Elsevier

Signal Processing

Volume 91, Issue 4, April 2011, Pages 821-831
Signal Processing

Direction finding via biquaternion matrix diagonalization with vector-sensors

https://doi.org/10.1016/j.sigpro.2010.08.015Get rights and content

Abstract

Direction-of-arrival (DOA) estimation based on the array of three-component electromagnetic vector-sensors is considered within a biquaternion framework. A relationship is established between the biquaternion covariance and the combination of both complex covariance and cross-product. By exploiting this relationship, the DOA estimates can finally be obtained by diagonalizing the biquaternion covariance matrix of the array outputs in a trilinear PARAFAC manner. This method does not require any a priori knowledge on the position of each sensor, and is shown to offer high robustness to colored noise for direction finding of non-linearly polarized signals. Simulations are provided to illustrate the performance of the proposed method.

Introduction

Direction-of-arrival (DOA) estimation with electromagnetic (EM) vector-sensors has received growing interests in the past decades. A ‘complete’ EM vector-sensor consists of six EM sensors (for example, orthogonally oriented short dipoles and small loops arranged in a collocated or distributed manner), and provides complete electric and magnetic field measurements induced by an EM incidence [1], [2], [3]. An ‘incomplete’ EM vector-sensor such as tripole and crossed dipole comprises only a subset of the above mentioned six EM sensors. Compared with the complete EM vector-sensor, the incomplete one has simpler structure and lower cost, and thus is of great interests in some practical applications [4], [5]. Numerous algorithms for DOA estimation with one or more EM vector-sensors were proposed. For example, maximum likelihood strategy based on vector-sensors was discussed in [6], [7], [8], [9], multiple signal classification (MUSIC [10]) was extended for both incomplete and complete EM vector-sensor arrays in [11], [12], [13], [14], [15], [16], subspace fitting technique was reconsidered for incomplete EM vector-sensors in [17], [18], estimation of signal parameters via rotational invariance techniques (ESPRIT [19]) was revised for EM vector-sensor(s) in [2], [20], [21], [22], [23], [24], [25], [26], and cross-product based direction finding/tracking methods were developed for both incomplete and complete EM vector-sensors in [3], [27], [28]. Tensor decompositions (such as parallel factor analysis and higher order eigenvalue decomposition) for DOA estimation and beamforming were considered in [29], [30], [31], [32]. The identifiability issue of EM vector-sensor based DOA estimation has been discussed in [33], [34], [35] and some other related work can be found in [36], [37], [38], [39], [40], [41].

More recently, some efforts have been made on characterizing the output of vector-sensors within a hypercomplex framework, wherein the vectorial structure of each vector-sensor is encapsulated into a hypercomplex scalar with one real part and multiple imaginary parts [42], [43], [44], [45], [46]. In particular, a quaternion version of singular value decomposition was applied to real-valued polarized wave separation [42] with three-component vector-sensors, MUSIC was extended to quaternion, biquaternion, quad-quaternion domains in [43], [44], [45], respectively, and ESPRIT was revised within the quaternion framework for a spatially shift-invariant array of crossed dipoles [46]. In these applications, the local vector components of a vector-sensor array are retained and operated in a compact hypercomplex manner, resulting in a more concise formalism and a better robustness to array modelization errors [43], [44], [45], [46]. However, to our knowledge, the geometric sense of hypercomplex operations is scarcely considered in existing hypercomplex based direction finding methods, and thus the potential advantages of using hypercomplex algebras in DOA estimation have not been fully exploited.

In this paper, we consider to incorporate the geometric sense of biquaternion operations into the current biquaternion DOA estimation framework with tripoles. More exactly, by interpreting the multiplication of two biquaternion scalars as the cross-product of two complex vectors, the biquaternion covariance could be related to the combination of both complex covariance and cross-product, and the biquaternion factors carrying DOA information of incident sources could then be extracted from the output signals, by diagonalizing the biquaternion covariance matrix via a trilinear parallel factor analysis (PARAFAC) scheme.

The rest of the paper is organized as follows. Section 2 introduces some biquaternion algebra prerequisites with emphasis on the geometric sense of biquaternion multiplications. The biquaternion measurement model is set up in Section 3 and the proposed algorithm is presented in Section 4. In Section 5, the performance of the new algorithm is demonstrated with simulations. Finally, this paper is concluded in Section 6.

Section snippets

Biquaternion algebra prerequisites

Biquaternions are an eight-dimensional algebra consisting of quaternion numbers with complex coefficients, and were first considered by Hamilton [47]. In this section, we review some definitions and results of biquaternion algebra that are used in the following sections. A detailed introduction of biquaternion algebra could be found in [44], [47].

Definition 1

Biquaternion and biquaternion matrix: A biquaternion bHC is defined as

b(b00+eb01)+i(b10+eb11)+j(b20+eb21)+k(b30+eb31)where bmnR(m=0,1,2,3, n

Measurement model and statistics

In this section, the biquaternion model for the output of a tripole array is firstly established, and then the biquaternion covariance is introduced and further related to the combination of cross-product and complex covariance. Based on this relationship, we show that the DOA information of impinging signals is implicit in the diagonalization structure of the biquaternion covariance matrix. The details are as follows.

Biquaternion matrix diagonalization via PARAFAC

Definition 5

The biquaternion matrix diagonalization (BMD) of a Hermitian biquaternion matrix BHCN×N (a square biquaternion matrix B is said to be Hermitian if BH=B), is to find a complex matrix U=[u1,u2,,uR]CN×R and a biquaternion diagonal matrix Λ=diag(μ1,,μR)HCR×R, such that B=UΛUH=r=1RurμrurH.

We propose the following biquaternion least-squares principle to look for the desired components for BMD:{u˜r,μ˜r|r=1,2,,R}=argminur,μrBr=1RururHμr2where ‘’ denotes the norm of a biquaternion matrix,

Simulation results

In this section, we present some simulations to compare the performance of the proposed method (termed as the BMD method) with existing biquaternion based method and trilinear methods. All the statistics shown below are the average of the results obtained from totally 200 independent trials. We use the Overall Root Mean Squared Angular Error (RMSAE) [3] to measure the accuracy of DOA estimation as follows:χ1Mm=1ME(arccos(pθm,φmTpθ˜mφ˜m))where pθmφm and pθ˜mφ˜m are the mth true and estimated

Conclusion

In this paper, we have proposed a new DOA estimator for three-component electromagnetic vector-sensor arrays (tripole arrays) within a biquaternion framework. This method exploits the link between biquaternion covariance and cross-product plus complex covariance, and provides closed form DOA estimates by performing biquaternion matrix diagonalization (BMD) to the biquaternion covariance matrix of the array outputs. In addition, the proposed method does not require any knowledge on the array’s

Acknowledgments

The authors would like to thank Associated Editor Dr. N. D. Sidiropoulos and the anonymous reviewers for their useful suggestions. This work was supported by the National Natural Science Foundation of China under contract nos. 60672084, 61072098, 60736006.

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