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| In [[probability theory]], the family of '''complex normal distributions''' characterizes [[complex random variable]]s whose real and imaginary parts are jointly [[Multivariate normal distribution|normal]],<ref>{{harvtxt|Goodman|1963}}</ref> i.e., normally distributed and independent. The complex normal family has three parameters: ''location'' parameter ''μ'', ''covariance'' matrix Γ, and the ''relation'' matrix ''C''. The '''standard complex normal''' is the univariate distribution with ''μ'' = 0, Γ = 1, and ''C'' = 0. | | In short, the amazing My Little Pony Costume. Arms were often a young girl would be your design featured on our online assistance and ladies halloween costumes place one bag inside the belt. Not ladies halloween costumes all costumes are just plain fun.<br><br>Feel free to surf to my blog post :: [http://Global.theknot.com/SitePreview/Default.aspx?externalurl=http://alicehalloweencostume.net Party Costumes] |
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| An important subclass of complex normal family is called the '''circularly-symmetric complex normal''' and corresponds to the case of zero relation matrix and zero mean: <math> \mu = 0 \ \text{and} \ C=0 </math>.<ref>[http://www.rle.mit.edu/rgallager/documents/CircSymGauss.pdf ''bookchapter, Gallager.R''], pg9.</ref> Circular symmetric complex normal random variables are used extensively in [[signal processing]], and are sometimes referred to as just '''complex normal''' in signal processing literature.
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| ==Definition==
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| Suppose ''X'' and ''Y'' are random vectors in '''R'''<sup>''k''</sup> such that vec[''X Y''] is a 2''k''-dimensional [[normal random vector]]. Then we say that the complex random vector
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| : <math>
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| Z = X + iY \,
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| </math>
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| has the '''complex normal distribution'''. This distribution can be described with 3 parameters:<ref name="picinbono">{{harvtxt|Picinbono|1996}}</ref>
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| : <math>
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| \mu = \operatorname{E}[Z], \quad
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| \Gamma = \operatorname{E}[(Z-\mu)(\overline{Z}-\overline\mu)'], \quad
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| C = \operatorname{E}[(Z-\mu)(Z-\mu)'],
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| </math>
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| where ''Z'' ′ denotes [[matrix transpose]], and <i style="text-decoration:overline">Z</i> denotes [[complex conjugate]]. Here the ''location'' parameter ''μ'' can be an arbitrary k-dimensional complex vector; the ''covariance'' matrix Γ must be [[Hermitian matrix|Hermitian]] and [[non-negative definite]]; the ''relation'' matrix ''C'' should be [[symmetric matrix|symmetric]]. Moreover, matrices Γ and ''C'' are such that the matrix
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| : <math>
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| P = \overline\Gamma - \overline{C}'\Gamma^{-1}C
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| </math>
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| is also non-negative definite.<ref name="picinbono"/>
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| Matrices Γ and ''C'' can be related to the covariance matrices of ''X'' and ''Y'' via expressions
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| : <math>\begin{align} | |
| & V_{xx} \equiv \operatorname{E}[(X-\mu_x)(X-\mu_x)'] = \tfrac{1}{2}\operatorname{Re}[\Gamma + C], \quad
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| V_{xy} \equiv \operatorname{E}[(X-\mu_x)(Y-\mu_y)'] = \tfrac{1}{2}\operatorname{Im}[-\Gamma + C], \\
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| & V_{yx} \equiv \operatorname{E}[(Y-\mu_y)(X-\mu_x)'] = \tfrac{1}{2}\operatorname{Im}[\Gamma + C], \quad\,
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| V_{yy} \equiv \operatorname{E}[(Y-\mu_y)(Y-\mu_y)'] = \tfrac{1}{2}\operatorname{Re}[\Gamma - C],
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| \end{align}</math>
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| and conversely
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| : <math>\begin{align}
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| & \Gamma = V_{xx} + V_{yy} + i(V_{yx} - V_{xy}), \\
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| & C = V_{xx} - V_{yy} + i(V_{yx} + V_{xy}).
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| \end{align}</math>
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| ==Density function==
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| The probability density function for complex normal distribution can be computed as
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| : <math>\begin{align} | |
| f(z) &= \frac{1}{\pi^k\sqrt{\det(\Gamma)\det(P)}}\,
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| \exp\!\left\{-\frac12 \begin{pmatrix}(\overline{z}-\overline\mu)' & (z-\mu)'\end{pmatrix}
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| \begin{pmatrix}\Gamma&C\\\overline{C}'&\overline\Gamma\end{pmatrix}^{\!\!-1}\!
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| \begin{pmatrix}z-\mu \\ \overline{z}-\overline{\mu}\end{pmatrix}
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| \right\} \\[8pt]
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| &= \tfrac{\sqrt{\det\left(\overline{P^{-1}}-\overline{R}'P^{-1}R\right)\det(P^{-1})}}{\pi^k}\,
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| e^{ -(\overline{z}-\overline\mu)'\overline{P^{-1}}(z-\mu) +
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| \operatorname{Re}\left((z-\mu)'R'\overline{P^{-1}}(z-\mu)\right)},
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| \end{align}</math>
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| where ''R'' = <i style="text-decoration:overline">C</i>′ Γ<sup> −1</sup> and ''P'' = <span style="text-decoration:overline">Γ</span> − ''RC''.
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| ==Characteristic function==
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| The [[characteristic function (probability theory)|characteristic function]] of complex normal distribution is given by <ref name="picinbono"/>
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| : <math>
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| \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\},
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| </math>
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| where the argument <math>w</math> is a ''k''-dimensional complex vector.
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| ==Properties==
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| * If ''Z'' is a complex normal ''k''-vector, ''A'' an ''ℓ×k'' matrix, and ''b'' a constant '' ℓ''-vector, then the linear transform {{nowrap|''AZ + b''}} will be distributed also complex-normally:
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| : <math>
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| Z\ \sim\ \mathcal{CN}(\mu,\, \Gamma,\, C) \quad\Rightarrow\quad AZ+b\ \sim\ \mathcal{CN}(A\mu+b,\, A\Gamma\overline{A}',\, ACA')
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| </math>
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| * If ''Z'' is a complex normal ''k''-vector, then
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| : <math>
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| 2\Big[ (\overline{Z}-\overline\mu)'\overline{P^{-1}}(Z-\mu) -
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| \operatorname{Re}\big((Z-\mu)'R'\overline{P^{-1}}(Z-\mu)\big)
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| \Big]\ \sim\ \chi^2(2k)
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| </math>
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| * '''Central limit theorem'''. If ''z''<sub>1</sub>, …, ''z''<sub>''T''</sub> are independent and identically distributed complex random variables, then
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| : <math>
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| \sqrt{T}\Big( \tfrac{1}{T}\textstyle\sum_{t=1}^Tz_t - \operatorname{E}[z_t]\Big) \ \xrightarrow{d}\
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| \mathcal{CN}(0,\,\Gamma,\,C),
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| </math>
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| where Γ = E[ ''z''<i style="text-decoration:overline">z′</i> ] and ''C'' = E[ ''zz′'' ].
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| ==Circularly-symmetric complex normal distribution==
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| The circularly-symmetric complex normal distribution <ref>[http://www.rle.mit.edu/rgallager/documents/CircSymGauss.pdf ''bookchapter, Gallager.R'']</ref> corresponds to the case of zero mean and zero relation matrix, ''μ=0'', ''C=0''. If {{nowrap|1=''Z'' = ''X'' + ''iY''}} is circularly-symmetric complex normal, then the vector vec[''X Y''] is multivariate normal with covariance structure
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| : <math>
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| \begin{pmatrix}X \\ Y\end{pmatrix} \ \sim\
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| \mathcal{N}\Big( \begin{bmatrix}
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| \operatorname{Re}\,\mu \\
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| \operatorname{Im}\,\mu
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| \end{bmatrix},\
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| \tfrac{1}{2}\begin{bmatrix}
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| \operatorname{Re}\,\Gamma & \operatorname{Im}\,\Gamma \\
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| \operatorname{Im}\,\Gamma & \operatorname{Re}\,\Gamma
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| \end{bmatrix}\Big)
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| </math>
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| where {{nowrap|1=''μ'' = E[ ''Z'' ] = 0}} and {{nowrap|1=Γ = E[ ''Z<span style="text-decoration:overline">Z′</span>'' ]}}. This is usually denoted
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| :<math>Z \sim \mathcal{CN}(0,\,\Gamma)</math>
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| and its distribution can also be simplified as
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| : <math>
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| f(z) = \tfrac{1}{\pi^k\det(\Gamma)}\, e^{ -\overline{z}'\; \Gamma^{-1}\; z }.
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| </math>
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| The '''standard complex normal''' corresponds to the distribution of a scalar random variable with ''μ'' = 0, ''C'' = 0 and Γ = 1. Thus, the standard complex normal distribution has density
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| : <math>
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| f(z) = \tfrac{1}{\pi} e^{-\overline{z}z} = \tfrac{1}{\pi} e^{-|z|^2}.
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| </math>
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| This expression demonstrates why the case ''C'' = 0, ''μ'' = 0 is called “circularly-symmetric”. The density function depends only on the magnitude of ''z'' but not on its [[Arg (mathematics)|argument]]. As such, the magnitude ''|z|'' of standard complex normal random variable will have the [[Rayleigh distribution]] and the squared magnitude ''|z|''<sup>2</sup> will have the [[Exponential distribution]], whereas the argument will be distributed [[Uniform distribution (continuous)|uniformly]] on [−''π'', ''π''].
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| If {''z''<sub>1</sub>, …, ''z''<sub>''n''</sub>} are independent and identically distributed ''k''-dimensional circular complex normal random variables with ''μ'' = 0, then random squared norm
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| : <math>
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| Q = \sum_{j=1}^n \overline{z_j'} z_j = \sum_{j=1}^n \| z_j \|^2
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| </math>
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| has the [[Generalized chi-squared distribution]] and the random matrix
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| : <math>
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| W = \sum_{j=1}^n z_j\overline{z_j'}
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| </math>
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| has the [[complex Wishart distribution]] with ''n'' degrees of freedom. This distribution can be described by density function
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| : <math>
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| f(w) = \frac{\det(\Gamma^{-1})^n\det(w)^{n-k}}{\pi^{k(k-1)/2}\prod_{j=1}^p(n-j)!}\
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| e^{-\operatorname{tr}(\Gamma^{-1}w)}
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| </math>
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| where ''n ≥ k'', and ''w'' is a ''k×k'' nonnegative-definite matrix.
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| ==See also==
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| * [[Directional statistics#Distribution of the mean]]
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| * [[Normal distribution]]
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| * [[Multivariate normal distribution]]
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| * [[Generalized chi-squared distribution]]
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| * [[Wishart distribution]]
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| ==References==
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| {{reflist}}
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| {{refbegin}}
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| * {{cite journal
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| | first = N.R.
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| | last = Goodman
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| | year = 1963
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| | title = Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction)
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| | journal = The Annals of Mathematical Statistics
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| | volume = 34
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| | issue = 1
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| | pages = 152–177
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| | jstor = 2991290
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| }}
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| * {{cite journal
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| | last = Picinbono
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| | first = Bernard
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| | year = 1996
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| | title = Second-order complex random vectors and normal distributions
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| | journal = IEEE Transactions on Signal Processing
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| | volume = 44
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| | issue = 10
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| | pages = 2637–2640
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| }}
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| {{refend}}
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| {{ProbDistributions|continuous-infinite}}
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| [[Category:Continuous distributions]]
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| [[Category:Multivariate continuous distributions]]
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| [[Category:Complex numbers]]
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| [[Category:Probability distributions]]
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