In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
, the family of complex normal distributions, denoted
or
, characterizes
complex random variable In probability theory and statistics, complex random variables are a generalization of real-valued random variables to complex numbers, i.e. the possible values a complex random variable may take are complex numbers. Complex random variables can a ...
s whose real and imaginary parts are jointly
normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
. 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
. This case is used extensively in
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
, 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
.
Formally,
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
In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional ( univariate) normal distribution to higher dimensions. One ...
. Then the complex random variable
is called complex normal random variable or complex Gaussian random variable.
[
]
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.[
That is a standard complex normal random vector is denoted .
]
Complex normal random vector
If and are random vectors in such that normal random vector
In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional ( univariate) normal distribution to higher dimensions. One ...
with components. Then we say that the complex random vector In probability theory and statistics, a complex random vector is typically a tuple of complex-valued random variables, and generally is a random variable taking values in a vector space over the field of complex numbers. If Z_1,\ldots,Z_n are com ...
:
is a complex normal random vector or a complex Gaussian random vector.
Mean, covariance, and relation
The complex Gaussian distribution can be described with 3 parameters:
:
where denotes matrix transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
of , and denotes conjugate transpose
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex c ...
.[
Here the ]location parameter
In geography, location or place are used to denote a region (point, line, or area) on Earth's surface or elsewhere. The term ''location'' generally implies a higher degree of certainty than ''place'', the latter often indicating an entity with an ...
is a n-dimensional complex vector; the covariance matrix
In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements o ...
is Hermitian {{Short description, none
Numerous things are named after the French mathematician Charles Hermite (1822–1901):
Hermite
* Cubic Hermite spline, a type of third-degree spline
* Gauss–Hermite quadrature, an extension of Gaussian quadrature m ...
and non-negative definite
In mathematics, a symmetric matrix M with real entries is positive-definite if the real number z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, a c ...
; and, the ''relation matrix
In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of ...
'' or ''pseudo-covariance matrix'' is symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
. The complex normal random vector can now be denoted asMoreover, matrices and are such that the matrix
:
is also non-negative definite where denotes the complex conjugate of .[
]
Relationships between covariance matrices
As for any complex random vector, the matrices and can be related to the covariance matrices of and via expressions
: Gamma + C
Gamma (uppercase , lowercase ; ''gámma'') is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter r ...
\quad
V_ \equiv \operatorname \mathbf-\mu_X)(\mathbf-\mu_Y)^\mathrm T= \tfrac\operatorname \Gamma + C \\
& V_ \equiv \operatorname \mathbf-\mu_Y)(\mathbf-\mu_X)^\mathrm T= \tfrac\operatornameGamma + C
Gamma (uppercase , lowercase ; ''gámma'') is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter r ...
\quad\,
V_ \equiv \operatorname \mathbf-\mu_Y)(\mathbf-\mu_Y)^\mathrm T= \tfrac\operatornameGamma - C
Gamma (uppercase , lowercase ; ''gámma'') is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter re ...
\end
and conversely
:
Density function
The probability density function for complex normal distribution can be computed as
:
where and .
Characteristic function
The characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
* The indicator function of a subset, that is the function
::\mathbf_A\colon X \to \,
:which for a given subset ''A'' of ''X'', has value 1 at point ...
of complex normal distribution is given by[
:
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:
:
* If is a complex normal ''n''-vector, then
:
* Central limit theorem. If are independent and identically distributed complex random variables, then
:
:where Hoyt distribution
The Nakagami distribution or the Nakagami-''m'' distribution is a probability distribution related to the gamma distribution. The family of Nakagami distributions has two parameters: a shape parameter m\geq 1/2 and a second parameter controlling ...
.
Circularly-symmetric central case
Definition
A complex random vector \mathbf is called circularly symmetric if for every deterministic \varphi \in [-\pi,\pi) the distribution of e^\mathbf equals the distribution of \mathbf .[
Central normal complex random vectors that are circularly symmetric are of particular interest because they are fully specified by the covariance matrix \Gamma.
The ''circularly-symmetric (central) complex normal distribution'' corresponds to the case of zero mean and zero relation matrix, i.e. \mu = 0 and C=0.][ ''bookchapter, Gallager.R'']
/ref> This is usually denoted
:\mathbf \sim \mathcal(0,\,\Gamma)
Distribution of real and imaginary parts
If \mathbf=\mathbf+i\mathbf is circularly-symmetric (central) complex normal, then the vector [\mathbf, \mathbf] is multivariate normal with covariance structure
:
\begin\mathbf \\ \mathbf\end \ \sim\
\mathcal\Big( \begin
\operatorname\,\mu \\
\operatorname\,\mu
\end,\
\tfrac\begin
\operatorname\,\Gamma & -\operatorname\,\Gamma \\
\operatorname\,\Gamma & \operatorname\,\Gamma
\end\Big)
where \mu = \operatorname mathbf= 0 and \Gamma=\operatorname mathbf \mathbf^/math>.
Probability density function
For nonsingular covariance matrix \Gamma, its distribution can also be simplified as[
:
f_(\mathbf) = \tfrac\, e^
.
Therefore, if the non-zero mean \mu and covariance matrix \Gamma are unknown, a suitable log likelihood function for a single observation vector z would be
:
\ln(L(\mu,\Gamma)) = -\ln (\det(\Gamma)) -\overline' \Gamma^ (z - \mu) -n \ln(\pi).
The standard complex normal (defined in )corresponds to the distribution of a scalar random variable with \mu = 0, C=0 and \Gamma=1. Thus, the standard complex normal distribution has density
:
f_Z(z) = \tfrac e^ = \tfrac e^.
]
Properties
The above expression demonstrates why the case C=0, \mu = 0 is called “circularly-symmetric”. The density function depends only on the magnitude of z but not on its argument
An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialect ...
. As such, the magnitude , z, of a standard complex normal random variable will have the Rayleigh distribution
In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. Up to rescaling, it coincides with the chi distribution with two degrees of freedom.
The distribu ...
and the squared magnitude , z, ^2 will have the exponential distribution
In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant averag ...
, whereas the argument will be distributed uniformly
Uniform distribution may refer to:
* Continuous uniform distribution
* Discrete uniform distribution
* Uniform distribution (ecology)
* Equidistributed sequence
See also
*
* Homogeneous distribution
In mathematics, a homogeneous distribution is ...
on \pi,\pi/math>.
If \left\ are independent and identically distributed ''n''-dimensional circular complex normal random vectors with \mu = 0, then the random squared norm
:
Q = \sum_^k \mathbf_j^ \mathbf_j = \sum_^k \, \mathbf_j \, ^2
has the generalized chi-squared distribution
In probability theory and statistics, the generalized chi-squared distribution (or generalized chi-square distribution) is the distribution of a quadratic form of a multinormal variable (normal vector), or a linear combination of different nor ...
and the random matrix
:
W = \sum_^k \mathbf_j \mathbf_j^
has the complex Wishart distribution
In statistics, the complex Wishart distribution is a complex version of the Wishart distribution. It is the distribution of n times the sample Hermitian covariance matrix of n zero-mean independent Gaussian random variables. It has support for ...
with k degrees of freedom. This distribution can be described by density function
:
f(w) = \frac\
e^
where k \ge n, and w is a n \times n nonnegative-definite matrix.
See also
* Complex normal ratio distribution
* Directional statistics#Distribution of the mean (polar form)
* Normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu i ...
* Multivariate normal distribution
In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional ( univariate) normal distribution to higher dimensions. One ...
(a complex normal distribution is a bivariate normal distribution)
* Generalized chi-squared distribution
In probability theory and statistics, the generalized chi-squared distribution (or generalized chi-square distribution) is the distribution of a quadratic form of a multinormal variable (normal vector), or a linear combination of different nor ...
* Wishart distribution
In statistics, the Wishart distribution is a generalization to multiple dimensions of the gamma distribution. It is named in honor of John Wishart, who first formulated the distribution in 1928.
It is a family of probability distributions defin ...
* Complex random variable In probability theory and statistics, complex random variables are a generalization of real-valued random variables to complex numbers, i.e. the possible values a complex random variable may take are complex numbers. Complex random variables can a ...
References
Further reading
*
*
* Wollschlaeger, Daniel. "ShotGroups." ''Hoyt''. RDocumentation, n.d. Web. https://www.rdocumentation.org/packages/shotGroups/versions/0.7.1/topics/Hoyt.
* 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.
{{ProbDistributions, continuous-infinite
Continuous distributions
Multivariate continuous distributions
Complex distributions