The Multivariate Complex Normal Distribution

Andersen, Henning Haahr; Højbjerre, Malene; Sørensen, D.; Eriksen, Poul Svante

doi:10.1007/978-1-4612-4240-6_2

Cited by 45 publications

(79 citation statements)

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“…The data can be represented using a real-valued representation and a multivariate normal distribution (Rowe, 2003). The real-valued representation used here is very general and within this framework contains the particular representation used to represent the complex-valued multivariate normal distribution (Anderson et al, 1995;Wooding, 1956). The transformation from complex-valued spatial frequency space to image space modifies properties of both the true noiseless signal and the measurement noise.…”

Section: Discussionmentioning

confidence: 99%

“…(A.3), let the 2p x dimensional vector be multivariate normally distributed (Rowe, 2003) with mean and covariance matrix: (A.4) Complex multivariate normal structure occurs when Λ 11 = Λ 22 = Ψ, -Λ 12 = Υ, and . That is, when the covariance matrix is of a skew-symmetric form as (Wooding, 1956;Anderson et al, 1995). The current representation is more general and less restrictive than multivariate complex normal structure.…”

Section: Discussionmentioning

confidence: 99%

“…By carrying out a multivariate transformation of variable with the real-valued representation from s to ρ through ρ = Ω x s, the statistical distribution of ρ is also multivariate normally distributed but with mean ρ 0 given by and covariance matrix , given by (A.5) where Ω x is of full rank if it is a Fourier matrix. Again, this representation is more general and less restrictive than multivariate complex normal structure (Anderson et al, 1995;Wooding, 1956). In the multivariate complex normal case (Anderson et al, 1995;Wooding, 1956) where Λ 11 = Λ 22 = Ψ, -Λ 12 = Υ , and , the covariance matrix Δ is (A.6) where Υ is a skew symmetric matrix, Υ T = -Υ .…”

Section: Discussionmentioning

confidence: 99%

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Signal and noise of Fourier reconstructed fMRI data

Rowe

Nencka

Hoffmann

2007

Journal of Neuroscience Methods

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In magnetic resonance imaging, complex-valued measurements are acquired in time corresponding to spatial frequency measurements in space generally placed on a Cartesian rectangular grid. These complex-valued measurements are transformed into a measured complexvalued image by an image reconstruction method. The most common image reconstruction method is the inverse Fourier transform. It is known that image voxels are spatially correlated. A property of the inverse Fourier transformation is that uncorrelated spatial frequency measurements yield spatially uncorrelated voxel measurements and vice versa. Spatially correlated voxel measurements result from correlated spatial frequency measurements. This paper describes the resulting correlation structure between voxel measurements when inverse Fourier reconstructing correlated spatial frequency measurements. A real-valued representation for the complex-valued measurements is introduced along with an associated multivariate normal distribution. One potential application of this methodology is that there may be a correlation structure introduced by the measurement process or adjustments made to the spatial frequencies. This would produce spatially correlated voxel measurements after inverse Fourier transform reconstruction that have artificially inflated spatial correlation. One implication of these results is that one source of spatial correlation between voxels termed connectivity may be attributed to correlated spatial frequencies. The true voxel connectivity may be less than previously thought. This methodology could be utilized to characterize noise correlation in its original form and adjust for it. The exact statistical relationship between spatial frequency measurements and voxel measurements has now been established.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

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Signal and noise of Fourier reconstructed fMRI data

Rowe

Nencka

Hoffmann

2007

Journal of Neuroscience Methods

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“…First studies of models combining symmetry constraints with conditional independence relations are given in [13][14][15]. The models we study have been introduced in [1] and contain the models in [13] and [14] as special cases. The types of restrictions are: equality between specified elements of the concentration matrix (RCON) and equality between specified partial correlations (RCOR).…”

Section: Introductionmentioning

confidence: 99%

Lattices of Graphical Gaussian Models with Symmetries

Gehrmann

2011

Symmetry

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In order to make graphical Gaussian models a viable modelling tool when the number of variables outgrows the number of observations, [1] introduced model classes which place equality restrictions on concentrations or partial correlations. The models can be represented by vertex and edge coloured graphs. The need for model selection methods makes it imperative to understand the structure of model classes. We identify four model classes that form complete lattices of models with respect to model inclusion, which qualifies them for an Edwards-Havránek model selection procedure [2]. Two classes turn out most suitable for a corresponding model search. We obtain an explicit search algorithm for one of them and provide a model search example for the other.

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“…The probability density function (PDF) of the complex Gaussian random variable  is written as [13] The PDF of the second term  in  is given by…”

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confidence: 99%

A Block Transmission Method for Converting a Non-Gaussian Channel into Parallel Gaussian Noise Channels

Murakami¹

2013

IJCCE

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Abstract-In many communication systems, information carrying symbols are transmitted in blocks with guard intervals between blocks for avoiding inter-block interference. This paper proposes a block transmission system that can be decomposed into a set of parallel subblock transmission systems having additive white Gaussian noise (AWGN) channels with equal noise powers even when the channel additive noise is non-Gaussian. The block transmission system has the advantages that standard signal design, block codes, and symbol detection methods for an AWGN channel are easily incorporated into the block transmission system and that the system is easily implemented by FFT processors.

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The Multivariate Complex Normal Distribution

Cited by 45 publications

References 0 publications

Signal and noise of Fourier reconstructed fMRI data

Signal and noise of Fourier reconstructed fMRI data

Lattices of Graphical Gaussian Models with Symmetries

A Block Transmission Method for Converting a Non-Gaussian Channel into Parallel Gaussian Noise Channels

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