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Complex normal distribution

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. The complex normal family has three parameters: location parameter μ, covariance matrix Γ…

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CF
exp { i Re ⁡ ( w ¯ ′ μ ) − 1 4 ( w ¯ ′ Γ w + Re ⁡ ( w ¯ ′ C w ¯ ) ) } {\displaystyle \exp \!{\big \{}i\operatorname {Re} ({\overline {w}}'\mu )-{\tfrac {1}{4}}{\big (}{\overline…
Mean
μ {\displaystyle \mathbf {\mu } }
Mode
μ {\displaystyle \mathbf {\mu } }
Parameters
μ ∈ C n {\displaystyle \mathbf {\mu } \in \mathbb {C} ^{n}} — location Γ ∈ C n × n {\displaystyle \Gamma \in \mathbb {C} ^{n\times n}} — covariance matrix (positive semi-definit…
PDF
complicated, see text
Support
C n {\displaystyle \mathbb {C} ^{n}}

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Overview

Definitions

Mean, covariance, and relation

Characteristic function

Properties

Circularly-symmetric central case

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Complex normal distribution

Nodes32
Edges31
Triples16
Avg. degree1.94
Density0.0625
Components1

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Complex normal distribution

Top relations

see also · 4
Complex normal distribution → Complex, Distribution, Generalized, Normal
related to Density function · 3
Complex normal distribution → Gamma, RC, The
CF · 1
Complex normal distribution → exp { i Re ⁡ ( w ¯ ′ μ ) − 1 4 ( w ¯ ′ Γ w + Re ⁡ ( w ¯ ′ C w ¯ ) ) } {\displaystyle \exp \!{\big \{}i\operatorname {Re} ({\overline {w}}'\mu )-{\tfrac {1}{4}}{\big (}{\overline…
Mean · 1
Complex normal distribution → μ {\displaystyle \mathbf {\mu } }
Mode · 1
Complex normal distribution → μ {\displaystyle \mathbf {\mu } }
Parameters · 1
Complex normal distribution → μ ∈ C n {\displaystyle \mathbf {\mu } \in \mathbb {C} ^{n}} — location Γ ∈ C n × n {\displaystyle \Gamma \in \mathbb {C} ^{n\times n}} — covariance matrix (positive semi-definit…
PDF · 1
Complex normal distribution → complicated, see text
Support · 1
Complex normal distribution → C n {\displaystyle \mathbb {C} ^{n}}
Variance · 1
Complex normal distribution → Γ {\displaystyle \Gamma }
is a · 1
Complex normal distribution → bivariate normal distribution

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Important terminology

complex displaystyle normal random distribution matrix vector mathbf gamma mu standard covariance mean relation variable central case mathrm function circularly-symmetric

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Complex normal distributionCFexp { i Re ⁡ ( w ¯ ′ μ ) − 1 4 ( w ¯ ′ Γ w + Re ⁡ ( w ¯ ′ C w ¯ ) ) } {\displaystyle \exp \!{\big \{}i\operatorname {Re} ({\overline {w}}'\mu )-{\tfrac {1}{4}}{\big (}{\overline…1.00infobox
Complex normal distributionMeanμ {\displaystyle \mathbf {\mu } }1.00infobox
Complex normal distributionModeμ {\displaystyle \mathbf {\mu } }1.00infobox
Complex normal distributionParametersμ ∈ C n {\displaystyle \mathbf {\mu } \in \mathbb {C} ^{n}} — location Γ ∈ C n × n {\displaystyle \Gamma \in \mathbb {C} ^{n\times n}} — covariance matrix (positive semi-definit…1.00infobox
Complex normal distributionPDFcomplicated, see text1.00infobox
Complex normal distributionSupportC n {\displaystyle \mathbb {C} ^{n}}1.00infobox
Complex normal distributionVarianceΓ {\displaystyle \Gamma }1.00infobox
Complex normal distributionis abivariate normal distribution0.90text
Complex normal distributionrelated to Characteristic functionThe0.60section
Complex normal distributionrelated to Density functionThe0.60section
Complex normal distributionrelated to Density functionGamma0.60section
Complex normal distributionrelated to Density functionRC0.60section

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