Direction finding with a single spatially stretched vector sensor in the presence of mutual coupling
 Ting Shu^{1}Email authorView ORCID ID profile,
 Kun Wang^{1},
 Jin He^{1} and
 Zhong Liu^{2}
https://doi.org/10.1186/s1363401805379
© The Author(s) 2018
Received: 28 July 2017
Accepted: 12 February 2018
Published: 27 February 2018
Abstract
This paper is concerned with DOA estimation using a singleelectromagnetic vector sensor in the presence of mutual coupling. Firstly, we apply the temporally smoothing technique to improve the identifiability limit of a singlevector sensor. In particular, we establish sufficient conditions for constructing temporally smoothed matrices to resolve K > 2 incompletely polarized (IP) monochromatic signals with a singlevector sensor. Then, we propose an efficient ESPRITbased method, which does not require any calibration signals or iterative operations, to jointly estimate the azimuthelevation angles and the mutual coupling coefficients. Finally, we derive the CramérRao bound (CRB) for the problem under consideration.
Keywords
1 Introduction
Direction finding using a singleelectromagnetic vector sensor (EMVS) has played an important role in applications such as radar, wireless communications and seismic exploration. An EMVS consists of six components, three identical but orthogonally oriented electrically short dipoles, and another three identical but orthogonally oriented magnetically small loops. An EMVS can therefore measure all the six electromagnetic field components induced by any electromagnetic incidence. After Li [1], and Nehorai and Paldi [2] first introduced the EMVS measurement model to the signal processing community, a variety of studies regarding signal processing with a single EMVS [2–8] have been extensively carried out. These methods ignore the mutual coupling across the six antenna component, which ultimately destroys the underlying model assumptions needed for their efficient implementations. Consequently, ignoring this mutual coupling effect can seriously degrade the performance the above mentioned algorithms. Therefore, it is of great significance to develop algorithms for simultaneous mutual coupling calibration and parameter estimation.
In the last few years, many advanced array calibration methods have been reported. These algorithms include the maximum likelihood algorithm [9], the iterative autocalibration method [10], the auxiliary sensorbased methods [11–14], the cumulantbased method [15], the Rankreduction (RARE)based calibration methods [16, 17], the sparse representationbased methods [18–20], and the matrix reconstruction method [21]. However, some of these methods require a set of calibration signals/auxiliary sensors [9, 11–14] or iterative/high order statistics/nonlinear optimization computations [10, 15–20]. Moreover, all such methods are designed for scalar sensor arrays and are not applicable to the vector sensor arrays. Calibration of mutual coupling for vector sensors has been studied recently in [22] and [23]. These two methods can offer closedform solutions for coupling matrix and parameter estimation. However, they require a couplingfree auxiliary vector sensor and design of a reference signal.
The aforementioned scalar sensor array calibration methods have been a strong motivation for us to develop new joint calibration and estimation methods for vector sensor arrays, and the contribution of the work lies in that direction. The proposed method is outlined as follows: the temporal smoothing technique is firstly applied to improve the identifiability limit of a single vector. In particular, sufficient conditions for constructing temporally smoothed matrices to resolve K incompletely polarized (IP) monochromatic signals with a singlevector sensor are established. An ESPRITbased method is then developed for jointly estimating the azimuthelevation angles and the mutual coupling coefficients. This method does not require any calibration signals or iterative operations. The CramérRao bound (CRB) for the problem under consideration is also derived.
Throughout the paper, scalar quantities are denoted by lowercase letters. Lowercase bold type faces are used for vectors and uppercase letters for matrices. Superscripts T, H, ∗, and † represent the transpose, conjugate transpose, complex conjugate and pseudo inverse, respectively, while ⊗ and ⊙, respectively, symbolize the Kroneckerproduct operator and the KhatriRao (columnwise Kronecker) matrix product. I_{ m } and 0_{m,n}, respectively, stand for the m×m identity matrix and m×n zero matrix.
2 Mathematical data model and assumptions

A_{ k } is the 6 × 2 EMVS response of the kth electromagnetic signal.

A is the 6×2K EMVS steering matrix.

s_{ k }(t) is a 2 × 1 vector, representing the two entries of the kth transmitted signal.

In (7), β_{k,i}, ω_{k,i} and ψ_{k,i}, i,=1,2, respectively, represent the energy, frequency, and the uniformly distributed random phase of the ith entry of the kth signal.

n(t) is the 6 × 1 noise vector.
where γ_{ k } and η_{ k } are polarization parameters referred to as the auxiliary polarization angle and polarization phase difference, respectively, and \(\phantom {\dot {i}\!}s_{k}(t) = \beta _{k} e^{j(\omega _{k} t + \psi _{k})}\) is the kth transmitted signal. Thus, for the case of K CP signals, the data vector in (1) reduces to the one used in [3].
where c_{1} and c_{2} represent the self and mutual coupling coefficients, respectively.
 1.
The parameters (θ_{1},ϕ_{1}),⋯,(θ_{ K },ϕ_{ K }) are pairwise distinct.
 2.
The value of K is known or correctly estimated.
 3.
The coupling coefficients c_{1} ≠ c_{2} so that the coupling matrix C is nonsingular.
 4.
The impinging signals are IP and are uncorrelated with one another. This implies that the frequencies ω_{1,1}≠ω_{1,2}≠⋯≠ω_{K,1}≠ω_{K,2}.
 5.
The noise is zeromean, complex Gaussian, and is statistically independent of all the signals.
3 Joint angle and mutual coupling matrix estimation
3.1 Temporal smoothing
The authors in [4] have found that the maximum number of arbitrary electromagnetic sources uniquely identifiable by one vector sensor is two. That is, the data matrix in (11) is rank deficient if the number of incoming signals is greater than two. In this subsection, we will apply the temporal smoothing technique [26] to deal with this rank deficiency problem. We will also show that under certain conditions, the temporal smoothing technique can restore the rank of the data matrix.
Theorem 1
If P≥2K and N≥4K−1, then the temporally smoothed data matrix Z_{TS} is of full rank 2K.
Proof
Finally, combining P ≥ 2K with (N − P + 1) ≥2K, we have Z_{TS} to be of full rank 2K, if P ≥ 2K and N≥4K − 1, since (Ψ⊙MA) is of full column rank and S^{ T } is of full row rank. This concludes the proof. □
Theorem 1 establishes sufficient but not necessary conditions for constructing temporally smoothed matrices to resolve K IP monochromatic signals with a singlevector sensor. Specially, on the basis of Theorem 1, an infinite number of uncorrelated signals with distinct frequencies may potentially be resolved as N approaches infinity.
3.2 Angle and mutual coupling matrix estimation
where q_{2k − 1}, q_{2k} and \(\bar {\mathbf {q}}_{2k~~1}\), \(\bar {\mathbf {q}}_{2k}\), k = 1,⋯,K, respectively, denote the top three and bottom three rows of the (2k − 1)th and (2k)th columns of \(\hat {\mathbf {Q}}\), α_{i,j} and \(\bar \alpha _{i,j}\), i = 1,2,j = 1,⋯,K represent the unknown scalars. Note that since q_{ k } ≠ 1, α_{i,j} is in general unequal to \(\bar \alpha _{i,j}\).
Note that the estimation of \(\hat \theta _{k}\) and \(\hat \phi _{k}\) are automatically paired without any additional processing.
In practice, apart from the scaling ambiguities, the estimated \(\hat {\mathbf {Q}}\) may also suffer from some permutation ambiguities. In this case, q_{2k} may not be the estimate of α_{2,k}Cv_{2,k}. Thus, the estimation of \(\hat \phi _{k}\) and \(\hat c\) obtained by using (33) and (34) from q_{2k} may be erroneous. These may further result in the erroneous estimation of \(\hat \theta _{k}\). Unlike the scaling ambiguities, the permutation ambiguities are not resolvable. Here, we provide a solution to deal with this permutation ambiguity problem as follows: first, for all k = 1,⋯,2K, obtain a set of 2K different azimuth angle estimates from q_{ k }. Each of these 2K azimuth angle estimates is then used to produce its own coupling coefficient and elevation angle estimates. Thus, the kth azimuth angle, elevation angle, and coupling coefficient estimates are automatically matched. We know that only a set of K estimates are true estimates. Theoretically, the K true coupling coefficient estimates are identical, while the K erroneous coupling coefficient estimates are, in general, distinct from one another and from the K true estimates. Therefore, we can take homogeneity in coupling coefficient estimates as a criterion for determining the true estimates of the angles and coupling coefficients, i.e., we take a set of K angle estimates associated with K identical coupling coefficient estimates as the true estimates. Without loss of generality, let us assume that the first K estimates are true and the last K estimates are erroneous; then, we have \(\hat c_{1} = \cdots = \hat c_{K}~=~\hat c~\neq ~\hat c_{K~+~1} \neq \cdots ~\neq ~\hat c_{2K}\). Finally, we obtain the estimates \((\hat \theta _{k}, \hat \phi _{k}), k~=~1, \cdots, K\) as the angle estimates of the K signals.
3.3 Remarks
In the presence of noise, the estimation procedures in Section 3.2 becomes approximate. Specially, with noise, the set of K coupling coefficient estimates are in general different. Nevertheless, we can search for a set of K coupling coefficient estimates with “most similar values” as the “identical” estimates.
Also note that the vector cross product estimator has been widely used for direction finding with a singlevector sensor [2, 3, 7]. However, this estimator cannot be exploited directly in the presence of mutual coupling among the vector sensor components. Obviously, with the estimation of \(\hat c\), the vector sensor can be calibrated by using the calibration matrix defined as \(\hat {\mathbf {M}}~=~\mathbf {I}_{2} \otimes \hat {\mathbf {C}}\). Therefore, the vector cross product estimator can be applied to the calibrated data matrix \(\hat {\mathbf {M}} \mathbf {Z}\) to extract the angle estimates of the incoming signals. Although the proposed method is designed for vector sensors with mutual coupling, it can also be applied to ideal vector sensors, where the measurement of each component is independent of the others.
The proposed method shares all the advantages indicated in [3]. For example, it offers automatically paired azimuth and elevation angle estimates, does not restrict Δ_{ T } to be constricted by the Nyquist sampling rate, does not need the signal frequencies to be known a priori, and suffers no frequencyDOA ambiguity. It should be noted that the method in [3] assumes CP signals, whereas the proposed method assumes IP ones.
Lastly, it should be pointed out that the application of ESPRIT technique for vector sensor mutual coupling calibration has been studied in works [22] and [23]. However, the differences between these two works and the present work are that (1) the former requires a couplingfree auxiliary vector sensor and design of a reference signal, while the latter does not, (2) the former does not apply the temporally smoothing technique to improve the identifiability limit of a vector sensor, and (3) the former assumes the incoming signals are completely polarized, while the latter considers the incompletely polarized signals.
4 CramérRao bound
where \(\boldsymbol {\mu }(n)~=~\sum _{k = 1}^{K} \mathbf {M}\mathbf {A}_{k} \mathbf {s}_{k}(n\Delta _{T})\), n = 1,⋯,N.
5 Simulation results and discussion
5.1 Simulation results
In this section, we provide simulation results to illustrate the performance of the proposed ESPRITbased method. In all the simulations, the vector sensor is assumed to be spatially collocated with the mutual coupling model defined in (12). The mutual coupling coefficient used is c=0.1e^{−jπ/4}. The additive noise is assumed to be spatial white complex Gaussian, and the SNR is defined relative to each signal. The result in each of the examples below is obtained from 500 independent MonteCarlo trials. For comparison purposes, three different methods are considered. The first method is to apply the vector cross product estimator to the measured data directly. This method is hereafter referred to as “VCP Estimator without calibration.” The second method is based on the condition that the mutual coupling coefficient is known a priori, and the vector cross product estimator is applied to the perfectly calibrated data. This method is referred to as “VCP Estimator perfect calibration.” The third method is the auxiliary sensor calibration method presented in [22]. The performance metric used is the root mean squared error (RMSE) of the two signals.
5.2 Discussion
or \(\left (\hat c_{2}, \hat \theta _{2}, \hat \phi _{2}\right)\) otherwise, where · denotes the Frobenius norm.
6 Conclusions
The present paper has considered, for the first time, the direction finding using a singlevector sensor in the presence of mutual coupling. The temporal smoothing technique has been applied to improve the identifiability limit of a single vector. In particular, sufficient conditions for constructing temporally smoothed matrices to resolve K>2 incompletely polarized (IP) monochromatic signals with a single vector sensor have been established. An efficient ESPRITbased method, which does not require any calibration sources or iterative operations, has been developed to jointly estimate the azimuthelevation angles and the mutual coupling coefficients. The CRB for the considered problem has also been derived. Simulation results have been presented showing the superiorities of the proposed method.
Declarations
Acknowledgements
The authors would like to thank the anonymous reviewers for their valuable comments and suggestions that helped improve the quality of this manuscript.
Funding
This work was supported by the National Natural Science Foundation of China (no. 61771302) and Shanghai Association of Science and Technology (no. 16511103004)
Availability of data and materials
The datasets generated and/or analyzed during the current study are not publicly available but are available from the corresponding author on reasonable request.
Authors’ contributions
All authors contributed extensively to the study presented in this manuscript. TS presented the main idea, carried out the simulation, interpreted the results, and wrote the paper. KW and JH conceived of the experiments. They also provided many valuable suggestions to this study. ZL supervised the main idea and edited the manuscript. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
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