[Retracted] 2D-DOA Estimation for EMVS Array with Nonuniform Noise

Cai, Chang-Xin; Huang, Guan-Jun; Wen, Fang-Qing; Wang, Xin-Hai; Wang, Lin

doi:https://doi.org/10.1155/2021/9053864

International Journal of Antennas and Propagation

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Abstract Introduction Preliminaries Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Research Article | Open Access

Volume 2021 | Article ID 9053864 | https://doi.org/10.1155/2021/9053864

[Retracted] 2D-DOA Estimation for EMVS Array with Nonuniform Noise

Chang-Xin Cai,¹Guan-Jun Huang,¹Fang-Qing Wen,²Xin-Hai Wang,³and Lin Wang⁴

Academic Editor: Jin He

Received21 Jun 2021

Revised24 Jul 2021

Accepted09 Aug 2021

Published19 Aug 2021

Abstract

Electromagnetic vector sensor (EMVS) array is one of the most potential arrays for future wireless communications and radars because it is capable of providing two-dimensional (2D) direction-of-arrival (DOA) estimation as well as polarization angles of the source signal. It is well known that existing subspace algorithm cannot directly be applied to a nonuniform noise scenario. In this paper, we consider the 2D-DOA estimation issue for EMVS array in the presence of nonuniform noise and propose an improved subspace-based algorithm. Firstly, it recasts the nonuniform noise issue as a matrix completion problem. The noiseless array covariance matrix is then recovered via solving a convex optimization problem. Thereafter, the shift invariant principle of the EMVS array is adopted to construct a normalized polarization steering vector, after which 2D-DOA is easily estimated as well as polarization angles by incorporating the vector cross-product technique and the pseudoinverse method. The proposed algorithm is effective to EMVS array with arbitrary sensor geometry. Furthermore, the proposed algorithm is free from the nonuniform noise. Several simulations verify the improvement of the proposed method.

1. Introduction

Sensor array is one of the most important infrastructures in wireless communication and radar detection [1–4]. Among the various branches in array signal processing, direction-of-arrival (DOA) estimation is the most canonical one and has aroused much attention. The principle of DOA estimation is to estimate the direction of the incoming source via the phase characteristics between sensors, and it is a highly nonlinear problem. Many efforts have been devoted to tackling this issue, for instance, an estimation approach to signal parameters with the rotational invariance technique (ESPRIT) [5, 6], Capon, multiple signal classification (MUSIC) [7–9], propagator method (PM) [10], maximum-likelihood (ML) [11], and tensor method [12–14]. Usually, the spectrum search counterparts, such as MUSIC, are always inefficient. Besides, they hardly avoid the off-grid problem. ESPRIT, however, is much more efficient than the spectrum search frameworks because it can acquire closed-form solutions to the parameter estimation issue.

A majority of the current studies focus on how to estimate the one-dimensional (1D) DOA from the scalar sensor arrays, e.g., uniform linear array (ULA). In practice, two-dimensional (2D) DOA may be more attractive. To pursue 2D-DOA estimation with the traditional scalar sensors, nonlinear sensor geometries are necessary [15–17], e.g., L-shape array, circular geometry, and rectangular manifold. Unfortunately, scalar sensor arrays often suffer from the sensor position error. Thus, complex array calibration is indispensable. Unlike scalar sensors, a single electromagnetic vector sensor (EMVS) is capable of providing 2D-DOA estimation [18]. Moreover, it is able to offer additional polarization angles of the source, and such characteristic may be very helpful in detecting stealth source [19]. Besides, an EMVS array with N sensors occupies more degree of freedom (DOF) than a scalar array, and thus, it provides more accurate estimation result than the latter. Furthermore, it has been proven that parameter estimation using EMVS array is insensitive to sensor positions [20], giving rise to the fact that the EMVS is more flexible than the traditional scalar sensor arrays.

It should be emphasized that the angle estimation issue using EMVS array is often more complex than that using the scalar sensors, since it involves 2D-DOA (azimuth angle and elevation angle) and 2D polarization angle (polarization phase difference and auxiliary polarization angle). In [19], the vector cross product was proposed, and the angles therein were obtained from the Poynting vector of the polarization steering vector. In [20], the ESPRIT-like algorithm was introduced. Therein, the concept of normalized Poynting vector was proposed to estimate the 2D-DOA, which was insensitive to the sensor positions and free from the sensor position error. In [21], the ULA-configured EMVS architecture was presented, and another ESPRIT estimator was derived. Unlike [20], the elevation angle was achieved by using ESPRIT, and the azimuth angle was estimated by using vector cross product. Likewise, in [22–26], the methods of combining the subspace approach and vector cross product were investigated. To avoid the eigendecompositon in the subspace approaches, a PM-like algorithm was presented in [27]. To further exploit the multidimensional structure in EMVS array, tensor algorithms were also been investigated in [28]. Besides, some efforts have been devoted to the active radar system with EMVS arrays [29–31], which brings new insights to target detection.

Nevertheless, it should be noticed that the subspace-based approaches achieve good performance with Gaussian white noise. In practice, the array noise may be nonuniform due to hardware nonideality. The nonuniform noise issue has been extensively stressed in scalar sensor array [32–35], but little attention has been paid to EMVS array. Therefore, we revisit the 2D-DOA estimation in EMVS array with nonuniform noise in this paper. An improved ESPRIT algorithm is presented. It eliminates the nonuniform noise via constructing a reduced covariance matrix, after which both noise covariance and parts of the signal covariance are removed. The recovery of the noiseless covariance matrix is recast as a matrix completion issue and is accomplished via solving a convex optimization problem. Thereafter, the ESPRIT idea is adopted to construct the normalized polarization steering vector. 2D-DOA, as well as polarization parameters, is then achieved by combining the vector cross product and the least squares (LS) technique. Our algorithm is effective in the scenario with arbitrary array geometry. Numerical simulations are designed to verify its effectiveness.

2. Preliminaries and the Data Model

2.1. EMVS Preliminaries

For a complete EMVS, it consists of six colocated antennas: three electric dipoles, and three magnetic loops. The dipoles and loops, respectively, sense the information of the electric field and magnetic field. Considering that a far-field source signal impinges on a single EMVS, the polarization responses of the six components can be expressed aswhere denotes transpose. D and p are, respectively, given byandwhere and denote the electric steering vector and the magnetic steering vector, respectively; , , , and denote, respectively, the elevation angle, the azimuth angle, the auxiliary polarization angle, and the polarization phase difference. denotes the direction-only matrix, and denotes the polarization-only vector, respectively. Moreover, the Poynting vector between and satisfies [20]where denotes the conjugate, returns the absolute value, and denotes the vector cross product.

2.2. Data Model

Let us consider an N-element EMVS array. Without loss of generality, let the coordinate of the n-th EMVS be . Suppose that K far-field signals appear in the array. Let , , , and stand for the k-th () angle parameters. The array signal can be written as [19]where t is the time index; denotes the Kronecker product, , where denotes the k-th spatial response (steering) vector with and is the carrier wavelength; and denotes the polarization response vector associated with the k-th target. accounts for the k-th signal; denotes the array noise. Let and . Equation (6) can also be formulated aswhere the symbol stands for the Khatri–Rao product, , and . Suppose that the noise is uncorrelated with the signal ; then, the covariance matrix of is given bywhere is to acquire the mathematical expectation and denotes Hermitian transpose., , and . In the presence of uncorrelated source signals, , where accounts for the diagonalization operation and is the power of the k-th source. Moreover, since the noise is nonuniform, its covariance matrix is then given bywhere denotes the noise power corresponding to the q-th component of the n-th EMVS. In practical applications, we can estimate via L samples as

Our objective here is to estimate the angles from .

3. The Proposed Approach

3.1. Principle of Traditional Eigendecompositon

It is well known that when Gaussian white noise (uniform noise) exists, the noise powers fulfillwhere is a constant, and then, the noise covariance becomeswhere stands for the identity matrix. If we ignore the noise item in equation (7), the eigendecomposition of is given bywhere is the k-th eigenvalue, is the associated eigenvector, and is for any ; denotes the eigenvalue matrix, and is called the signal subspace, which spans the same subspace as . Namely, there is a full-rank matrix such that

Also, the noiseless covariance matrix can be expressed aswhere is the eigenvector from the null subspace of , i.e., , and represents the full zero matrix. , and is called the noise subspace. Since the identity matrix can be formulated as the product of arbitrary unitary matrix and its Hermitian transpose, the noisy can be written aswhere . The results in equation (15) reveal that the uniform noise would not destroy the eigendistribution of the signal. However, in the presence of nonuniform noise, the noise power is not unique, so the conclusion in equation (15) will be untenable. It is necessary for us to denoise before further processing.

3.2. Denoising

Let be a set that records the nonzero entities of , i.e.,

We define a sampling operator that picks up the elements of the matrix in the blanket with indexes in , for example, such thatwhere denotes the (m, n)-th entity of R and is similar to others. Since is a diagonal matrix, we have . The effect of the noise can be easily removed via the following reduced covariance matrix:

The abovementioned denoising procedure is illustrated in Figure 1. However, the abovementioned denoising procedure can also destroy the structure of , so we need to recover from . Next, let us focus on the diagonal element of . It can be deduced that the m-th (m = 6 (n − 1) + q) diagonal entity of is

We define and let , and then, we have

Since is a low-rank matrix, the noiseless recovery problem can be formulated as [32]where returns the rank of R. To recover, the noiseless covariance coincides with the concept of matrix completion. Noting that the rank optimization is a nonconvex issue, it is usually replaced by the relaxed nuclear norm constraint , i.e.,

In practice, is replaced by its estimation, denoted by . Since there exists error between and , a boundis usually set, and the issue in equation (22) is transformed into

In practice, is usually chosen according to the noise tolerance; in this paper, it is set to . The abovementioned optimization can be easily accomplished via the convex toolboxes, e.g., cvx. After that, the eigendecompositon can be performed, and then, the estimation of the signal subspace is accomplished.

3.3. Parameter Estimation

Actually, the following rotational invariance relations exist:where returns a diagonal matrix, the diagonals of which are the n-th row of , and , , . Next, we definewhere accounts for the -th column of . The relations in equation (24) become

Inserting equation (26) into equation (13) yieldswhere denotes the inverse. In other words, we havewhere the superscript denotes the pseudoinverse. Performing eigendecomposition on , we can get the eigenvalues of the matrix and the associated eigenvectors, which reveal the estimation of and the estimation of (denoted as ). Calculating the left parts of equation (28) (except the first row), one can get the estimation of , , , and , respectively.

It has been pointed out in [20] that can be written as

It is easy to find that

Let and . Then, we have

According to equation (30), we can get

Obviously, is a constant, and it has been removed by normalizing calculation. Let the estimations of , , and be , , and , respectively. 2D-DOA can be estimated by

Once the 2D-DOA estimation has been accomplished, the polarization parameters can be estimated via the least squares approach in [30]. The details are omitted for simplicity.

4. Algorithmic Analyses

4.1. Important Remarks

Remark 1: as described in the context, the proposed method is insensitive to the , which means it is suitable for arbitrary sensor geometry. Besides, it is insensitive to the sensor position error as well. Remark 2: it is well known that the uniform noise is a special case of the nonuniform noise. Therefore, the proposed algorithm is effective in the white noise scenario. Remark 3: as explained in [30], all the estimated parameters are one-to-one paired. Remark 4: since the matrix completion would not hurt the rank and the dimension of the covariance matrix, the proposed algorithm can identify the same number of sources in [20].

4.2. Stochastic CRB

Let , where is a real vector that parameterizes . From the derivations in [35], we can get the stochastic CRB on 2D-DOA and polarization angle, which are given bywithwhere , with , And , where and with and , respectively, where denotes that with the k-th column of C. . , where denotes the k-th column of and denotes the vectorization. with , .

5. Simulation Results

In this subsection, the Monte Carlo simulation is utilized to assess the estimation accuracy. We consider an arbitrary N-element EMVS receives array configuration, and we assume that far-field sources impinge on the array, whose parameters are , , , and . Moreover, we suppose that L snapshots have been collected. Each result relies on 200 experiments. In the simulation, the signal-to-noise ratio (SNR) is defined as the ration of the power of the two components in equation (6). Two measures are adopted: one is the root mean square error (RMSE), and the other is the probability of successful detection (PSD). In all the simulations, the noise is randomly generated with powers which are uniformly chosen from [1 100000].

Example 1. We give the scattering figures of the proposed algorithm with N = 4 and L = 500, where the SNR is set to 20 dB. Herein, the EMVS is randomly placed in a three-dimensional space with () fulfilling a uniform distribution with interval [−0.5λ, 0.5λ]. Figure 2 shows the results of direction angle estimation and polarization parameters estimation. Clearly, all the angles can be correctly estimated and automatically paired. It is evident that our estimator is effective in a nonuniform noise scenario.

(a)

(b)

Example 2. We present the RMSE and PSD curves of the proposed estimator. Herein, a successful trial is recognized if the absolute error of the estimated angle is smaller than . In Figure 3, we plot the average estimation performance on the direction angle estimate (labelled with the suffix “−d”) and polarization angle estimate (labelled with the suffix “−p”), where N = 4 and L = 500. In contrast, the RMSE results of ESPRIT in [20] as well as the CRB are added. From the result, we can observe that the RMSE performance of both the proposed estimator and ESPRIT is improved with the increasing SNR, while the PSD of both estimators reaches 100%, once SNR is larger than a threshold (e.g., 25 dB). Besides, it depicts that the proposed estimator provides more accurate parameter estimation performance for direction angle estimation when SNR <10 dB. However, the improvement is not obvious as for polarization parameter estimation.

(a)

(b)

Example 3. We plot the average RMSE performance and the average PSD curves with different snapshot number L in Figure 4, while N and SNR are set to 4 and 10 dB, respectively. Notably, both the proposed estimator and ESPRIT provide better estimation performance with larger L. Similar to the previous observation, the proposed algorithm outperforms the ESPRIT algorithm. The improvement benefits from the fact that the proposed algorithm can eliminate the noise, while traditional ESPRIT algorithm cannot avoid the effect of nonuniform noise.

(a)

(b)

6. Conclusions

In this paper, we investigate the issue of angle estimation using EMVS array with arbitrary sensor geometry and nonuniform noise. A matrix completion-based algorithm has been proposed, which first eliminates the effect of nonuniform via solving a convex problem. After the noiseless covariance matrix has been recovered, the traditional subspace method is utilized to estimate the signal subspace, and the ESPRIT idea is adopted for 2D-DOA estimation. Our algorithm is robust to nonuniform noise and sensor position error. It should be pointed out that the tensor structure has not been exploited. More attention should be paid to this topic to further increase the estimation accuracy.

Data Availability

No data were used in this study.

Conflicts of Interest

The authors declare no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant no. 62071476.

References

G. Gui, M. Liu, F. Tang, N. Kato, and F. Adachi, “6G: opening new horizons for integration of comfort, security, and intelligence,” IEEE Wireless Communications, vol. 27, no. 5, pp. 126–132, 2020.
View at: Publisher Site | Google Scholar
H. Wang, L. Xu, Z. Yan, and T. A. Gulliver, “Low-complexity MIMO-FBMC sparse channel parameter estimation for industrial big data communications,” IEEE Transactions on Industrial Informatics, vol. 17, no. 5, pp. 3422–3430, 2021.
View at: Publisher Site | Google Scholar
H. Wang, X. Li, R. H. Jhaveri et al., “Sparse bayesian learning based channel estimation in FBMC/OQAM industrial IoT networks,” Computer and Communications, vol. 17, pp. 40–45, 2021.
View at: Google Scholar
L. Wan, K. Liu, Y.-C. Liang, and T. Zhu, “DOA and polarization estimation for non-circular signals in 3-D millimeter wave polarized massive MIMO systems,” IEEE Transactions on Wireless Communications, vol. 20, no. 5, pp. 3152–3167, 2021.
View at: Publisher Site | Google Scholar
D. Chen, B. Chen, and G. Qin, “Angle estimation using ESPRIT in MIMO radar,” Electronics Letters, vol. 44, no. 12, pp. 770-771, 2008.
View at: Google Scholar
J. Chen, H. Gu, and W. Su, “Angle estimation using ESPRIT without pairing in MIMO radar,” Electronics Letters, vol. 44, no. 24, pp. 1422-1423, 2008.
View at: Google Scholar
I. Bekkerman and J. Tabrikian, “Target detection and localization using MIMO radars and sonars,” IEEE Transactions on Signal Processing, vol. 54, no. 10, pp. 3873–3883, 2006.
View at: Publisher Site | Google Scholar
H. Yan, J. Li, and G. Liao, “Multitarget identification and localization using bistatic MIMO radar systems,” EURASIP Journal on Applied Signal Processing, vol. 2008, Article ID 8828610, 2008.
View at: Google Scholar
X. Zhang, L. Xu, L. Xu, and D. Xu, “Direction of departure (DOD) and direction of arrival (DOA) estimation in mimo radar with reduced-dimension music,” IEEE Communications Letters, vol. 14, no. 12, pp. 1161–1163, 2010.
View at: Publisher Site | Google Scholar
Z. D. Zheng and J. Y. Zhang, “Fast method for multi-target localisation in bistatic MIMO radar,” Electronics Letters, vol. 47, no. 2, pp. 138-139, 2011.
View at: Publisher Site | Google Scholar
B. Tang, J. Tang, Y. Zhang, and Z. Zheng, “Maximum likelihood estimation of DOD and DOA for bistatic MIMO radar,” Signal Processing, vol. 93, no. 5, pp. 1349–1357, 2013.
View at: Publisher Site | Google Scholar
Y. Cheng, R. Yu, H. Gu, and W. Su, “Multi-SVD based subspace estimation to improve angle estimation accuracy in bistatic MIMO radar,” Signal Processing, vol. 93, no. 7, pp. 2003–2009, 2013.
View at: Publisher Site | Google Scholar
X. Zhang, Z. Xu, L. Xu, and D. Xu, “Trilinear decomposition-based transmit angle and receive angle estimation for multiple-input multiple-output radar,” IET Radar, Sonar & Navigation, vol. 5, no. 6, pp. 626–631, 2011.
View at: Publisher Site | Google Scholar
J. Shi, F. Wen, and T. Liu, “Nested MIMO radar: coarrays, tensor modeling and angle estimation,” IEEE Transactions on Aerospace and Electronic Systems, vol. 57, no. 1, pp. 573–585, 2021.
View at: Google Scholar
W. Rao, D. Li, and J. Q. Zhang, “A tensor-based approach to L-shaped arrays processing with enhanced degrees of freedom,” IEEE Signal Processing Letters, vol. 25, no. 2, pp. 234–238, 2018.
View at: Publisher Site | Google Scholar
R. Goossens and H. Rogier, “A hybrid UCA-RARE/Root-MUSIC approach for 2-D direction of arrival estimation in uniform circular arrays in the presence of mutual coupling,” IEEE Transactions on Antennas and Propagation, vol. 55, no. 3, pp. 841–849, 2007.
View at: Google Scholar
M. D. Zoltowski, M. Haardt, and C. P. Mathews, “Closed-form 2-D angle estimation with rectangular arrays in element space or beamspace via unitary ESPRIT,” IEEE Transactions on Signal Processing, vol. 44, no. 2, pp. 316–328, 1996.
View at: Publisher Site | Google Scholar
J. He, L. Li, and T. Shu, “Sparse nested arrays with spatially spread orthogonal dipoles: high accuracy passive direction finding with less mutual coupling,” IEEE Transactions on Aerospace and Electronic Systems, 2021.
View at: Publisher Site | Google Scholar
K. T. Wong and X. Yuan, ““Vector cross-product direction-finding” with an electromagnetic vector-sensor of six orthogonally oriented but spatially noncollocating dipoles/loops,” IEEE Transactions on Signal Processing, vol. 59, no. 1, pp. 160–171, 2011.
View at: Publisher Site | Google Scholar
K. T. Wong and M. D. Zoltowski, “Closed-form direction finding and polarization estimation with arbitrarily spaced electromagnetic vector-sensors at unknown locations,” IEEE Transactions on Antennas and Propagation, vol. 48, no. 5, pp. 671–681, 2000.
View at: Publisher Site | Google Scholar
A. Nehorai and E. Paldi, “Vector-sensor array processing for electromagnetic source localization,” IEEE Transactions on Signal Processing, vol. 42, no. 2, pp. 376–398, 1994.
View at: Publisher Site | Google Scholar
J. Li, “Direction and polarization estimation using arrays with small loops and short dipoles,” IEEE Transactions on Antennas and Propagation, vol. 41, no. 3, pp. 379–387, 1993.
View at: Publisher Site | Google Scholar
K. T. Wong and M. D. Zoltowski, “Uni-vector-sensor ESPRIT for multisource azimuth, elevation, and polarization estimation,” IEEE Transactions on Antennas and Propagation, vol. 45, no. 10, pp. 1467–1474, 1997.
View at: Publisher Site | Google Scholar
K. T. Wong and M. D. Zoltowski, “Self-initiating music-based direction finding and polarization estimation in spatio-polarizational beamspace,” IEEE Transactions on Antennas and Propagation, vol. 48, no. 8, pp. 1235–1245, 2000.
View at: Publisher Site | Google Scholar
M. D. Zoltowski and K. T. Wong, “ESPRIT-based 2-D direction finding with a sparse uniform array of electromagnetic vector sensors,” IEEE Transactions on Signal Processing, vol. 48, no. 8, pp. 2195–2204, 2000.
View at: Publisher Site | Google Scholar
S. Miron, N. Le Bihan, and J. Mars, “Vector-sensor MUSIC for polarized seismic sources localization,” EURASIP Journal on Applied Signal Processing, vol. 48, no. 8, Article ID 280527, 2005.
View at: Publisher Site | Google Scholar
J. He and Z. Liu, “Computationally efficient 2D direction finding and polarization estimation with arbitrarily spaced electromagnetic vector sensors at unknown locations using the propagator method,” Digital Signal Processing, vol. 19, no. 3, pp. 491–503, 2009.
View at: Publisher Site | Google Scholar
T. Ahmed, Z. Xiaofei, and Z. Wang, “DOA estimation for coprime EMVS arrays via minimum distance criterion based on PARAFAC analysis,” IET Radar, Sonar & Navigation, vol. 13, no. 1, pp. 65–73, 2019.
View at: Publisher Site | Google Scholar
S. Chintagunta and P. Palanisamy, “2D-DOD and 2D-DOA estimation using the electromagnetic vector sensors,” Signal Processing, vol. 147, pp. 163–172, 2018.
View at: Publisher Site | Google Scholar
F. Wen, J. Shi, and Z. Zhang, “Closed-form estimation algorithm for EMVS-MIMO radar with arbitrary sensor geometry,” Signal Processing, vol. 186, Article ID 108117, 2021.
View at: Publisher Site | Google Scholar
X. Wang, M. Huang, and L. Wan, “Joint 2D-DOD and 2D-DOA estimation for coprime EMVS-MIMO radar,” Circuits, Systems, and Signal Processing, 2021.
View at: Publisher Site | Google Scholar
B. Liao, H. Lei, C. Guo et al., “New approaches to direction-of-arrival estimation with sensor arrays in unknown nonuniform noise,” IEEE Sensors Journal, vol. 16, no. 24, pp. 8982–8989, 2016.
View at: Publisher Site | Google Scholar
B. Liao, S. C. Chan, H. Lei et al., “Iterative methods for subspace and DOA estimation in nonuniform noise,” IEEE Transactions on Signal Processing, vol. 64, no. 12, pp. 3008–3020, 2016.
View at: Publisher Site | Google Scholar
B. Liao, “Fast angle estimation for MIMO radar with nonorthogonal waveforms,” IEEE Transactions on Aerospace and Electronic Systems, vol. 54, no. 4, pp. 2091–2096, 2018.
View at: Publisher Site | Google Scholar
M. Pesavento and A. B. Gershman, “Maximum-likelihood direction-of-arrival estimation in the presence of unknown nonuniform noise fields,” IEEE Transactions on Signal Processing, vol. 49, no. 7, pp. 1310–1324, 2001.
View at: Publisher Site | Google Scholar

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Copyright © 2021 Chang-Xin Cai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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