Blind Direction-of-Arrival Estimation with Uniform Circular Array in Presence of Mutual Coupling

A blind direction-of-arrival (DOA) estimation algorithm based on the estimation of signal parameters via rotational invariance techniques (ESPRIT) is proposed for a uniform circular array (UCA) when strong electromagnetic mutual coupling is present. First, an updated UCA model with mutual coupling in a discrete Fourier transform (DFT) beam space is deduced, and the new manifoldmatrix is equal to the product of a centrosymmetric diagonal matrix and a Vandermonde-structurematrix.Thenwe carry out blind DOA estimation through a modified ESPRIT method, thus avoiding the need for spatial angular searching. In addition, two mutual coupling parameter estimation methods are presented after the DOAs have been estimated. Simulation results show that the new algorithm is reliable and effective especially for closely spaced signals.


Introduction
Uniform circular array based DOA estimation methods are always attractive because the UCA has a special symmetric structure that provides almost the same resolution ability along the 360 ∘ azimuth angle domain.Except for conventional DOA estimation methods such as beam space searching or the Capon method [1], many other methods can be applied, such as the commonly used multiple signal classification (MUSIC) [2] or the well-known ESPRIT [3] in phase mode space.More effective methods such as UCA-RB-MUSIC and UCA-ESPRIT [4] have been introduced, and the mapping error reducing method was developed [5].
However, the electromagnetic mutual coupling effect cannot be ignored in a real array.Generally, this effect will severely degrade the performance of the above methods [6].A classic DOA estimation method based on an iterative search technique was proposed to estimate the DOA and the mutual coupling matrix (MCM) parameters jointly [7], but it has a high computation cost.The rank reduction (RARE) method introduced by [7] was further developed to obtain the DOA estimate for a UCA [8][9][10], but an angular search is still needed.Moreover, angular ambiguities are challenges for RARE-based blind methods.For example, two closely spaced signals cannot be differentiated according to the UCA-RARE [8] spectrum because there is a spurious peak at the middle of the two true angles.
In this paper, we design a new modified ESPRIT method for UCA to estimate the azimuth angle when mutual coupling is present.The proposed method is blind to mutual coupling and can completely avoid angular searching.Reliable DOA estimates can be obtained, especially for closely spaced signals.In addition, we will introduce two methods to estimate the MCM parameters once the DOA values are calculated.

UCA Model with Mutual Coupling
Suppose -element UCA has a radius  (Figure 1).All of the antenna elements are identical, and there are  far-field narrow signals impinging from {  ,  = 1, . . ., } which are the parameters to be estimated.The snapshot can be written as  is the total sampling number and A is the manifold matrix: where  = 1, .
⌊⋅⌋ is the flooring function and  0 is normalized as 1.For ideal dipole antenna array (Figure 1), induced EMF (Electromotive Force) method can give a close form of self-impedance and mutual impedance and thus MCM can be calculated according to Gupta and Ksienski's formulation [6].Suppose the signals are uncorrelated and n() is white Gaussian noise: (5)

Algorithm
3.1.UCA Model in DFT Beam Space.First we introduce the DFT of the MCM for UCA.
Lemma 1 (see [11][12][13]).If C is a circular matrix with its first column vector as c = [ 0 ,  1 , . . .,  −1 ]  and F is a Fourier matrix where  = 0, . . .,  − 1, then Λ = FCF  is a diagonal matrix with entries as C's eigenvalues: in which  = 0, 1, . . .,  − 1.For the Fourier matrix If C is also symmetric, this means that Then we can rewrite Λ as and we have where  = 1, . . .,  − 1.From ( 8) and ( 12), we get a linear equation which c  and   should satisfy for even : where Θ fl Therefore, if we determine the estimate of   , then we can obtain the estimate of the mutual coupling parameters c  by (13).In addition, there are similar equations where  is odd.
According to [4], we set where  is the number of excited phase modes.We define another Fourier matrix F  , which is different from (6): Then the snapshot in DFT beam space is if we write F   CF  as where (see (11), (12), and ( 18)) In addition, we can obtain [4] with (⋅) is  order first-kind Bessel function.According to its property we get where Ã is the updated manifold matrix with a Vandermonde structure Finally, we obtain the snapshot in DFT beam space in which (29)

DOA Estimation and MCM Calculation.
We carry out the eigendecomposition on the covariance matrix R z and obtain the signal subspace Ũ , which consists of eigenvectors corresponding to  maximum eigenvalues.Select the first  + 1 rows of Ũ as Ũ1 and the second  + 1 rows as Ũ2 .Select the first  + 1 rows of Ã as Ã1 and the second  + 1 rows as Ã2 .Select the first ( + 1) × ( + 1) diagonal matrix of Γ as Γ 1 and the second ( + 1) × ( + 1) diagonal matrix as Γ 2 .Use the same notations for other matrices Γ 1 , Γ 2 , Γ 1 , and Γ 2 .
Then we have where T is a nonsingular matrix.Thus we get with Since we use sampled data, we should replace Ũ with Û and define the object function as We use the solution to the above equations from [14,15].Consider with where and ⊙ is the Hadamard product.If  min is the eigenvector of Q corresponding to its minimum eigenvalue, then we have Equation (37) indicates that there is a phase  ambiguity for , and thus it will introduce a  ambiguity to the DOA values through the eigenvalues of Ψ.However, this ambiguity can be cleared by comparing the RARE spectrum [9] values on θ and θ +.We mark the estimated vector as dopt without  ambiguity.
Finally, we can determine the DOA estimates through the eigenvalues of Ψ.Following is the detailed procedure of the blind method for DOA estimation of {  }: (1) Calculate the sample covariance matrix R and do eigendecomposition.Get the estimated signal subspace Û and noise subspace Û .
(4) Compare the blind RARE spectral values [9] on θ and θ +  and clear the ambiguity.Output the final DOA estimates.
We label the above mentioned method as "Blind-m1-half" because we only select  + 1 rows of Û .In addition, we can select the first 2 and the second 2 rows from Û as Û1 and Û2 .This method is labeled "Blind-m1-full."Furthermore, we can select the first  and the third  rows from Û as Û1 and Û2 or select the first 2 − 1 and the third 2 − 1 rows from Û as Û1 and Û2 .We label the two methods as "Blind-m2-half" and "Blind-m2-full," respectively.We will carry out simulations on these four methods in Section 4. Now we can estimate the MCM parameters once we get θ.
Method 1 can only estimate  mutual coupling parameters.

Simulations
Consider a 16-element half-wave dipole antenna UCA with  0 = 1.032GHz and  = 0.7 .Set  = 4.The mutual coupling vector c is listed in Table 1.The theoretical parameters are calculated according to Gupta and Ksienski's formulation [6], and small values are treated as   = 0,  = 3, . . ., 8. Real parameters should be estimated by array calibration method.Two signals are impinging from 35 ∘ and 45 ∘ with the same signal noise ratio (SNR).We apply the classic MUSIC, blind RARE [9], blind R-RARE [10], and UCA-RARE [8] methods to obtain azimuth estimates.The results are shown in Figure 2.
It shows that the MUSIC method and RARE-based blind methods cannot differentiate these two signals because RARE-based methods will introduce spurious estimates.We should obtain the initial estimates from the MUSIC spectrum to start the iterative method [7], but it is difficult to find two different spectral peaks from the spectrum of "MUSIC without MCM." We apply the proposed blind method to the above example.The estimated average DOA absolute bias and root mean square error (RMSE) versus SNR are illustrated in Figures 3-6 ( = 10000, 100 trials).Method in [7] ( = 0.01) and the Cramér-Rao bound (CRB) with a known MCM are also presented [16].This shows that all the four proposed methods can give satisfactory estimates and that the tendency of the RMSE is the same as the CRB with an increase of SNR.Method Blind-m1-full and method Blind-m2-full are more effective than the other two methods because more rows of Û are involved when we calculate the fitting matrix Ψ (see (35)).A comparison of simulations versus sampling number will give similar results.Method in [7] gives a biased estimate.

Conclusion
In DFT beam space, we utilize a modified ESPRIT algorithm to obtain a reliable DOA estimate when there is a severe mutual coupling effect.The new blind method is efficient because it avoids searching for the spectral peaks.For closely spaced signals, neither the classic MUSIC nor RARE-based methods provide a good estimate, while the proposed new method can produce an accurate estimate.Moreover, these direction estimates can be used for further MCM parameter estimation.

Table 2
lists the MCM parameter estimates based on the two methods introduced in Section 3.This shows that Method 2 can give more accurate results.