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Complex normal distribution
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<h2 id="definitions">Definitions</h2> <h3>Complex standard normal random variable</h3> <p>The standard complex normal random variable or standard complex Gaussian random variable is a complex random variable Z {\displaystyle Z} whose real and imaginary parts are independent normally distributed random variables with mean zero and variance 1 / 2 {\displaystyle 1/2} .<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>: p. 494 <a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>: pp. 501 Formally, </p> <table><tbody><tr><td> Z ∼ C N ( 0 , 1 ) ⟺ ℜ ( Z ) ⊥ ⊥ ℑ ( Z ) and ℜ ( Z ) ∼ N ( 0 , 1 / 2 ) and ℑ ( Z ) ∼ N ( 0 , 1 / 2 ) {\displaystyle 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)} </td> <td></td> <td>Eq.1</td></tr></tbody></table> <p>where Z ∼ C N ( 0 , 1 ) {\displaystyle Z\sim {\mathcal {CN}}(0,1)} denotes that Z {\displaystyle Z} is a standard complex normal random variable. </p> <h3>Complex normal random variable</h3> <p>Suppose X {\displaystyle X} and Y {\displaystyle Y} are real random variables such that ( X , Y ) T {\displaystyle (X,Y)^{\mathrm {T} }} is a 2-dimensional <a href="/facts/Normal_random_vector/2Xfegqz2">normal random vector</a>. Then the complex random variable Z = X + i Y {\displaystyle Z=X+iY} is called complex normal random variable or complex Gaussian random variable.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>: p. 500 </p> <table><tbody><tr><td> Z complex normal random variable ⟺ ( ℜ ( Z ) , ℑ ( Z ) ) T real normal random vector {\displaystyle Z{\text{ complex normal random variable}}\quad \iff \quad (\Re (Z),\Im (Z))^{\mathrm {T} }{\text{ real normal random vector}}} </td> <td></td> <td>Eq.2</td></tr></tbody></table> <h3>Complex standard normal random vector</h3> <p>A n-dimensional complex random vector Z = ( Z 1 , … , Z n ) T {\displaystyle \mathbf {Z} =(Z_{1},\ldots ,Z_{n})^{\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.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a>: p. 502 <a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a>: pp. 501 That Z {\displaystyle \mathbf {Z} } is a standard complex normal random vector is denoted Z ∼ C N ( 0 , I n ) {\displaystyle \mathbf {Z} \sim {\mathcal {CN}}(0,{\boldsymbol {I}}_{n})} . </p> <table><tbody><tr><td> Z ∼ C N ( 0 , I n ) ⟺ ( Z 1 , … , Z n ) independent and for 1 ≤ i ≤ n : Z i ∼ C N ( 0 , 1 ) {\displaystyle \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)} </td> <td></td> <td>Eq.3</td></tr></tbody></table> <h3>Complex normal random vector</h3> <p>If X = ( X 1 , … , X n ) T {\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{n})^{\mathrm {T} }} and Y = ( Y 1 , … , Y n ) T {\displaystyle \mathbf {Y} =(Y_{1},\ldots ,Y_{n})^{\mathrm {T} }} are <a href="/facts/Random_vector/qMfooyVf">random vectors</a> in R n {\displaystyle \mathbb {R} ^{n}} such that [ X , Y ] {\displaystyle [\mathbf {X} ,\mathbf {Y} ]} is a <a href="/facts/Normal_random_vector/2Xfegqz2">normal random vector</a> with 2 n {\displaystyle 2n} components. Then we say that the <a href="/facts/Complex_random_vector/orfynJII">complex random vector</a> </p> Z = X + i Y {\displaystyle \mathbf {Z} =\mathbf {X} +i\mathbf {Y} \,} <p>is a complex normal random vector or a complex Gaussian random vector. </p> <table><tbody><tr><td> Z complex normal random vector ⟺ ( ℜ ( Z T ) , ℑ ( Z T ) ) T = ( ℜ ( Z 1 ) , … , ℜ ( Z n ) , ℑ ( Z 1 ) , … , ℑ ( Z n ) ) T real normal random vector {\displaystyle \mathbf {Z} {\text{ complex normal random vector}}\quad \iff \quad (\Re (\mathbf {Z} ^{\mathrm {T} }),\Im (\mathbf {Z} ^{\mathrm {T} }))^{\mathrm {T} }=(\Re (Z_{1}),\ldots ,\Re (Z_{n}),\Im (Z_{1}),\ldots ,\Im (Z_{n}))^{\mathrm {T} }{\text{ real normal random vector}}} </td> <td></td> <td>Eq.4</td></tr></tbody></table> <h2 id="mean-covariance-and-relation">Mean, covariance, and relation</h2> <p>The complex Gaussian distribution can be described with 3 parameters:<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> </p> μ = E [ Z ] , Γ = E [ ( Z − μ ) ( Z − μ ) H ] , C = E [ ( Z − μ ) ( Z − μ ) T ] , {\displaystyle \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} }],} <p>where Z T {\displaystyle \mathbf {Z} ^{\mathrm {T} }} denotes <a href="/facts/Matrix_transpose/8wmsagGS">matrix transpose</a> of Z {\displaystyle \mathbf {Z} } , and Z H {\displaystyle \mathbf {Z} ^{\mathrm {H} }} denotes <a href="/facts/Conjugate_transpose/7eBWtqVG">conjugate transpose</a>.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a>: p. 504 <a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a>: pp. 500 </p><p>Here the <a href="/facts/Location_parameter/UQTyS3JU">location parameter</a> μ {\displaystyle \mu } is a n-dimensional complex vector; the <a href="/facts/Covariance_matrix/kLhgjHC6">covariance matrix</a> Γ {\displaystyle \Gamma } is <a href="/facts/Hermitian_matrix/qArPblYo">Hermitian</a> and <a href="/facts/Non-negative_definite/Hbr6AuPS">non-negative definite</a>; and, the <i><a href="/facts/Relation_matrix/kLhgjHC6">relation matrix</a></i> or <i>pseudo-covariance matrix</i> C {\displaystyle C} is <a href="/facts/Symmetric_matrix/08CGwjNl">symmetric</a>. The complex normal random vector Z {\displaystyle \mathbf {Z} } can now be denoted as Z ∼ C N ( μ , Γ , C ) . {\displaystyle \mathbf {Z} \ \sim \ {\mathcal {CN}}(\mu ,\ \Gamma ,\ C).} Moreover, matrices Γ {\displaystyle \Gamma } and C {\displaystyle C} are such that the matrix </p> P = Γ ¯ − C H Γ − 1 C {\displaystyle P={\overline {\Gamma }}-{C}^{\mathrm {H} }\Gamma ^{-1}C} <p>is also non-negative definite where Γ ¯ {\displaystyle {\overline {\Gamma }}} denotes the complex conjugate of Γ {\displaystyle \Gamma } .<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> </p> <h2 id="relationships-between-covariance-matrices">Relationships between covariance matrices</h2> <p class="note">Main article: <a href="/facts/Complex_random_vector/orfynJII">Complex random vector § Covariance matrix and pseudo-covariance matrix</a></p> <p>As for any complex random vector, the matrices Γ {\displaystyle \Gamma } and C {\displaystyle C} can be related to the covariance matrices of X = ℜ ( Z ) {\displaystyle \mathbf {X} =\Re (\mathbf {Z} )} and Y = ℑ ( Z ) {\displaystyle \mathbf {Y} =\Im (\mathbf {Z} )} via expressions </p> V X X ≡ E [ ( X − μ X ) ( X − μ X ) T ] = 1 2 Re [ Γ + C ] , V X Y ≡ E [ ( X − μ X ) ( Y − μ Y ) T ] = 1 2 Im [ − Γ + C ] , V Y X ≡ E [ ( Y − μ Y ) ( X − μ X ) T ] = 1 2 Im [ Γ + C ] , V Y Y ≡ E [ ( Y − μ Y ) ( Y − μ Y ) T ] = 1 2 Re [ Γ − C ] , {\displaystyle {\begin{aligned}&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{aligned}}} <p>and conversely </p> Γ = V X X + V Y Y + i ( V Y X − V X Y ) , C = V X X − V Y Y + i ( V Y X + V X Y ) . {\displaystyle {\begin{aligned}&\Gamma =V_{XX}+V_{YY}+i(V_{YX}-V_{XY}),\\&C=V_{XX}-V_{YY}+i(V_{YX}+V_{XY}).\end{aligned}}} <h2 id="density-function">Density function</h2> <p>The probability density function for complex normal distribution can be computed as </p> f ( z ) = 1 π n det ( Γ ) det ( P ) exp { − 1 2 [ z − μ z ¯ − μ ¯ ] H [ Γ C C ¯ Γ ¯ ] − 1 [ z − μ z ¯ − μ ¯ ] } = det ( P − 1 ¯ − R ∗ P − 1 R ) det ( P − 1 ) π n e − ( z − μ ) ∗ P − 1 ¯ ( z − μ ) + Re ( ( z − μ ) ⊺ R ⊺ P − 1 ¯ ( z − μ ) ) , {\displaystyle {\begin{aligned}f(z)&={\frac {1}{\pi ^{n}{\sqrt {\det(\Gamma )\det(P)}}}}\,\exp \!\left\{-{\frac {1}{2}}{\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{aligned}}} <p>where R = C H Γ − 1 {\displaystyle R=C^{\mathrm {H} }\Gamma ^{-1}} and P = Γ ¯ − R C {\displaystyle P={\overline {\Gamma }}-RC} . </p> <h2 id="characteristic-function">Characteristic function</h2> <p>The <a href="/facts/Characteristic_function_(probability_theory)/rl3uAGKM">characteristic function</a> of complex normal distribution is given by<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> </p> φ ( w ) = exp { i Re ( w ¯ ′ μ ) − 1 4 ( w ¯ ′ Γ w + Re ( w ¯ ′ C w ¯ ) ) } , {\displaystyle \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 \}},} <p>where the argument w {\displaystyle w} is an <i>n</i>-dimensional complex vector. </p> <h2 id="properties">Properties</h2> <ul><li>If Z {\displaystyle \mathbf {Z} } is a complex normal <i>n</i>-vector, A {\displaystyle {\boldsymbol {A}}} an <i>m×n</i> matrix, and b {\displaystyle b} a constant <i>m</i>-vector, then the linear transform A Z + b {\displaystyle {\boldsymbol {A}}\mathbf {Z} +b} will be distributed also complex-normally:</li></ul> Z ∼ C N ( μ , Γ , C ) ⇒ A Z + b ∼ C N ( A μ + b , A Γ A H , A C A T ) {\displaystyle Z\ \sim \ {\mathcal {CN}}(\mu ,\,\Gamma ,\,C)\quad \Rightarrow \quad AZ+b\ \sim \ {\mathcal {CN}}(A\mu +b,\,A\Gamma A^{\mathrm {H} },\,ACA^{\mathrm {T} })} <ul><li>If Z {\displaystyle \mathbf {Z} } is a complex normal <i>n</i>-vector, then</li></ul> 2 [ ( Z − μ ) H P − 1 ¯ ( Z − μ ) − Re ( ( Z − μ ) T R T P − 1 ¯ ( Z − μ ) ) ] ∼ χ 2 ( 2 n ) {\displaystyle 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)} <ul><li>Central limit theorem. If Z 1 , … , Z T {\displaystyle Z_{1},\ldots ,Z_{T}} are independent and identically distributed complex random variables, then</li></ul> T ( 1 T ∑ t = 1 T Z t − E [ Z t ] ) → d C N ( 0 , Γ , C ) , {\displaystyle {\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 Γ = E [ Z Z H ] {\displaystyle \Gamma =\operatorname {E} [ZZ^{\mathrm {H} }]} and C = E [ Z Z T ] {\displaystyle C=\operatorname {E} [ZZ^{\mathrm {T} }]} . <ul><li>The modulus of a complex normal random variable follows a <a href="/facts/Hoyt_distribution/HoyxF3AF">Hoyt distribution</a>.<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a></li></ul> <h2 id="circularly-symmetric-central-case">Circularly-symmetric central case</h2> <h3>Definition</h3> <p>A complex random vector Z {\displaystyle \mathbf {Z} } is called circularly symmetric if for every deterministic φ ∈ [ − π , π ) {\displaystyle \varphi \in [-\pi ,\pi )} the distribution of e i φ Z {\displaystyle e^{\mathrm {i} \varphi }\mathbf {Z} } equals the distribution of Z {\displaystyle \mathbf {Z} } .<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a>: pp. 500–501 </p> <p class="note">Main article: <a href="/facts/Complex_random_vector/orfynJII">Complex random vector § Circular symmetry</a></p> <p>Central normal complex random vectors that are circularly symmetric are of particular interest because they are fully specified by the covariance matrix Γ {\displaystyle \Gamma } . </p><p>The <i>circularly-symmetric (central) complex normal distribution</i> corresponds to the case of zero mean and zero relation matrix, i.e. μ = 0 {\displaystyle \mu =0} and C = 0 {\displaystyle C=0} .<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a>: p. 507 <a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> This is usually denoted </p> Z ∼ C N ( 0 , Γ ) {\displaystyle \mathbf {Z} \sim {\mathcal {CN}}(0,\,\Gamma )} <h3>Distribution of real and imaginary parts</h3> <p>If Z = X + i Y {\displaystyle \mathbf {Z} =\mathbf {X} +i\mathbf {Y} } is circularly-symmetric (central) complex normal, then the vector [ X , Y ] {\displaystyle [\mathbf {X} ,\mathbf {Y} ]} is multivariate normal with covariance structure </p> ( X Y ) ∼ N ( [ 0 0 ] , 1 2 [ Re Γ − Im Γ Im Γ Re Γ ] ) {\displaystyle {\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 )}} <p>where Γ = E [ Z Z H ] {\displaystyle \Gamma =\operatorname {E} [\mathbf {Z} \mathbf {Z} ^{\mathrm {H} }]} . </p> <h3>Probability density function</h3> <p>For nonsingular covariance matrix Γ {\displaystyle \Gamma } , its distribution can also be simplified as<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a>: p. 508 </p> f Z ( z ) = 1 π n det ( Γ ) e − ( z − μ ) H Γ − 1 ( z − μ ) {\displaystyle f_{\mathbf {Z} }(\mathbf {z} )={\tfrac {1}{\pi ^{n}\det(\Gamma )}}\,e^{-(\mathbf {z} -\mathbf {\mu } )^{\mathrm {H} }\Gamma ^{-1}(\mathbf {z} -\mathbf {\mu } )}} . <p>Therefore, if the non-zero mean μ {\displaystyle \mu } and covariance matrix Γ {\displaystyle \Gamma } are unknown, a suitable log likelihood function for a single observation vector z {\displaystyle z} would be </p> ln ( L ( μ , Γ ) ) = − ln ( det ( Γ ) ) − ( z − μ ) ¯ ′ Γ − 1 ( z − μ ) − n ln ( π ) . {\displaystyle \ln(L(\mu ,\Gamma ))=-\ln(\det(\Gamma ))-{\overline {(z-\mu )}}'\Gamma ^{-1}(z-\mu )-n\ln(\pi ).} <p>The standard complex normal (defined in Eq.1) corresponds to the distribution of a scalar random variable with μ = 0 {\displaystyle \mu =0} , C = 0 {\displaystyle C=0} and Γ = 1 {\displaystyle \Gamma =1} . Thus, the standard complex normal distribution has density </p> f Z ( z ) = 1 π e − z ¯ z = 1 π e − | z | 2 . {\displaystyle f_{Z}(z)={\tfrac {1}{\pi }}e^{-{\overline {z}}z}={\tfrac {1}{\pi }}e^{-|z|^{2}}.} <h3>Properties</h3> <p>The above expression demonstrates why the case C = 0 {\displaystyle C=0} , μ = 0 {\displaystyle \mu =0} is called “circularly-symmetric”. The density function depends only on the magnitude of z {\displaystyle z} but not on its <a href="/facts/Arg_(mathematics)/JSjfxb5U">argument</a>. As such, the magnitude | z | {\displaystyle |z|} of a standard complex normal random variable will have the <a href="/facts/Rayleigh_distribution/vLFh4zHr">Rayleigh distribution</a> and the squared magnitude | z | 2 {\displaystyle |z|^{2}} will have the <a href="/facts/Exponential_distribution/yY3EdV0u">exponential distribution</a>, whereas the argument will be distributed <a href="/facts/Uniform_distribution_(continuous)/XbnlVljT">uniformly</a> on [ − π , π ] {\displaystyle [-\pi ,\pi ]} . </p><p>If { Z 1 , … , Z k } {\displaystyle \left\{\mathbf {Z} _{1},\ldots ,\mathbf {Z} _{k}\right\}} are independent and identically distributed <i>n</i>-dimensional circular complex normal random vectors with μ = 0 {\displaystyle \mu =0} , then the random squared norm </p> Q = ∑ j = 1 k Z j H Z j = ∑ j = 1 k ‖ Z j ‖ 2 {\displaystyle Q=\sum _{j=1}^{k}\mathbf {Z} _{j}^{\mathrm {H} }\mathbf {Z} _{j}=\sum _{j=1}^{k}\|\mathbf {Z} _{j}\|^{2}} <p>has the <a href="/facts/Generalized_chi-squared_distribution/jm2A9dWu">generalized chi-squared distribution</a> and the random matrix </p> W = ∑ j = 1 k Z j Z j H {\displaystyle W=\sum _{j=1}^{k}\mathbf {Z} _{j}\mathbf {Z} _{j}^{\mathrm {H} }} <p>has the <a href="/facts/Complex_Wishart_distribution/ct6sBK9W">complex Wishart distribution</a> with k {\displaystyle k} degrees of freedom. This distribution can be described by density function </p> f ( w ) = det ( Γ − 1 ) k det ( w ) k − n π n ( n − 1 ) / 2 ∏ j = 1 k ( k − j ) ! e − tr ( Γ − 1 w ) {\displaystyle 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)}} <p>where k ≥ n {\displaystyle k\geq n} , and w {\displaystyle w} is a n × n {\displaystyle n\times n} nonnegative-definite matrix. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Complex_normal_ratio_distribution/5rUEmWkG">Complex normal ratio distribution</a></li> <li><a href="/facts/Directional_statistics/ucnUv52T">Directional statistics § Distribution of the mean</a> (polar form)</li> <li><a href="/facts/Normal_distribution/UapjjPyQ">Normal distribution</a></li> <li><a href="/facts/Multivariate_normal_distribution/2Xfegqz2">Multivariate normal distribution</a> (a complex normal distribution is a bivariate normal distribution)</li> <li><a href="/facts/Generalized_chi-squared_distribution/jm2A9dWu">Generalized chi-squared distribution</a></li> <li><a href="/facts/Wishart_distribution/MEcqTump">Wishart distribution</a></li> <li><a href="/facts/Complex_random_variable/h2v6W3rf">Complex random variable</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Goodman, N.R. (1963). "Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction)". The Annals of Mathematical Statistics. 34 (1): 152–177. doi:10.1214/aoms/1177704250. JSTOR 2991290. <a href="https://doi.org/10.1214%2Faoms%2F1177704250" target="_blank">https://doi.org/10.1214%2Faoms%2F1177704250</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>bookchapter, Gallager.R, pg9. <a href="http://www.rle.mit.edu/rgallager/documents/CircSymGauss.pdf" target="_blank">http://www.rle.mit.edu/rgallager/documents/CircSymGauss.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Lapidoth, A. (2009). A Foundation in Digital Communication. Cambridge University Press. ISBN 9780521193955. <a href="9780521193955" target="_blank">9780521193955</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Tse, David (2005). Fundamentals of Wireless Communication. Cambridge University Press. ISBN 9781139444668. <a href="9781139444668" target="_blank">9781139444668</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Lapidoth, A. (2009). A Foundation in Digital Communication. Cambridge University Press. ISBN 9780521193955. <a href="9780521193955" target="_blank">9780521193955</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Lapidoth, A. (2009). A Foundation in Digital Communication. Cambridge University Press. ISBN 9780521193955. <a href="9780521193955" target="_blank">9780521193955</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Tse, David (2005). Fundamentals of Wireless Communication. Cambridge University Press. ISBN 9781139444668. <a href="9781139444668" target="_blank">9781139444668</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Picinbono, Bernard (1996). "Second-order complex random vectors and normal distributions". IEEE Transactions on Signal Processing. 44 (10): 2637–2640. Bibcode:1996ITSP...44.2637P. doi:10.1109/78.539051. <a href="https://ieeexplore-ieee-org.ezp1.lib.umn.edu/document/539051" target="_blank">https://ieeexplore-ieee-org.ezp1.lib.umn.edu/document/539051</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Lapidoth, A. (2009). A Foundation in Digital Communication. Cambridge University Press. ISBN 9780521193955. <a href="9780521193955" target="_blank">9780521193955</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Tse, David (2005). Fundamentals of Wireless Communication. Cambridge University Press. ISBN 9781139444668. <a href="9781139444668" target="_blank">9781139444668</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Picinbono, Bernard (1996). "Second-order complex random vectors and normal distributions". IEEE Transactions on Signal Processing. 44 (10): 2637–2640. Bibcode:1996ITSP...44.2637P. doi:10.1109/78.539051. <a href="https://ieeexplore-ieee-org.ezp1.lib.umn.edu/document/539051" target="_blank">https://ieeexplore-ieee-org.ezp1.lib.umn.edu/document/539051</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Picinbono, Bernard (1996). "Second-order complex random vectors and normal distributions". IEEE Transactions on Signal Processing. 44 (10): 2637–2640. Bibcode:1996ITSP...44.2637P. doi:10.1109/78.539051. <a href="https://ieeexplore-ieee-org.ezp1.lib.umn.edu/document/539051" target="_blank">https://ieeexplore-ieee-org.ezp1.lib.umn.edu/document/539051</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Daniel Wollschlaeger. "The Hoyt Distribution (Documentation for R package 'shotGroups' version 0.6.2)".[permanent dead link] <a href="http://finzi.psych.upenn.edu/usr/share/doc/library/shotGroups/html/hoyt.html" target="_blank">http://finzi.psych.upenn.edu/usr/share/doc/library/shotGroups/html/hoyt.html</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Tse, David (2005). Fundamentals of Wireless Communication. Cambridge University Press. ISBN 9781139444668. <a href="9781139444668" target="_blank">9781139444668</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Lapidoth, A. (2009). A Foundation in Digital Communication. Cambridge University Press. ISBN 9780521193955. <a href="9780521193955" target="_blank">9780521193955</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>bookchapter, Gallager.R <a href="http://www.rle.mit.edu/rgallager/documents/CircSymGauss.pdf" target="_blank">http://www.rle.mit.edu/rgallager/documents/CircSymGauss.pdf</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>Lapidoth, A. (2009). A Foundation in Digital Communication. Cambridge University Press. ISBN 9780521193955. <a href="9780521193955" target="_blank">9780521193955</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> </ol>