '''complex Gaussian random variable'''.<ref name=Lapidoth/>{{rp|p. 500}}

{{Equation box 1
|indent =
|title=
|equation = {{NumBlk||<math>Z \text{ complex normal random variable} \quad \iff \quad (\Re(Z),\Im(Z))^{\mathrm T} \text{ real normal random vector} </math>|{{EquationRef|Eq.2}}}}
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===Complex standard normal random vector===
A n-dimensional complex random vector <math>\mathbf{Z}=(Z_1,\ldots,Z_n)^{\mathrm T}</math> 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.<ref name=Lapidoth/>{{rp|p. 502}}<ref name=TseViswanath/>{{rp|pp. 501}}
That <math>\mathbf{Z}</math> is a standard complex normal random vector is denoted <math>\mathbf{Z} \sim \mathcal{CN}(0,\boldsymbol{I}_n)</math>.

{{Equation box 1
|indent =
|title=
|equation = {{NumBlk||<math>\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)</math>|{{EquationRef|Eq.3}}}}
|cellpadding= 6
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|border colour = #0073CF
|background colour=#F5FFFA}}

===Complex normal random vector===
If <math>\mathbf{X}=(X_1,\ldots,X_n)^{\mathrm T}</math> and <math>\mathbf{Y}=(Y_1,\ldots,Y_n)^{\mathrm T}</math> are [[random vector|random vectors]] in <math>\mathbb{R}^n</math> such that <math>[\mathbf{X},\mathbf{Y}]</math> is a [[normal random vector]] with <math>2n</math> components. Then we say that the [[complex random vector]]
: <math>
    \mathbf{Z} = \mathbf{X} + i \mathbf{Y} \,
  </math>
has the is a '''complex normal random vector''' or a '''complex Gaussian random vector'''.

{{Equation box 1
|indent =
|title=
|equation = {{NumBlk||<math>\mathbf{Z} \text{ complex normal random vector} \quad \iff \quad (\Re(Z_1),\ldots,\Re(Z_n),\Im(Z_1),\ldots,\Im(Z_n))^{\mathrm T} \text{ real normal random vector} </math>|{{EquationRef|Eq.4}}}}
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F5FFFA}}

==Notation==
The symbol <math>\mathcal{N}_{\mathcal{C}}</math> is also used for the complex normal distribution.

==Mean and covariance==
The complex Gaussian distribution can be described with 3 parameters:<ref name="picinbono">{{harvtxt|Picinbono|1996}}</ref>
: <math>
    \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}],
  </math>
where <math>\mathbf{Z}^{\mathrm T}</math> denotes [[matrix transpose]] of <math>\mathbf{Z}</math>, and <math>\mathbf{Z}^{\mathrm H}</math> denotes [[conjugate transpose]].<ref name=Lapidoth/>{{rp|p. 504}}<ref name=TseViswanath/>{{rp|pp. 500}}

Here the [[location parameter]] <math>\mu</math> is a n-dimensional complex vector; the [[covariance matrix]] <math>\Gamma</math> is [[Hermitian matrix|Hermitian]] and [[non-negative definite]]; and, the ''relation matrix'' or '''[[Complex random vector#Covariance matrix and pseudo-covariance matrix|pseudo-covariance matrix]]''' <math>C</math> is [[symmetric matrix|symmetric]]. The complex normal random vector <math>
    \mathbf{Z}
  </math> can now be denoted as<math display="block">
    \mathbf{Z}\ \sim\ \mathcal{CN}(\mu,\ \Gamma,\ C).
  </math>Moreover, matrices <math>\Gamma</math> and <math>C</math> are such that the matrix
: <math>
    P = \overline{\Gamma} - {C}^{\mathrm H}\Gamma^{-1}C
  </math>
is also non-negative definite where <math>\overline{\Gamma}</math> denotes the complex conjugate of <math>\Gamma</math>.<ref name="picinbono"/>

==Relationships between covariance matrices==
{{main|Complex random vector#Covariance matrix and pseudo-covariance matrix}}

As for any complex random vector, the matrices <math>\Gamma</math> and <math>C</math> can be related to the covariance matrices of <math>\mathbf{X} = \Re(\mathbf{Z})</math> and <math>\mathbf{Y} = \Im(\mathbf{Z})</math> via expressions
: <math>\begin{align}
  & 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_Y)(\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{align}</math>
and conversely
: <math>\begin{align}
    & \Gamma = V_{XX} + V_{YY} + i(V_{YX} - V_{XY}), \\
    & C = V_{XX} - V_{YY} + i(V_{YX} + V_{XY}).
  \end{align}</math>

==Density function==
The probability density function for complex normal distribution can be computed as

: <math>\begin{align}
    f(z) &= \frac{1}{\pi^n\sqrt{\det(\Gamma)\det(P)}}\, 
            \exp\!\left\{-\frac12 \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^k}\,
            e^{ -(z-\mu)^\ast\overline{P^{-1}}(z-\mu) + 
                \operatorname{Re}\left((z-\mu)^\intercal R^\intercal\overline{P^{-1}}(z-\mu)\right)},
  \end{align}</math>

where <math>R=C^{\ast} \Gamma^{-1}</math> and <math>P=\overline{\Gamma}-RC</math>.

==Characteristic function==
The [[characteristic function (probability theory)|characteristic function]] of complex normal distribution is given by<ref name="picinbono"/>
: <math>
    \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\},
  </math>
where the argument <math>w</math> is a ''n''-dimensional complex vector.

==Properties==
* If <math>\mathbf{Z}</math> is a complex normal ''n''-vector, <math>\boldsymbol{A}</math> an ''m×n'' matrix, and <math>b</math> a constant ''m''-vector, then the linear transform <math>\boldsymbol{A}\mathbf{Z}+b</math> will be distributed also complex-normally:
: <math>
    Z\ \sim\ \mathcal{CN}(\mu,\, \Gamma,\, C) \quad \Rightarrow \quad AZ+b\ \sim\ \mathcal{CN}(A\mu+b,\, A \Gamma A^{\mathrm H},\, A C A^{\mathrm T})
  </math>

* If <math>\mathbf{Z}</math> is a complex normal ''n''-vector, then
: <math>
    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)
  </math>

* '''Central limit theorem'''. If <math>Z_1,\ldots,Z_T</math> are independent and identically distributed complex random variables, then
: <math>
    \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),
  </math>
:where <math>\Gamma = \operatorname{E}[Z Z^{\mathrm H}]</math> and <math>C = \operatorname{E}[Z Z^{\mathrm T}]</math>.

* The modulus of a complex normal random variable follows a [[Hoyt distribution]].<ref>{{cite web |title=The Hoyt Distribution (Documentation for R package 'shotGroups' version 0.6.2) |author=Daniel Wollschlaeger |url=http://finzi.psych.upenn.edu/usr/share/doc/library/shotGroups/html/hoyt.html }}{{Dead link|date=July 2019 |bot=InternetArchiveBot |fix-attempted=yes }}</ref>

==Circularly-symmetric normal distribution==
===Definition===
A complex random vector <math> \mathbf{Z} </math> is called circularly symmetric if for every deterministic <math> \varphi \in [-\pi,\pi) </math> the distribution of <math> e^{\mathrm i \varphi}\mathbf{Z} </math> equals the distribution of <math> \mathbf{Z} </math>.<ref name=TseViswanath/>{{rp|pp. 500–501}}.
{{main|Complex random vector#Circular symmetry}}

Gaussian complex random vectors that are circularly symmetric are of particular interest because they are fully specified by the covariance matrix <math>\Gamma</math>.

The '''circularly-symmetric normal distribution'' corresponds to the case of zero mean and zero relation matrix, i.e. <math>\mu = 0</math> and <math>C=0</math><ref name=Lapidoth/>{{rp|p. 507}}<ref>[http://www.rle.mit.edu/rgallager/documents/CircSymGauss.pdf  ''bookchapter, Gallager.R'']</ref>. This is usually denoted 
:<math>\mathbf{Z} \sim \mathcal{CN}(0,\,\Gamma)</math>

===Distribution of real and imaginary parts===
If <math>\mathbf{Z}=\mathbf{X}+i\mathbf{Y}</math> is circularly-symmetric complex normal, then the vector <math>[\mathbf{X}, \mathbf{Y}]</math> is multivariate normal with covariance structure
: <math>
    \begin{pmatrix}\mathbf{X} \\ \mathbf{Y}\end{pmatrix} \ \sim\  
    \mathcal{N}\Big( \begin{bmatrix}
                       \operatorname{Re}\,\mu \\
                       \operatorname{Im}\,\mu
                     \end{bmatrix},\ 
                     \tfrac{1}{2}\begin{bmatrix}
                       \operatorname{Re}\,\Gamma & -\operatorname{Im}\,\Gamma \\
                       \operatorname{Im}\,\Gamma &  \operatorname{Re}\,\Gamma
                     \end{bmatrix}\Big)
  </math>
where <math>\mu = \operatorname{E}[\mathbf{Z}] = 0</math> and <math>\Gamma=\operatorname{E}[\mathbf{Z} \mathbf{Z}^{\mathrm H}]</math>.

===Probability density function===
For nonsingular covariance matrix <math>\Gamma</math>,its distribution can also be simplified as<ref name=Lapidoth/>{{rp|p. 508}}
: <math>
    f_{\mathbf{Z}}(\mathbf{z}) = \tfrac{1}{\pi^n \det(\Gamma)}\, e^{ -\mathbf{z}^{\mathrm H} \Gamma^{-1} \mathbf{z}}
  </math>.

Therefore, if the non-zero mean <math>\mu</math> and covariance matrix <math>\Gamma</math> are unknown, a suitable log likelihood function for a single observation vector <math>z</math> would be
: <math>
    \ln(L(\mu,\Gamma)) = -\ln (\det(\Gamma)) -\overline{(z - \mu)}' \Gamma^{-1} (z - \mu) -n \ln(\pi).
  </math>

The '''standard complex normal''' (defined in {{EquationNote|Eq.1}})corresponds to the distribution of a scalar random variable with <math>\mu = 0</math>, <math>C=0</math> and <math>\Gamma=1</math>. Thus, the standard complex normal distribution has density

: <math>
    f_Z(z) = \tfrac{1}{\pi} e^{-\overline{z}z} = \tfrac{1}{\pi} e^{-|z|^2}.
  </math>

===Properties===
The above expression demonstrates why the case <math>C=0</math>, <math>\mu = 0</math> is called “circularly-symmetric”. The density function depends only on the magnitude of <math>z</math> but not on its [[Arg (mathematics)|argument]]. As such, the magnitude <math>|z|</math> of a standard complex normal random variable will have the [[Rayleigh distribution]] and the squared magnitude <math>|z|^2</math> will have the [[exponential distribution]], whereas the argument will be distributed [[Uniform distribution (continuous)|uniformly]] on <math>[-\pi,\pi]</math>.

If <math>\left\{ \mathbf{Z}_1,\ldots,\mathbf{Z}_k \right\}</math> are independent and identically distributed ''n''-dimensional circular complex normal random vectors with <math>\mu = 0</math>, then the random squared norm
: <math>
    Q = \sum_{j=1}^k \mathbf{Z}_j^{\mathrm H} \mathbf{Z}_j = \sum_{j=1}^k \| \mathbf{Z}_j \|^2
  </math>
has the [[generalized chi-squared distribution]] and the random matrix
: <math>
    W = \sum_{j=1}^k \mathbf{Z}_j \mathbf{Z}_j^{\mathrm H}
  </math>
has the [[complex Wishart distribution]] with <math>k</math> degrees of freedom. This distribution can be described by density function
: <math>
    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)}
  </math>
where <math>k \ge n</math>, and <math>w</math> is a <math>n \times n</math> nonnegative-definite matrix.

==See also==
* [[Directional statistics#Distribution of the mean]]
* [[Normal distribution]]
* [[Multivariate normal distribution]] (a complex normal distribution is a bivariate normal distribution)
* [[Generalized chi-squared distribution]]
* [[Wishart distribution]]
* [[Complex random variable]]

==References==
{{More footnotes|date=July 2011}}
{{reflist}}

==Further reading==
{{refbegin}}
* {{cite journal
  | first = N.R.
  | last = Goodman
  | year = 1963
  | title = Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction)
  | journal = The Annals of Mathematical Statistics
  | volume = 34
  | issue = 1
  | pages = 152–177
  | jstor = 2991290
  | doi=10.1214/aoms/1177704250
  }}
* {{cite journal
  | last = Picinbono
  | first = Bernard
  | year = 1996
  | title = Second-order complex random vectors and normal distributions
  | journal = IEEE Transactions on Signal Processing
  | volume = 44
  | issue = 10
  | pages = 2637–2640
  | doi=10.1109/78.539051
  | url = https://hal-centralesupelec.archives-ouvertes.fr/hal-01736682/file/Reconstr.%20Normal%20Vect..pdf
 }}
{{refend}}
* <small>Wollschlaeger, Daniel. "ShotGroups." ''Hoyt''. RDocumentation, n.d. Web. https://www.rdocumentation.org/packages/shotGroups/versions/0.7.1/topics/Hoyt.</small>
* <small>Gallager, Robert G (2008). "Circularly-Symmetric Gaussian Random Vectors." (n.d.): n. pag. Pre-print. Web. '''9''' http://www.rle.mit.edu/rgallager/documents/CircSymGauss.pdf.</small>
{{ProbDistributions|continuous-infinite}}

[[Category:Continuous distributions]]
[[Category:Multivariate continuous distributions]]
[[Category:Complex numbers]]
[[Category:Complex distributions]]