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Augmented second-order statistics of quaternion random signals

Augmented second-order statistics of quaternion random signals,10.1016/j.sigpro.2010.06.024,Signal Processing,Clive Cheong Took,Danilo P. Mandic

Augmented second-order statistics of quaternion random signals   (Citations: 8)
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Second order statistics of quaternion random variables and signals are revisited in order to exploit the complete second order statistical information available. The conditions for Q-proper (second order circular) random processes are presented, and to cater for the non-vanishing pseudocovariance of such processes, the use of i-E-k-covariances is investigated. Next, the augmented statistics and the corresponding widely linear model are introduced, and a generic multivariate Gaussian distribution is subsequently derived for both Q-proper and Q-improper processes. The maximum entropy bound and an extension of mutual information to multivariate processes are derived in order to provide a complete description of joint information theoretic properties of general quaternion valued processes. A comparative analysis with the corresponding second order statistics of quadrivariate real valued processes supports the approach.
Journal: Signal Processing , vol. 91, no. 2, pp. 214-224, 2011
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    • ...Examples of papers dealing explicitly with maximally improper signalsare[1],[2],[49],[107].Inaddition,wewouldliketonote the growing interest in the extension of these results to hypercomplex numbers, in particular quaternions (see, e.g., [9], [17], [120], [121], [124], [130], and [131])...

    Tülay Adaliet al. Complex-Valued Signal Processing: The Proper Way to Deal With Impropri...

    • ...A unifying framework has recently been proposed in [5] which defines a set of four bases from which to construct augmented quaternion statistics, with a similar approach given in [6]...
    • ...The quaternion widely linear model uses those bases to allow for the optimal minimum mean square error modelling of both Q-proper and Qimproper quaternion signals [5, 6, 7]. Existing blind source separation methodologies for the quaternion domain include a semi-blind block-based algorithm in [8], based on the calculation of rotation angle of whitened quaternion data, and the maximum likelihood approach in [9] where the choice of ...
    • ...Consider the quaternion signal y(k )= ya(k )+ ıyb(k )+ jyc(k )+ κyd(k), where ya(k) ,y b(k) ,y c(k) and yd(k) are real-valued scalars, and ı, j and κ are orthogonal unit vectors, where ı 2 = j 2 = κ 2 = −1. Its optimal linear mean square estimate in terms of the observation x(k) ∈ H N is given by the widely linear model [5]...
    • ...A detailed account of the quaternion augmented statistics and WL model can be found in [5, 6, 7]...

    S. Javidiet al. Blind extraction of improper quaternion sources

    • ...We also show that full second-order statistical information in the quaternion domain can be exploited by combining the proposed nonlinear models with the so-called augmented quaternion statistics and the widely linear model [21], [22]...
    • ...These complementary covariance matrices are termed the ı -covariance Cqı , j -covariance Cqj ,a nd κ-covariance Cqκ , and are given by [21] and [22]...
    • ...The basis proposed in [21] and used here, q a =[ q T q ıT q j T q κ T ] T , provides most convenient representation, as shown in the augmented covariance structure for H-circular signals in (12) and (15)...
    • ...The quaternion widely linear model is based on the augmented basis that builds the matrix C a q (12), and can be described by [21], [28] and [22]...
    • ...We shall now extend the QNGD to fully capture the secondorder statistics of the signal by incorporating the quaternion widely linear model [21], [22], [28] into its derivation, resulting in the augmented quaternion nonlinear gradient...

    Bukhari Che Ujanget al. Quaternion-Valued Nonlinear Adaptive Filtering

    • ...(A more general look at the bases used in the regressor vector is presented in [7], where a block-diagonal covariance matrix is obtained...

    Fernando G. Almeida Netoet al. A novel reduced-complexity widely linear QLMS algorithm

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