A Quaternion Widely Linear Model for Nonlinear Gaussian Estimation

Navarro-Moreno, J.; Fernández-Alcalá, Rosa M.; Ruiz-Molina, Juan Carlos

doi:10.1109/tsp.2014.2364790

Cited by 28 publications

(9 citation statements)

References 49 publications

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“…In the simulations for the SWL-QLMS, the desired signal was generated from the semi-widely linear model in (10) and (12). The variance of the additive Gaussian noise υ n was σ 2 υ = 0.01, and the Euclidean norm of the Gaussian weight variation q n was 0.0008. x n is H-proper and when x n is improper with r = 0.5.…”

Section: B Swl-qlmsmentioning

confidence: 99%

Performance Analysis of Quaternion-Valued Adaptive Filters in Nonstationary Environments

Xiang

Kanna

Mandic

2018

IEEE Trans. Signal Process.

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Quaternion adaptive filters have been widely used for the processing of 3D and 4D phenomena, but complete analysis of their performance is still lacking, partly due to the cumbersomeness of multivariate quaternion analysis. This causes difficulties in both understanding their behaviour and designing optimal filters. Based on a thorough exploration of the augmented statistics of quaternion random vectors, this paper extends an analysis framework for real-valued adaptive filters to the mean and mean square convergence analyses of general quaternion adaptive filters in non-stationary environments. The extension is non-trivial, considering the non-commutative quaternion algebra, only recently resolved issues with quaternion gradient, and the multidimensional augmented quaternion statistics. Also, for rigour, in order to model a non-stationary environment, the system weights are assumed to vary according to a first-order random-walk model. Transient and steady-state performance of a general class of quaternion adaptive filters is provided by exploiting the augmented quaternion statistics. An innovative quaternion decorrelation technique allows us to develop intuitive closed-form expressions for the performance of quaternion least mean square (QLMS) filters with Gaussian inputs, which provide new insights into the relationship between the filter behaviour and the complete second-order statistics of the input signal, that is, quaternion noncircularity. The closed-form expressions for the performance of strictly linear, semi-widely linear, and widely linear QLMS filters are investigated in detail, while numerical simulations for the three classes of QLMS filters with correlated Gaussian inputs support the theoretical analysis.

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Section: B Swl-qlmsmentioning

confidence: 99%

Performance Analysis of Quaternion-Valued Adaptive Filters in Nonstationary Environments

Xiang

Kanna

Mandic

2018

IEEE Trans. Signal Process.

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“…Of particular interest to this work are quaternion involutions around the imaginary units i , j , k , given by [ 21 ]

which allows us to express the four real-valued components of a quaternion q as [ 13 , 21 ]

This is analogous to the complex case, where

and y =−( i /2)( z − z *) for any

[ 22 ]. Note that the quaternion conjugation operator (⋅)* is also an involution and can be written in terms of q , q i , q j and q k as

…”

Section: Background On Quaternionsmentioning

confidence: 99%

“…The widely linear QLMS (WL-QLMS) algorithm is based on the quaternion widely linear model y ( n )= w T ( n ) p ( n ) which deals with the generality of quaternion signals (both proper and improper) [ 8 , 22 , 35 ], where p =( x T ( n ), x iT ( n ), x jT ( n ), x kT ( n )) T is the augmented input vector and w =( h T ( n ), g T ( n ), u T ( n ), v T ( n )) T is the associated weight (parameter) vector. The cost function to be minimized is a real-valued function of quaternion variables

where e ( n )= d ( n )− y ( n ) is the error between the desired signal d ( n ) and the filter output y ( n ).…”

Section: Applications Of the Generalized Hr Calculusmentioning

confidence: 99%

Enabling quaternion derivatives: the generalized HR calculus

Jahanchahi

Took

et al. 2015

R. Soc. open sci.

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Quaternion derivatives exist only for a very restricted class of analytic (regular) functions; however, in many applications, functions of interest are real-valued and hence not analytic, a typical case being the standard real mean square error objective function. The recent HR calculus is a step forward and provides a way to calculate derivatives and gradients of both analytic and non-analytic functions of quaternion variables; however, the HR calculus can become cumbersome in complex optimization problems due to the lack of rigorous product and chain rules, a consequence of the non-commutativity of quaternion algebra. To address this issue, we introduce the generalized HR (GHR) derivatives which employ quaternion rotations in a general orthogonal system and provide the left- and right-hand versions of the quaternion derivative of general functions. The GHR calculus also solves the long-standing problems of product and chain rules, mean-value theorem and Taylor's theorem in the quaternion field. At the core of the proposed GHR calculus is quaternion rotation, which makes it possible to extend the principle to other functional calculi in non-commutative settings. Examples in statistical learning theory and adaptive signal processing support the analysis.

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“…In fact, there are several applications which require higher dimensional representation such as quaternions, which is convenient to represent the rotations of three-dimensional space. Quaternions are used to characterize data of several systems/applications including aerospace [132], computer graphics [133], signal array processing [134], Fourier transforms of images [135], design of orthogonal polarized STBC [136], wave separation [137], wind forecasting [138], nonlinear estimation [139], adaptive filtering [140], [141], and vector sensors [142]. One compelling application is the unified treatment of the relative position and orientation in hand-eye calibration of a robot [143].…”

Section: Overviewmentioning

confidence: 99%

A Journey From Improper Gaussian Signaling to Asymmetric Signaling

Javed

Amin

Shihada

et al. 2020

IEEE Commun. Surv. Tutorials

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The deviation of continuous and discrete complex random variables from the traditional proper and symmetric assumption to a generalized improper and asymmetric characterization (accounting correlation between a random entity and its complex conjugate), respectively, introduces new design freedom and various potential merits. As such, the theory of impropriety has vast applications in medicine, geology, acoustics, optics, image and pattern recognition, computer vision, and other numerous research fields with our main focus on the communication systems. The journey begins from the design of improper Gaussian signaling in the interference-limited communications and leads to a more elaborate and practically feasible asymmetric discrete modulation design. Such asymmetric shaping bridges the gap between theoretically and practically achievable limits with sophisticated transceiver and detection schemes in both coded/uncoded wireless/optical communication systems. Interestingly, introducing asymmetry and adjusting the transmission parameters according to some design criterion render optimal performance without affecting the bandwidth or power requirements of the systems. This dual-flavored article initially presents the tutorial base content covering the interplay of reality/complexity, propriety/impropriety and circularity/non-circularity and then surveys majority of the contributions in this enormous journey.

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A Quaternion Widely Linear Model for Nonlinear Gaussian Estimation

Cited by 28 publications

References 49 publications

Performance Analysis of Quaternion-Valued Adaptive Filters in Nonstationary Environments

Performance Analysis of Quaternion-Valued Adaptive Filters in Nonstationary Environments

Enabling quaternion derivatives: the generalized HR calculus

A Journey From Improper Gaussian Signaling to Asymmetric Signaling

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