Lesson plan
Complex normal distribution
{{Probability distribution | name = Complex normal | type = multivariate | pdf_image = | cdf_image = | notation = | parameters = \mathbf{\mu} \in \mathbb{C}^n — location \Gamma \in \mathbb{C}^{n \times n} — covariance matrix (positive semi-definite matrix) C \in \mathbb{C}^{n \times n} — relation matrix (complex symmetric matrix) | support = \mathbb{C}^n | pdf = complicated, see text | mean = \mathbf{\mu} | mode = \mathbf{\mu} | variance = \Gamma | cf = \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\} }} In probability theory, the family of complex normal distributions, denoted \mathcal{CN} or \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 \Gamma , and the relation matrix C . The standard complex normal is the univariate distribution with \mu = 0 , \Gamma=1 , and C=0 . 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: \mu = 0 and C=0 . This case is used extensively in signal processing, where it is sometimes referred to as just complex normal in the literature.
Article summary
{{Probability distribution | name = Complex normal | type = multivariate | pdf_image = | cdf_image = | notation = | parameters = \mathbf{\mu} \in \mathbb{C}^n — location \Gamma \in \mathbb{C}^{n \times n} — covariance matrix (positive semi-definite matrix) C \in \mathbb{C}^{n \times n} — relation matrix (complex symmetric matrix) | support = \mathbb{C}^n | pdf = complicated, see text | mean = \mathbf{\mu} | mode = \mathbf{\mu} | variance = \Gamma | cf = \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\} }} In probability theory, the family of complex normal distributions, denoted \mathcal{CN} or \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 \Gamma , and the relation matrix C . The standard complex normal is the univariate distribution with \mu = 0 , \Gamma=1 , and C=0 . 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: \mu = 0 and C=0 . This case is used extensively in signal processing, where it is sometimes referred to as just complex normal in the literature.
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Discussion questions
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