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(11)
Comparative Analysis
Gaussian Distribution
Maximum Entropy
Mutual Information
Process Support
Random Process
Random Variable
Second Order Statistics
Information Theoretic
Second Order
Widely Linear
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Augmented secondorder statistics of quaternion random signals
Augmented secondorder statistics of quaternion random signals,10.1016/j.sigpro.2010.06.024,Signal Processing,Clive Cheong Took,Danilo P. Mandic
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Augmented secondorder statistics of quaternion random signals
(
Citations: 8
)
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Clive Cheong Took
,
Danilo P. Mandic
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 Qproper (second order circular) random processes are presented, and to cater for the nonvanishing pseudocovariance of such processes, the use of iEkcovariances 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 Qproper and Qimproper 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. 214224, 2011
DOI:
10.1016/j.sigpro.2010.06.024
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Citation Context
(4)
...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 Adali
,
et al.
ComplexValued 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 Qproper and Qimproper quaternion signals [
5
, 6, 7]. Existing blind source separation methodologies for the quaternion domain include a semiblind blockbased 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 realvalued 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. Javidi
,
et al.
Blind extraction of improper quaternion sources
...We also show that full secondorder statistical information in the quaternion domain can be exploited by combining the proposed nonlinear models with the socalled 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 κcovariance 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 Hcircular 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 Ujang
,
et al.
QuaternionValued Nonlinear Adaptive Filtering
...(A more general look at the bases used in the regressor vector is presented in [
7
], where a blockdiagonal covariance matrix is obtained...
Fernando G. Almeida Neto
,
et al.
A novel reducedcomplexity widely linear QLMS algorithm
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