Direction finding via biquaternion matrix diagonalization with vector-sensors
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 is defined as
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 (a square biquaternion matrix B is said to be Hermitian if ), is to find a complex matrix and a biquaternion diagonal matrix , such that .
We propose the following biquaternion least-squares principle to look for the desired components for BMD:where ‘’ 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:where and 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.
References (56)
- et al.
Direction-of-arrival estimation via twofold mode-projection
Signal Process.
(May 2009) - et al.
Singular value decomposition of quaternion matrices: a new tool for vector-sensor signal processing
Signal Process.
(July 2004) Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics
Linear Algebra Appl.
(1977)- et al.
An enhanced line search scheme for complex-valued tensor decompositions. Application in DS-CDMA
Signal Process.
(March 2008) - et al.
A comparison of algorithms for fitting the PARAFAC model
Comput. Stat. Data Anal.
(April 2006) The tripole antenna: an adaptive array with full polarization flexibility
IEEE Trans. Antennas Propag.
(November 1981)Direction and polarization estimation using arrays with small loops and short dipoles
IEEE Trans. Antennas Propag.
(March 1993)- et al.
Vector-sensor array processing for electromagnetic source localization
IEEE Trans. Signal Process.
(February 1994) Direction finding/polarization estimation-dipole and/or loop triad(s)
IEEE Trans. Aerosp. Electron. Syst.
(April 2001)- et al.
An efficient vector sensor configuration for source localization
IEEE Signal Process. Lett.
(August 2004)
Maximum likelihood localization of diversely polarized sources by simulated annealing
IEEE Trans. Antennas Propag.
Maximum likelihood signal estimation for polarization sensitive arrays
IEEE Trans. Antennas Propag.
Efficient parameter estimation of partially polarized electromagnetic waves
IEEE Trans. Signal Process.
Maximum likelihood methods for determining the direction of arrival for a single electromagnetic source with unknown polarization
IEEE Trans. Signal Process.
Direction finding with an array of antennas having diverse polarizations
IEEE Trans. Antennas Propag.
A pencil-MUSIC algorithm for finding two-dimensional angles and polarizations using crossed dipoles
IEEE Trans. Antennas Propag.
Direction finding for diversely polarized signals using polynomial rooting
IEEE Trans. Signal Process.
Self-initiating MUSIC-based direction finding and polarization estimation in spatio-polarizational beamspace
IEEE Trans. Antennas Propag.
Root-MUSIC-based direction-finding and polarization estimation using diversely polarized possibly collocated antennas
IEEE Antennas Wireless Propag. Lett.
Closed-form eigenstructure-based direction finding using arbitrary but identical subarrays on a sparse uniform Cartesian array grid
IEEE Trans. Signal Process.
Subspace fitting with diversely polarized antenna arrays
IEEE Trans. Antennas Propag.
Efficient direction and polarization estimation with a COLD array
IEEE Trans. Antennas Propag.
ESPRIT-estimation of signal parameters via rotational invariance techniques
IEEE Trans. Acoust. Speech Signal Process.
Angle and polarization estimation using ESPRIT with a polarization sensitive array
IEEE Trans. Antennas Propag.
Angle estimation using a polarization sensitive array
IEEE Trans. Antennas Propag.
Two-dimensional angle and polarization estimation using the ESPRIT algorithm
IEEE Trans. Antennas Propag.
On polarization estimation using a crossed-dipole array
IEEE Trans. Signal Process.
Cited by (38)
A coarray processing technique for nested vector-sensor arrays with improved resolution capabilities
2022, Digital Signal Processing: A Review JournalCitation Excerpt :In particular, tensorial DOA estimation approaches mainly based on higher order singular value decomposition (HOSVD) and parallel factor (PARAFAC) analysis have been designed in [18–23]. On the other hand, hypercomplex numbers including quaternions and biquaternions have been employed to construct compact vector-sensor array signal models leading to computationally efficient DOA estimation algorithms in [24–27]. The organization of the present work is as follows.
Channel equalization and beamforming for quaternion-valued wireless communication systems
2017, Journal of the Franklin InstituteCitation Excerpt :Moreover, the dual-polarised antenna pair or an array of them has a similar structure to the well-studied vector sensors or sensor arrays [15–18], where they are used mainly for traditional array signal processing applications. Although the recently developed quaternion-valued array signal processing algorithms based on such traditional array applications employed a quaternion-valued array model [4–6], the desired signals are still traditional complex-valued signals, instead of quaternion-valued communication signals. In the following, the 4-D modulation scheme based on two orthogonally polarised antennas will be introduced in Section 2 and the required quaternion-valued equalisation and inter-channel interference suppression solution and their extension to multiple dual-polarised antennas are presented in Section 3.
Quaternion-based polarimetric array manifold interpolation
2015, Signal ProcessingCitation Excerpt :Using hypercomplex algebras it is possible to treat each vector sensor as a fundamental entity (i.e. a single hypercomplex number). To this end, quaternions (see [11]) as well as biquaternions (see [12,13]) have been applied to obtain a more natural model of vector sensor arrays. Sensors in the far-field of an emitter only encounter transversal components of the electro-magnetic field.
Quaternion-valued robust adaptive beamformer for electromagnetic vector-sensor arrays with worst-case constraint
2014, Signal ProcessingCitation Excerpt :In addition, another subspace-based approach – estimation of signal parameters via rotational invariance techniques (ESPRIT), was also extended to the quaternion domain [16], and this method outperforms the conventional ESPRIT, especially in the circumstances of short data length, low signal-to-noise ratio (SNR) and unknown model errors. For three-component EM vector-sensor arrays, bi-quaternion models were introduced accordingly [21–23]. In adaptive beamforming, the quaternionic version of the conventional MVDR beamformer has been derived with a two-component EM vector-sensor array in [17,18], where a better performance is obtained in the presence of steering vector mismatch errors.
Coherent sources direction finding and polarization estimation with various compositions of spatially spread polarized antenna arrays
2014, Signal ProcessingCitation Excerpt :The electromagnetic vector-sensor can resolve both the polarization and the direction-of-arrival (DOA) differences of the source [7,8]. The electromagnetic vector-sensor (array) has been investigated extensively for direction finding and polarization estimation: [7–10,15–19,24–42]. However, much of the literature models the sources as uncorrelated signals.
Coarray Interpolation for Direction Finding and Polarization Estimation Using Coprime EMVS Array via Atomic Norm Minimization
2023, IEEE Transactions on Vehicular Technology