Quad-Quaternion MUSIC for DOA Estimation Using Electromagnetic Vector Sensors

EURASIP Journal on Advances in Signal Processing, Apr 2009

A new quad-quaternion model is herein established for an electromagnetic vector-sensor array, under which a multidimensional algebra-based direction-of-arrival (DOA) estimation algorithm, termed as quad-quaternion MUSIC (QQ-MUSIC), is proposed. This method provides DOA estimation (decoupled from polarization) by exploiting the orthogonality of the newly defined "quad-quaternion" signal and noise subspaces. Due to the stronger constraints that quad-quaternion orthogonality imposes on quad-quaternion vectors, QQ-MUSIC is shown to offer high robustness to model errors, and thus is very competent in practice. Simulation results have validated the proposed method.

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Quad-Quaternion MUSIC for DOA Estimation Using Electromagnetic Vector Sensors

EURASIP Journal on Advances in Signal Processing Hindawi Publishing Corporation Quad-Quaternion MUSIC for DOA Estimation Using Electromagnetic Vector Sensors Xiaofeng Gong 0 Zhiwen Liu 0 Yougen Xu 0 0 Department of Electronic Engineering, Beijing Institute of Technology , Beijing 100081 , China A new quad-quaternion model is herein established for an electromagnetic vector-sensor array, under which a multidimensional algebra-based direction-of-arrival (DOA) estimation algorithm, termed as quad-quaternion MUSIC (QQ-MUSIC), is proposed. This method provides DOA estimation (decoupled from polarization) by exploiting the orthogonality of the newly defined “quadquaternion” signal and noise subspaces. Due to the stronger constraints that quad-quaternion orthogonality imposes on quadquaternion vectors, QQ-MUSIC is shown to offer high robustness to model errors, and thus is very competent in practice. Simulation results have validated the proposed method. 1. INTRODUCTION A “complete” electromagnetic (EM) vector sensor comprises six collocated and orthogonally oriented EM sensors (e.g., short dipole and small loop), and provides complete electric and magnetic field measurements induced by an EM incidence [ 1–3 ]. An “incomplete” EM vector sensor with one or more components removed is also of high interest in some practical applications [ 4, 5 ]. Numerous algorithms for direction-of-arrival (DOA) estimation using one or more EM vector sensors have been proposed. For example, vector sensor-based maximum likelihood strategy was addressed in [ 6–9 ], multiple signal classification (MUSIC [ 10 ]) was extended for both incomplete and complete EM vector-sensor arrays in [ 11–16 ], subspace fitting technique was reconsidered for incomplete EM vector sensors in [ 17, 18 ], and estimation of signal parameters via rotational invariance techniques (ESPRIT [ 19 ]) was revised for EM vector sensor(s) in [ 20–26 ]. The identifiability issue of EM vector sensor-based DOA estimation has been discussed in [ 27–29 ]. Some other related work can be found in [ 30–34 ]. In all the contributions mentioned above, complex-valued vectors are used to represent the output of each EM vector sensor in the array, and the collection of an EM vector-sensor array is arranged via concatenation of these vectors into a “long vector.” Consequently, the corresponding algorithms somehow destroy the vector nature of incident signals carrying multidimensional information in space, time, and polarization. More recently, a few efforts have been made on characterizing the output of vector sensors within a hypercomplex framework, wherein hypercomplex values, such as quaternions and biquaternions, are used to retain the vector nature of each vector sensor [ 35–37 ]. In particular, singular value decomposition technique was extended for quaternion matrices in [ 35 ] using three-component vector sensors. Quaternion-based MUSIC variant (Q-MUSIC) was proposed in [ 36 ] by using two-component vector sensors. Biquaternion-based MUSIC (BQ-MUSIC) was proposed in [ 37 ] by employing three-component vector sensors. The advantage of using quaternions and biquaternions for vector sensors is that the local vector nature of a vector-sensor array is preserved in multiple imaginary parts, and thus could result in a more compact formalism and a better estimation of signal subspace [ 36, 37 ]. More importantly, from the algebraic point of view, the algebras of quaternions and biquaternions are associative division algebras using specified norms [38], and therefore are convenient to use in the modeling and analysis of vector-sensor array processing. However, it is important to note that quaternions and biquaternions deal with only four-dimensional (4D) and (8D) algebras, respectively, while a full characterization of the sensor output for complete six-component EM vector sensors requires an algebra with dimensions equal to 12 or more. Unfortunately, not all algebras having 12 or more dimensions are associative division algebras. For example, sedenions, as a well-known 16D algebra, are neither an associative algebra nor a division algebra [ 39 ], and thus are not suitable for the modeling and analysis of vector sensors. In this paper, we use a specific 16D algebra— quad-quaternions algebra [ 40–42 ] to model the output of six-component EM vector sensor(s) [ 3 ]. This 16D quadquaternion algebra can be proved to be an associative division algebra, and thus is well adapted to the modeling and analysis of complete EM vector sensors. More precisely, We redefine the array manifold, signal subspace, and noise subspace from a quad-quaternion perspective, and propose a quad-quaternion-based MUSIC variant (QQMUSIC) for DOA estimation by recognizing and exploiting the quad-quaternion orthogonality between the quadquaternion signal and noise subspaces. QQ-MUSIC here is shown to be more attractive in the presence of two typical model errors, that is, s (...truncated)


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Xiaofeng Gong, Zhiwen Liu, Yougen Xu. Quad-Quaternion MUSIC for DOA Estimation Using Electromagnetic Vector Sensors, EURASIP Journal on Advances in Signal Processing, 2009, pp. 213293, Volume 2008, Issue 1, DOI: 10.1155/2008/213293