complex normal distribution

The standard complex normal random variable or standard complex Gaussian random variable is a complex random variable $Z$ whose real and imaginary parts are independent normally distributed random variables with mean zero and variance $1/2$.{{cite book | author=Lapidoth, A.| title=A Foundation in Digital Communication| publisher=Cambridge University Press | year=2009 | isbn=9780521193955}}{{rp|p. 494}}{{cite book |first=David |last=Tse |year=2005 |title=Fundamentals of Wireless Communication |publisher=Cambridge University Press|isbn=9781139444668 |url=https://books.google.com/books?id=GdsLAQAAQBAJ&q=%22random+variable%22}}{{rp|pp. 501}} Formally,

{{Equation box 1

|indent =

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|equation = {{NumBlk||$Z\; \backslash sim\; \backslash mathcal\{CN\}(0,1)\; \backslash quad\; \backslash iff\; \backslash quad\; \backslash Re(Z)\; \backslash perp\backslash !\backslash !\backslash !\backslash perp\; \backslash Im(Z)\; \backslash text\{\; and\; \}\; \backslash Re(Z)\; \backslash sim\; \backslash mathcal\{N\}(0,1/2)\; \backslash text\{\; and\; \}\; \backslash Im(Z)\; \backslash sim\; \backslash mathcal\{N\}(0,1/2)$|{{EquationRef|Eq.1}}}}

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

Suppose $X$ and $Y$ are real random variables such that $(X,Y)^\{\backslash mathrm\; T\}$ is a 2-dimensional normal random vector. Then the complex random variable $Z=X+iY$ is called complex normal random variable or complex Gaussian random variable.{{rp|p. 500}}

{{Equation box 1

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|equation = {{NumBlk||$Z\; \backslash text\{\; complex\; normal\; random\; variable\}\; \backslash quad\; \backslash iff\; \backslash quad\; (\backslash Re(Z),\backslash Im(Z))^\{\backslash mathrm\; T\}\; \backslash text\{\; real\; normal\; random\; vector\}$|{{EquationRef|Eq.2}}}}

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A n-dimensional complex random vector $\backslash mathbf\{Z\}=(Z\_1,\backslash ldots,Z\_n)^\{\backslash mathrm\; T\}$ 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.{{rp|p. 502}}{{rp|pp. 501}}

That $\backslash mathbf\{Z\}$ is a standard complex normal random vector is denoted $\backslash mathbf\{Z\}\; \backslash sim\; \backslash mathcal\{CN\}(0,\backslash boldsymbol\{I\}\_n)$.

{{Equation box 1

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|equation = {{NumBlk||$\backslash mathbf\{Z\}\; \backslash sim\; \backslash mathcal\{CN\}(0,\backslash boldsymbol\{I\}\_n)\; \backslash quad\; \backslash iff\; (Z\_1,\backslash ldots,Z\_n)\; \backslash text\{\; independent\}\; \backslash text\{\; and\; for\; \}\; 1\; \backslash leq\; i\; \backslash leq\; n\; :\; Z\_i\; \backslash sim\; \backslash mathcal\{CN\}(0,1)$|{{EquationRef|Eq.3}}}}

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If $\backslash mathbf\{X\}=(X\_1,\backslash ldots,X\_n)^\{\backslash mathrm\; T\}$ and $\backslash mathbf\{Y\}=(Y\_1,\backslash ldots,Y\_n)^\{\backslash mathrm\; T\}$ are random vectors in $\backslash mathbb\{R\}^n$ such that $[\backslash mathbf\{X\},\backslash mathbf\{Y\}]$ is a normal random vector with $2n$ components. Then we say that the complex random vector

: $$

\mathbf{Z} = \mathbf{X} + i \mathbf{Y} \,

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

{{Equation box 1

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|equation = {{NumBlk||$\backslash mathbf\{Z\}\; \backslash text\{\; complex\; normal\; random\; vector\}\; \backslash quad\; \backslash iff\; \backslash quad\; (\backslash Re(\backslash mathbf\{Z\}^\{\backslash mathrm\; T\}),\backslash Im(\backslash mathbf\{Z\}^\{\backslash mathrm\; T\}))^\{\backslash mathrm\; T\}\; =\; (\backslash Re(Z\_1),\backslash ldots,\backslash Re(Z\_n),\backslash Im(Z\_1),\backslash ldots,\backslash Im(Z\_n))^\{\backslash mathrm\; T\}\; \backslash text\{\; real\; normal\; random\; vector\}$|{{EquationRef|Eq.4}}}}

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The complex Gaussian distribution can be described with 3 parameters:{{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

| bibcode = 1996ITSP...44.2637P

| url = https://ieeexplore-ieee-org.ezp1.lib.umn.edu/document/539051

}}

: $$

\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 $\backslash mathbf\{Z\}^\{\backslash mathrm\; T\}$ denotes matrix transpose of $\backslash mathbf\{Z\}$, and $\backslash mathbf\{Z\}^\{\backslash mathrm\; H\}$ denotes conjugate transpose.{{rp|p. 504}}{{rp|pp. 500}}

Here the location parameter $\backslash mu$ is a n-dimensional complex vector; the covariance matrix $\backslash Gamma$ is Hermitian and non-negative definite; and, the relation matrix or pseudo-covariance matrix $C$ is symmetric. The complex normal random vector $$

\mathbf{Z}

can now be denoted as$$$$

\mathbf{Z}\ \sim\ \mathcal{CN}(\mu,\ \Gamma,\ C).

Moreover, matrices $\backslash Gamma$ and $C$ are such that the matrix

: $$

P = \overline{\Gamma} - {C}^{\mathrm H}\Gamma^{-1}C

is also non-negative definite where $\backslash overline\{\backslash Gamma\}$ denotes the complex conjugate of $\backslash Gamma$.

{{main|Complex random vector#Covariance matrix and pseudo-covariance matrix}}

As for any complex random vector, the matrices $\backslash Gamma$ and $C$ can be related to the covariance matrices of $\backslash mathbf\{X\}\; =\; \backslash Re(\backslash mathbf\{Z\})$ and $\backslash mathbf\{Y\}\; =\; \backslash Im(\backslash mathbf\{Z\})$ via expressions

: $\backslash begin\{align\}$

& 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_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{align}

and conversely

: $\backslash begin\{align\}$

& \Gamma = V_{XX} + V_{YY} + i(V_{YX} - V_{XY}), \\

& C = V_{XX} - V_{YY} + i(V_{YX} + V_{XY}).

\end{align}

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

: $\backslash begin\{align\}$

f(z) &= \frac{1}{\pi^n\sqrt{\det(\Gamma)\det(P)}}\,

\exp\!\left\{-\frac12 \begin{bmatrix} z - \mu \\ \overline z -\overline \mu\end{bmatrix}^{\mathrm H}

\begin{bmatrix}\Gamma & C \\ \overline{C}&\overline\Gamma\end{bmatrix}^{\!\!-1}\!

\begin{bmatrix}z-\mu \\ \overline{z}-\overline{\mu}\end{bmatrix}

\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{align}

where $R=C^\{\backslash mathrm\; H\}\; \backslash Gamma^\{-1\}$ and $P=\backslash overline\{\backslash Gamma\}-RC$.

The characteristic function of complex normal distribution is given by

: $$

\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 $w$ is an n-dimensional complex vector.

• If $\backslash mathbf\{Z\}$ is a complex normal n-vector, $\backslash boldsymbol\{A\}$ an m×n matrix, and $b$ a constant m-vector, then the linear transform $\backslash boldsymbol\{A\}\backslash mathbf\{Z\}+b$ will be distributed also complex-normally:

: $$

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})

• If $\backslash mathbf\{Z\}$ is a complex normal n-vector, then

: $$

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 $Z\_1,\backslash ldots,Z\_T$ are independent and identically distributed complex random variables, then

: $$

\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 $\backslash Gamma\; =\; \backslash operatorname\{E\}[Z\; Z^\{\backslash mathrm\; H\}]$ and $C\; =\; \backslash operatorname\{E\}[Z\; Z^\{\backslash mathrm\; T\}]$.

• The modulus of a complex normal random variable follows a Hoyt distribution.{{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 }}

A complex random vector $\backslash mathbf\{Z\}$ is called circularly symmetric if for every deterministic $\backslash varphi\; \backslash in\; [-\backslash pi,\backslash pi)$ the distribution of $e^\{\backslash mathrm\; i\; \backslash varphi\}\backslash mathbf\{Z\}$ equals the distribution of $\backslash mathbf\{Z\}$.{{rp|pp. 500–501}}

{{main|Complex random vector#Circular symmetry}}

Central normal complex random vectors that are circularly symmetric are of particular interest because they are fully specified by the covariance matrix $\backslash Gamma$.

The circularly-symmetric (central) complex normal distribution corresponds to the case of zero mean and zero relation matrix, i.e. $\backslash mu\; =\; 0$ and $C=0$.{{rp|p. 507}}[http://www.rle.mit.edu/rgallager/documents/CircSymGauss.pdf bookchapter, Gallager.R] This is usually denoted

:$\backslash mathbf\{Z\}\; \backslash sim\; \backslash mathcal\{CN\}(0,\backslash ,\backslash Gamma)$

If $\backslash mathbf\{Z\}=\backslash mathbf\{X\}+i\backslash mathbf\{Y\}$ is circularly-symmetric (central) complex normal, then the vector $[\backslash mathbf\{X\},\; \backslash mathbf\{Y\}]$ is multivariate normal with covariance structure

: $$

\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 $\backslash Gamma=\backslash operatorname\{E\}[\backslash mathbf\{Z\}\; \backslash mathbf\{Z\}^\{\backslash mathrm\; H\}]$.

For nonsingular covariance matrix $\backslash Gamma$, its distribution can also be simplified as{{rp|p. 508}}

: $$

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 $\backslash mu$ and covariance matrix $\backslash Gamma$ are unknown, a suitable log likelihood function for a single observation vector $z$ would be

: $$

\ln(L(\mu,\Gamma)) = -\ln (\det(\Gamma)) -\overline{(z - \mu)}' \Gamma^{-1} (z - \mu) -n \ln(\pi).

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

: $$

f_Z(z) = \tfrac{1}{\pi} e^{-\overline{z}z} = \tfrac{1}{\pi} e^{-|z|^2}.

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

If $\backslash left\backslash \{\; \backslash mathbf\{Z\}\_1,\backslash ldots,\backslash mathbf\{Z\}\_k\; \backslash right\backslash \}$ are independent and identically distributed n-dimensional circular complex normal random vectors with $\backslash mu\; =\; 0$, then the random squared norm

: $$

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 = \sum_{j=1}^k \mathbf{Z}_j \mathbf{Z}_j^{\mathrm H}

has the complex Wishart distribution with $k$ degrees of freedom. This distribution can be described by density function

: $$

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 $k\; \backslash ge\; n$, and $w$ is a $n\; \backslash times\; n$ nonnegative-definite matrix.

• Complex normal ratio distribution
• {{section link|Directional statistics|Distribution of the mean}} (polar form)
• Normal distribution
• Multivariate normal distribution (a complex normal distribution is a bivariate normal distribution)
• Generalized chi-squared distribution
• Wishart distribution
• Complex random variable

{{More footnotes needed|date=July 2011}}

{{reflist}}

{{ProbDistributions|continuous-infinite}}

Category:Continuous distributions

Category:Multivariate continuous distributions

Category:Complex distributions