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Twodimensional DOA estimation of coherent sources using two parallel uniform linear arrays
EURASIP Journal on Wireless Communications and Networking volume 2017, Article number: 60 (2017)
Abstract
A novel twodimensional (2D) directionofarrival (DOA) estimation approach based on matrix reconstruction is proposed for coherent signals impinging on two parallel uniform linear arrays (ULAs). In the proposed algorithm, the coherency of incident signals is decorrelated through two equivalent covariance matrices, which are constructed by utilizing crosscorrelation information of received data between the two parallel ULAs and the changing reference element. Then, the 2D DOA estimation can be estimated by using eigenvalue decomposition (EVD) of the new constructed matrix. Compared with the previous works, the proposed algorithm can offer remarkably good estimation performance. In addition, the proposed algorithm can achieve automatic parameter pairmatching without additional computation. Simulation results demonstrate the effectiveness and efficiency of the proposed algorithm.
1 Background
2D directionofarrival (DOA) estimation of incident coherent source signals has received increasing attention in radar, sonar, and seismic exploration [1–5]. Many highresolution techniques, such as MUSIC [6] and ESPRIT [7], have achieved exciting estimation performance. However, the aforementioned methods assume the incident signals are independent, which would encounter performance degradation due to the rank deficiency when coherent signals exist. To decorrelate coherent signals, the spatial smoothing (SS) [8] or forwardbackward spatial smoothing (FBSS) [9] are especially noteworthy. However, this technique generally reduces the effective array aperture, and the maximum number of resolvable signals cannot exceed the number of array sensors. In [10], an effective matrix decomposition method utilizing crosscorrelation matrix is proposed to decorrelate coherent signals. Chen et al. [11] have proposed a 2D ESPRITlike method that realizes decorrelation by reconstructing a Toeplitz matrix. With the help of three correlation matrices, Wang et al. [12] have presented a 2D DOA estimation method. Recently, Nie et al. [13] have introduced an efficient subspace algorithm for 2D DOA estimation. In [14], a novel 2D DOA estimation method using a sparse Lshaped array is proposed to obtain high performance and less complexity. Xia et al. [15] have proposed a polynomial rootfindingbased method for 2D DOA estimation by using two parallel uniform linear arrays (ULAs), which has less computational burden. Some decorrelation algorithms are proposed in [16–18] to achieve 2D DOA estimation by utilizing two parallel ULAs. However, the limitation of the abovementioned algorithms is that the estimation performance cannot be satisfactory due to the fact that the structure of the array is not being fully exploited.
For the purpose of description, the following notations are used. Boldface italic lower/uppercase letters denote vectors/matrices. (·)*, (·)^{T}, (·)^{†}, and (·)^{H} stand for the conjugation, transpose, MoorePenrose pseudoinverse, and conjugate transpose of a vector/matrix, respectively. The notation E(x) and diag (·) separately denote the expectation operator and the diagonal matrix, respectively.
2 Date model
As illustrated in Fig. 1, the antenna array consists of two parallel ULAs (X_{ a } and Y_{ a }) in the x − y plane. Each ULA has N omnidirectional sensors with spacing d _{ x }, and the interelement spacing between the two ULAs is d _{ y }. Suppose that M farfield narrowband coherent signals impinge on the two parallel ULAs from 2D distinct directions (α _{ i }, β _{ i })(1 ≤ i ≤ M), where α _{ i } and β _{ i } are measured relatively to the x axis and to the y axis, respectively.
Let the kth element of the subarray X_{ a } be the phase reference and then the observed signals \( {x}_m^k(t) \) at the mth element can be expressed as
where s _{ i }(t) denotes the complex envelope of the ith coherent signal, λ is the signal wavelength, and d _{ x } represents the spacing between two adjacent sensors. The superscript k(k = 1, 2, ⋯, N) of the \( {x}_m^k(t) \) stands for the number of the reference element in subarray X_{ a }, and the subscript m(m = 1, 2, ⋯, N) of the \( {x}_m^k(t) \) denotes the number of the element along the x positive axis in subarray X_{ a }. n _{ x,m }(t) is the additive Gaussian white noise (AGWN) of the mth element in subarray X_{ a }.
Note that when m = k, the observed signals at the kth element can be expressed as
With a similar processing, employing the kth element of the subarray Y_{ a } as the phase reference and then the observed signals \( {y}_m^k(t) \) at the mth element can be expressed as
Similarly as in (1), the superscript k(k = 1, 2, ⋯, N) of the \( {y}_m^k(t) \) stands for the number of the reference element in subarray Y_{ a }, and the subscript m(m = 1, 2, ⋯, N) of the \( {y}_m^k(t) \) denotes the number of the element along the x positive axis in subarray Y_{ a }. n _{ y,m }(t) is the AGWN of the mth element in subarray Y_{ a } _{.}
The observed vectors X ^{k}(t) and Y ^{k}(t) can be written as
3 The proposed algorithm
For the subarray X_{ a }, the autocorrelation calculation is defined as follows:
where
Assume that the kth element of the subarray X_{ a } is the phase reference. Thus, the autocorrelation vectors \( {\mathbf{r}}_{{\mathbf{X}}^k{\left({x}_k^k\right)}^{\ast}}^k \) between X ^{k}(t) and the corresponding reference element \( {x}_k^k(t) \) can be defined as follows:
It is obvious that N column vectors will be achieved as the superscript k of the \( {\mathbf{r}}_{{\mathbf{X}}^k{\left({x}_k^k\right)}^{\ast}}^k \) is changed from 1 to N. Therefore, we construct an equivalent autocovariance matrix R _{ xx } as follows:
Similarly as in (6), for the subarray Y_{ a }, the crosscorrelation calculation \( {\tilde{r}}_{y_m^k{\left({x}_k^k\right)}^{\ast}}^k \) can be written as
Then, the crosscorrelation vectors \( {\tilde{\mathbf{r}}}_{{\mathbf{Y}}^k{\left({x}_k^k\right)}^{\ast}}^k \) between Y ^{k}(t) and the reference element \( {x}_k^k(t) \) in subarray X_{ a } can be expressed as
Obviously, we can obtain another N column vectors when the superscript k of the \( {\tilde{\mathbf{r}}}_{{\mathbf{Y}}^k{\left({x}_k^k\right)}^{\ast}}^k \) is varied from 1 to N. Based on the N column vectors, an equivalent crosscovariance matrix R _{ yx } can be given by
In order to obtain the final matrix form of the equivalent autocovariance matrix R _{ xx } as in (10), we need to further investigate the autocorrelation calculation \( {r}_{x_m^k{\left({x}_k^k\right)}^{\ast}}^k \) in (6).
where
It can be seen from (16) that \( {\mathtt{a}}_m\left(\alpha \right) \) is the mth row of the steering matrix in covariance matrix with the scenario when the first element of the subarray X_{ a } is set to be the reference element. According to (14), (15), and (16), Eq. (9) can be rewritten as
where \( \mathbf{A}\left(\alpha \right)=\left[\begin{array}{cccc}\hfill \mathtt{a}\left({\alpha}_1\right)\hfill & \hfill \mathtt{a}\left({\alpha}_2\right)\hfill & \hfill \cdots \hfill & \hfill \mathtt{a}\left({\alpha}_M\right)\hfill \end{array}\right] \) is the steering matrix of the covariance matrix along the subarray X_{ a }, and \( \mathtt{a}\left({\alpha}_i\right)={\left[\begin{array}{cccc}\hfill 1\hfill & \hfill {e}^{ j\left(2\pi /\lambda \right){d}_x \cos {\alpha}_i}\hfill & \hfill \cdots \hfill & \hfill {e}^{ j\left(2\pi /\lambda \right)\left( N1\right){d}_x \cos {\alpha}_i}\hfill \end{array}\right]}^T \).
Based on (17), the matrix R _{ xx } in (10) can be rewritten as
where \( {\sigma}_i^2 \) is the noise power on the ith element of the subarray X_{ a }.
Similar to the equivalent autocovariance matrix R _{ xx } in (18), the equivalent crosscovariance matrix R _{ yx } in (13) can be rewritten as
where
From (18) and (19), it is easy to see that since α _{ i } ≠ α _{ j }, (i ≠ j), A(α) is a full column rank matrix with rank (A(α)) = M. Similarly, since β _{ i } ≠ β _{ j }, (i ≠ j), Ψ(β) is a fullrank diagonal matrix with rank (Ψ(β)) = M. According to (7) and (15), note that the incident signals s _{ i }(t) ≠ 0, (i = 1, 2 ⋯ M), so g _{ i }(t) ≠ 0. As a result, G is a fullrank diagonal matrix, namely, rank(G) = M. That is, if the narrowband farfield signals are statistically independent, the diagonal element g _{ i }(t) of the matrix G represents the power of the ith incident signal. If the narrowband farfield signals are fully coherent, the diagonal element g _{ i }(t) of the matrix G denotes the sum of the powers of the M incident signals. Notice that if the narrowband farfield signals are the coexistence of the uncorrelated and coherent signals, which means there are K coherent signals, the remaining are M − K statistically independent signals. Then, the diagonal element g _{ i }(t) of the matrix G stands for the sum of the powers of the K coherent signals when the subscript of the diagonal element g _{ i }(t) in matrix G corresponding to the source signal belongs to one of the K coherent signals. If the subscript of the diagonal element g _{ i }(t) in matrix G corresponding to the source signal belongs to one of the remaining M − K mutually independent signals, the diagonal element g _{ i }(t) of the matrix G denotes the power of the ith independent signal.
From the above theoretical analysis, the coherency of incident signals is decorrelated through matrices constructing no matter whether the signals are uncorrelated, coherent, or partially correlated.
From (18), we can obtain the noiseless autocovariance matrix \( {\widehat{\mathbf{R}}}_{xx} \)
The eigenvalue decomposition (EVD) of \( {\widehat{\mathbf{R}}}_{xx} \) can be written
where {λ _{1} ≥ λ _{2} ≥ ⋯ ≥ λ _{ M }} and {U _{1}, U _{2}, ⋯, U _{ M }} are the nonzero eigenvalues and eigenvector of the noiseless autocovariance matrix \( {\widehat{\mathbf{R}}}_{xx} \), respectively. Then, the pseudoinverse of \( {\widehat{\mathbf{R}}}_{xx} \) is
Since A(α) is a column fullrank matrix, the Eq. (22) can be expressed as
According to (19) and (25), the matrix R _{ yx } can be rewritten as
Rightmultiplying both sides of (26) by \( {\mathbf{R}}_{xx}^{\dagger}\mathbf{A}\left(\alpha \right) \)
Substituting (23) and (24) into (27) yields
Notice that \( {\displaystyle \sum_{i=1}^M{\mathbf{U}}_i{\mathbf{U}}_i^H} \) is an identity matrix, that is, \( {\displaystyle \sum_{i=1}^M{\mathbf{U}}_i{\mathbf{U}}_i^H}=\mathbf{I} \). Based on (24) and (26), a new matrix R can be defined as follows:
From (29), the (28) can be further rewritten as
Obviously, the columns of A(α) are the eigenvectors corresponding to the major diagonal elements of diagonal matrix Ψ(β). Therefore, by performing the EVD of R, the A(α) and Ψ(β) can be achieved. Then, the DOA estimation of the coherent signals can be achieved according to \( \upsilon \left({\beta}_i\right)={e}^{j\left(2\pi /\lambda \right){d}_y \cos {\beta}_i} \) and \( \mathtt{a}\left({\alpha}_i\right)={\left[1,{e}^{ j\left(2\pi /\lambda \right){d}_x \cos {\alpha}_i},\cdots, {e}^{ j\left(2\pi /\lambda \right)\left( N1\right){d}_x \cos {\alpha}_i}\right]}^T \) without additional computations for parameter pairmatching and 2D peak searching.
Up to now, the steps of the proposed matrix reconstruction method with the finite sampling data are summarized as follows:

(1)
Calculate the column vectors \( {\mathbf{r}}_{{\mathbf{X}}^k{\left({x}_k^k\right)}^{\ast}}^k \) of the equivalent autocovariance matrix R _{ xx } by (6) and (9). Similarly, compute the column vectors \( {\tilde{\mathbf{r}}}_{{\mathbf{Y}}^k{\left({x}_k^k\right)}^{\ast}}^k \) of the equivalent crosscovariance matrix R _{ yx } according to (11) and (12)

(2)
Achieve the matrix R _{ xx } and the matrix R _{ yx } by (10) and (13)

(3)
Obtain the noiseless autocovariance matrix \( {\widehat{\mathbf{R}}}_{xx} \) by (22). Then, perform EVD to obtain pseudoinverse matrix \( {\mathbf{R}}_{xx}^{\dagger } \)

(4)
Construct the new matrix R by (29) and then get the A(α) and Ψ(β) by performing EVD of the new matrix R

(5)
Estimate the 2D DOAs θ _{ i } = (α _{ i }, β _{ i }) of incident coherent source signals via \( \upsilon \left({\beta}_i\right)={e}^{j\left(2\pi /\lambda \right){d}_y \cos {\beta}_i} \) and \( \mathtt{a}\left({\alpha}_i\right)={\left[1,{e}^{ j\left(2\pi /\lambda \right){d}_x \cos {\alpha}_i},\cdots, {e}^{ j\left(2\pi /\lambda \right)\left( N1\right){d}_x \cos {\alpha}_i}\right]}^T \).
4 Simulation result
In this section, computer simulations are performed to ascertain the performance of the proposed algorithm. The proposed method is compared with another efficient algorithm (DMRDOAM) in [17]. The number of sensors in each subarray is N = 7 with sensor displacement d _{ x } = d _{ y } = λ/2. Consider M = 4 coherent signals with carrier frequency f = 900MHz coming from α = (75^{o}, 100^{o}, 120^{o}, 60^{o}) and β = (65^{o}, 75^{o}, 90^{o}, 50^{o}). The phases of coherent signals are [π/5, π/3, π/3, π/3]. Results on each of the simulation are analyzed by 1000 Monte Carlo trials. Two performance indices, called the rootmeansquareerror (RMSE) and normalized probability of success (NPS), are defined to evaluate the performance of the proposed algorithm and DMRDOAM algorithm.
where \( {\widehat{\alpha}}_n(i) \) and \( {\widehat{\beta}}_n(i) \) are the estimates of α _{ n } and β _{ n } for the ith Monte Carlo trial respectively, and K is the source number.
where ϒ _{suc} and Τ _{total} denote the times of success and Monte Carlo trial, respectively. Furthermore, a successful experiment is that satisfies \( \max \left(\left{\widehat{\theta}}_n{\theta}_n\right\right)<\varepsilon \), and ε equals 0.5 for estimation of the coherent signals.
In the first simulation, we evaluate the performance of the two algorithms with respect to the input signaltonoise ratio (SNR). The number of snapshots is fixed at 1000, and the SNR varies from −10 to 10 dB. The RMSE of the DOAs versus the SNR is shown in Fig. 2. It can be seen from Fig. 2 that the proposed algorithm can provide better DOA estimation than the DMRDOAM algorithm no matter whether the RMSE curve of the α or the RMSE curve of the β. Fig. 3 shows the NPS of the DOAs versus SNR, which illustrates that the performance of the proposed algorithm is better than that of the DMRDOAM algorithm. Furthermore, even at low SNR, the proposed algorithm can still achieve better estimation performance. The reason is that the proposed algorithm takes full advantage of all the received data of the two parallel ULAs to construct the equivalent autocovariance matrix R _{ xx } and crosscovariance matrix R _{ yx }, which can improve the estimation precision. On the contrary, the DMRDOAM algorithm obtains the DOAs of signals at the cost of reduction in array aperture, which often leads to poorer DOA estimation.
In the second simulation, we investigate the performance of the two algorithms versus the number of snapshots. The simulation conditions are similar to those in the first simulation, except that the SNR is set at 5 dB, and the number of snapshots is varied from to 10 to 250. The RMSE of the DOAs versus the number of snapshots is depicted in Fig. 4. As shown in Fig. 4, the proposed algorithm behaves better performance than the DMRDOAM algorithm.
The result in Fig. 5 shows the NPS of the DOAs versus the number of snapshots. From Fig. 5, it can be observed that the proposed algorithm has much higher estimation performance than the DMRDOAM algorithm as the number of snapshots increases. Moreover, the superiority of the proposed algorithm is much more obvious than the DMRDOAM algorithm no matter whether the number of snapshots is small or large. This indicates that the proposed algorithm is more useful especially when the lowcomputational cost and highly realtime data process are inquired.
In the last simulation, we assess the performance of the proposed algorithm as the correlation factor ρ is varied from 0 to 1 between s _{1}(t) and s _{2}(t). The SNR is set at 5 dB, and the number of snapshots is 800. Note that the ε in (32) is set to be 0.6 in this simulation. The performance curves of the DOA estimation against correlation factor are shown in Figs. 6 and 7. From Figs. 6 and 7, we can see that the proposed algorithm outperforms the DMRDOAM algorithm.
5 Conclusions
A novel decoupling algorithm for 2D DOA estimation with two parallel ULAs has been presented. In the proposed algorithm, two equivalent covariance matrices are reconstructed to achieve the decorrelation of the coherent signals and the estimated angle parameters are pairmatched automatically. It has been shown that the proposed algorithm yields remarkably better estimation performance than the DMRDOAM algorithm.
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Acknowledgements
This research was supported by the National Natural Science Foundation of China (61602346), by the Key Talents Project for Tianjin University of Technology and Education (TUTE) (KYQD16001), by the Tianjin Municipal Science and Technology innovation platform, intelligent transportation coordination control technology service platform (16PTGCCX00150), and by the National Natural Science Foundation of China (61601494).
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The authors declare that they have no competing interests.
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Shi, H., Li, Z., Cao, J. et al. Twodimensional DOA estimation of coherent sources using two parallel uniform linear arrays. J Wireless Com Network 2017, 60 (2017). https://doi.org/10.1186/s1363801708440
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DOI: https://doi.org/10.1186/s1363801708440