Computationally Efficient Ambiguity-Free Two-Dimensional DOA Estimation Method for Coprime Planar Array: RD-Root-MUSIC Algorithm

Key Laboratory of Dynamic Cognitive System of Electromagnetic Spectrum Space (Nanjing University of Aeronautics and Astronautics), Ministry of Industry and Information Technology, Nanjing 211106, China College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China +e 28 Research Institute of China Electronics Technology Group Corporation, Nanjing 210007, China

In recent years, sparse arrays such as coprime arrays [15,17,18], nested arrays [19], and minimum redundancy arrays [20] have been proposed to tackle this issue. As a typical sparse array, the coprime arrays have inherent superiorities over the conventional compact arrays, including enlarged array aperture, increased DOFs, and reduced mutual coupling [21], employed the traditional 2D-MUSIC algorithm to CPA by exploring the transformation relation between true and ambiguous estimates, and further proposed a 2D partial spectrum search (2D-PSS) method [22], which considerably relieves the computational burden of 2D total spectrum search (TSS). By combining the reduceddimensional MUSIC (RD-MUSIC) [23] method with the PSS method, the reduced-dimension transformation is performed to further reduce complexity [24]. A generalized CPA array structure was designed in [25] based on the mechanism of ambiguity elimination method, which provides a more flexible array layout and significant increase in DOFs. e aforementioned methods [22,24,25] can be categorized as decomposition algorithms, which process the received data of each subarray separately and then the estimates are combined to determine the final DOAs, whereas the mutual information between the two subarrays is unfortunately neglected. e root-MUSIC method to CPA with low complexity is applied [26], while the estimation performance of this cascade approach depends heavily on the initial estimates, especially at low SNRs. An ambiguity-free MUSIC (AF-MUSIC) method was proposed in [27], where the output of two subarrays were stacked and processed jointly to avoid the ambiguous problem. Although the entire information of CPA is fully exploited, the 2D spectrum search leads to heavy computational burden.
In the above research studies for 2D DOA estimation methods with CPA, they either treat the two subarrays as individual arrays, which suffers performance degradation due to the loss of mutual information, or 2D spectrum search is required leading to expensive computational cost, or extra ambiguity elimination process is involved. To address these issues, we propose a computationally efficient ambiguity-free algorithm via reduced-dimensional polynomial rooting technique. Specifically, we first map the CPA into full array of the CPA (FCPA) using an extraction matrix, based on the characteristics of array configuration, which enables the sufficient utilization of the entire received data of the CPA. Meanwhile, the number of achievable DOFs is enhanced benefiting from the utilization of full information, as compared to the conventional decomposition algorithms. Furthermore, we transform the 2D spectrum search into 2D polynomial root-finding process and further perform reduced-dimension transformation to convert the 2D root-finding operation into two 1D one, which substantially reduces complementation complexity as well as the computational burden. In addition, extra ambiguity elimination can be avoided owing to the inherent ambiguity-free characteristic of the FCPA.
We summarize the major contributions of our work below: (1) We construct the FCPA corresponding to CPA, which processes ambiguity-free characteristic and thereby extra ambiguity elimination operation can be avoided (2) We exploit the received data of the two subarrays jointly, where improved DOA estimation performance as well as enhanced achievable DOFs can be achieved (3) We propose a reduced-dimensional polynomial root-finding algorithm with CPA for 2D DOA estimation, which transforms the 3D spectrum search into 1D polynomial rooting and hence reduces the complexity significantly while preserving the estimation accuracy We outline this paper as follows. Section 2 introduces the data model of CPA and its corresponding FCPA. e proposed algorithm is elaborated in Section 3, and we analyze the complexity and DOFs in Section 4. Section 5 provides simulation results to corroborate the effectiveness of the proposed algorithm, and Section 6 concludes this paper.
1.1. Notations. Bold uppercase (lowercase) characters represent matrices (vectors). (·) T , (·) H , (·) − 1 , and (·) * denote the transpose, conjugate transpose, inverse, and conjugate operation, respectively. ⊗ and ⊙ are Kronecker product and Khatri-Rao product, respectively. Rank (·) means the rank of the matrix. angle(·) represents the phase operator. det(·) denotes the determinant of the matrix. e total number of elements is T � M 2 1 + M 2 2 − 1 since the two subarrays share the same element at the origin. Define a transformation as u k � sin θ k sin ϕ k and v k � sin θ k cos ϕ k for simplification. A CPA configuration is displayed in Figure 1 as an example, where M 1 � 2, M 2 � 3, and T � 12.

Preliminaries
For the ith (i � 1, 2) subarray, the received signal can be expressed by [22] where S represents the source matrix and S � [s 1 , s 2 , . . . , s K ] T ∈ C K×L , s k denotes source vector and s k � [s k (1), . . . , s k (L)] T ∈ C L×1 , L is the number of snapshots, C represents a complex set, A i is the steering matrix of the ith (i � 1, 2) subarray, and and a xi (v k ) represent the steering vectors along the y-axis and x-axis, respectively, the specific forms can be expressed as Gaussian noise with mean value zero and variance σ 2 of the ith (i � 1, 2) subarray. e output of the whole CPA can be stacked as [27] X � X 1 where A represents the direction matrix of the whole CPA, 2 )×K , and N denotes the white Gaussian noise of the whole arrays and

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In practice, the covariance matrix of X can be calculated using L snapshots by Perform eigenvalue decomposition (EVD) and R can be decomposed by where Λ s is a diagonal matrix whose diagonal elements are the K largest eigenvalues, Λ n is a diagonal matrix composed of the rest eigenvalues, E s represents the signal subspace spanned of the eigenvectors corresponding to the K largest eigenvalues, and E n is the noise subspace composed of the remaining eigenvectors.

Full Array of CPA and Extraction.
e CPA can be extracted from a large nonuniform planar array, which can be denoted by the full array of CPA (FCPA). e sensor location of the FCPA can be expressed as where Ω xFCPA and Ω yFCPA represent the location sets of xaxis and y-axis, respectively. Figure 2 illustrates the FCPA corresponding to the CPA shown in Figure 1, where Ω xFCPA � 0, 2, 3, 4 { }d and Ω yFCPA � 0, 2, 3, 4 { }d, d � λ/2. It can be observed that the FCPA contains all elements of the CPA and has four additional elements with sensor number 7, 10, 12, and 13, respectively, which demonstrates that the CPA can be regarded as an extraction from FCPA.
According to the correspondence of the CPA and FCPA, we introduce an extraction matrix G ∈ 0, 1 { } to characterize the mapping relation as where denote the location sets of elements on the y-axis and x-axis, respectively.
To demonstrate the extraction more specifically, each element in the CPA and FCPA is labeled according to their order in the steering vectors, i.e., 1 ∼ M 2 1 for the subarray 1 and M 2 1 + 1 ∼ M 2 1 + M 2 2 for the subarray 2 in CPA, 1 ∼ (M 1 + M 2 − 1) 2 for the FCPA. If the ith sensor in the CPA and the jth sensor in the FCPA overlap; then, g ij � 1, otherwise g ij � 0, where g ij denotes the (i, j)th element of G. For the FCPA given in Figure 2, G is a 13 × 16 matrix with 3 columns of all zeros.
Definition 1 (extraction efficiency). e extraction efficiency is the proportion of nonzero elements in the extraction matrix.
e sensor location of CPA can be given by where and m 1 , m 2 ∈ Z. en, we construct a uniform planar array that has the same array aperture as the CPA with the location set: where e locations of the sensors in the FCPA corresponding to the CPA can be expressed as where Ω xCPA ⊆ Ω xFCPA ⊆ Ω xUPA and Ω yCPA ⊆ Ω yFCPA ⊆ Ω yUPA represent the location set of FCPA on the x-and y-axes, respectively.
For the CPA study in this paper, the corresponding FCPA can be constructed by the following four forms: According to Definition 1, the extraction efficiency of the above schemes is 0.0625, 0.05, 0.05, and 0.04, respectively. It is clearly seen that the FCPA we designed in (5) has the highest extraction efficiency.

2D-MUSIC Algorithm.
Derive from the orthogonal relationship between the noise subspace and the steering vector, and the spectral function of CPA can be represented by [26,27] where a(u, v) denotes the steering vector of CPA, � 1, 2), and E n is the noise subspace of CPA.
According to (6), we have a(u, v) � Ga F (u, v). en, (11) can be rewritten as [26,27] where a F (u, v) is the steering vector of the FCPA, Although the autopaired 2D DOA estimates can be obtained via performing spectral search on (12), it suffers from tremendously expensive computational cost. To tackle this issue, we first performed reduced-dimension transformation and then exploited 1D polynomial root-finding technique to estimate u and v.

Reduced-Dimensional Polynomial Root-Finding Process.
Construct the polynomial based on (12) as where and According to the relation of rank of the matrix product, the following constraint has to be satisfied, and then we have We can conclude that det Q(u) { } is nonzero polynomial from (16); thus, Q(u) is a factor of V(u, v). As Q(u) depends only on the variable u, the roots of det Q(u) { } � 0 can make the following equation hold: It is noteworthy that only 1D polynomial is involved to achieve the estimates of u. Similarly, we can get the estimates of v. Consequently, the problem of obtaining paired estimates of u and v from the 2D polynomial is transformed into two 1D root-finding process. en, we reconstruct (13) and (14) as 4 Mathematical Problems in Engineering Define z 1 � e j2πdu/λ , where d � λ/2. Define the steering vector of UPA that has the same array aperture as FCPA along the x-axis as a Ex (v) � [1, e j2πdv/λ , ×1 . We assume that M 1 < M 2 for simplification and then a Ex (v) can be expressed as a Ex (v) � [1, e j2πdv/λ , . . . , e j2πM 1 (M 2 − 1)dv/λ ] T ∈ C (M 1 (M 2 − 1)+1)×1 . Based on the correspondence between FCPA and UPA with same array aperture, we have where G 1 ∈ Z (M 1 +M 2 −1)×(M 1 (M 2 −1)+1) . To be specific, g 1ij � 1 holds when the ith sensor in the a Fx (v) and the jth sensor in the a Ex (v) overlap, otherwise g ij � 0, where g ij is the (i, j)th element of G 1 . For the FCPA displayed in Figure 2, G 1 is a 4 × 5 matrix with one columns of all zeros. Similarly, we can obtain the relation a Fy (u) � G 1 a Ey (u). Correspondingly, the steering vectors can be rewritten as Without loss of generality, substituting z Considering that det Q(z 1 ) and det Q(z 2 ) are polynomials of even degree, u k and v i can be obtained from the K roots distributed closest to the unit circle corresponding to (24) and (25), and the roots are denoted by z 11 , . . . , z 1k , . . . , z 1K and z 21 , . . . , z 2i , . . . , z 2K , respectively, i.e., For the conventional DOA estimation methods with CPA, the ambiguity elimination operation is required since the interelement spacing in the two subarrays is larger than half-wavelength. e FCPA is an unambiguous array which has at least one sensor pair with separation no larger than half-wavelength according to (5), and we can obtain the true DOA estimates directly after the parameter pairing without extra ambiguity elimination process.

Parameter Pairing and DOA Estimation.
In this part, we determine the pairing of u k and v i since the two root-finding procedures are conducted separately. Construct the cost function for pairing as where a(u k , v i ) represents the steering vector reconstructed u k and v i , which can be obtained according to (1). For each u k , we can obtain the value of i and k that minimize V k,i (1 ≤ i ≤ K), and we define the paired index as i ′ and k. Finally, the 2D DOAs can be calculated by Mathematical Problems in Engineering where v i′ is reconstructed by i ′ (1 ≤ i ′ ≤ K).

e Procedure of the Proposed Algorithm.
We summarize the major steps of the proposed algorithm as follows: Step1: calculate R of the received data X and perform EVD to obtain the noise space E n Step 2: reconstruct the spectral function P(u, v) according to (6) Step 3: construct the polynomial V(u, v) and conduct reduced-dimension transformation according to (12) Step 4: calculate u k and v i according to (26) and (27) Step 5: perform parameter matching to obtain θ k and ϕ k according to (28)-(30)

Complexity Analysis.
Herein, we compare the computational complexity of the proposed algorithm, 2D-PSS [22], RD-MUSIC [23], AF-MUSIC [27], and 2D-ROOT [26] methods in this section. For the proposed algorithm, calculating the covariance matrix requires O T 2 L and the complexity of eigenvalue decomposition is O T 3 . e rootfinding operation costs O 2(2M 1 (M 2 − 1)) 3 and parameter matching process requires O K 2 (T − K)(T + 1) . Consequently, the total complexity is O T 2 L + T 3 + 2(2(M 1 (M 2 − 1))) 3 + K 2 (T − K)(T + 1) . Table 1 lists the total complexity of above algorithms, where Δ � 0.0001 is the spectral search interval. In addition, Figure 3 displays the complexity comparison versus number of sensors, where K � 2 and L � 500, while the complexity comparison with different number of snapshots is shown in Figure 4, where K � 2, M 1 � 2, and M 2 � 3. It is clearly seen that the proposed algorithm owns the approximate low complexity to the 2D-ROOT method, which is significantly lower than that of the 2D-PSS, RD-MUSIC, and AF-MUSIC methods, as the spectral search process with heavy computational burden is transformed into computationally efficient polynomial root-finding.

Achievable DOFs.
As for the conventional decomposition-based 2D DOA estimation methods, such as RD-MUSIC, 2D-PSS, and other algorithms, the number of achievable DOFs is min M 2 1 − 1, M 2 2 − 1 . According [26], M 2 1 + M 2 2 − 2 signals at most can be resolved by utilizing AF-MUSIC algorithm which processes the received data of the two subarrays jointly, whereas extremely high computational complexity is involved due to the 2D spectral peak search. RD root-finding technique is employed in the proposed algorithm to deal with the complexity and the achievable DOFs can be obtained from (15). If u or v does not match any of the incident signals, we have the constraint where Consequently, the maximum number of signals which can be identified by the proposed algorithm is It is clear that the proposed algorithm can greatly improve the DOFs compared with the traditional 2D DOA estimation methods.

4.4.
Advantages. Based on the above discussion, the advantages of the proposed algorithm can be listed as follows: (1) It can achieve superior estimation performance and higher achievable DOFs than the decomposition methods, owing to the utilization of entire received data. Moreover, additional ambiguity elimination is no longer required due to the ambiguity-free characteristic of the FCPA.

Simulations
In this section, we perform 500 Monte Carlo simulations to validate the performance of the proposed algorithm. Assume that K uncorrelated far-field narrowband signals impinge on the CPA. Define root mean square error (RMSE) by where ϕ k,i and θ k,i are the estimates of the kth signal in the ith trial corresponding to the true azimuth ϕ k and elevation θ k , respectively.

RMSE Results Versus Snapshots.
In this part, we present the DOA estimation performance of the proposed algorithm with different number of snapshots in Figure 6, where e result demonstrates that the estimation accuracy improves as the number of snapshots increases, owing to the more accurate covariance.

RMSE Results Versus Number of Sensors.
Herein, we provide the RMSE results of the proposed algorithm versus number of sensors in Figure 7, where (θ 1 , ϕ 1 ) � (20 ∘ , 30 ∘ ), (θ 2 , ϕ 2 ) � (40 ∘ , 50 ∘ ), and L � 200. As the number of array elements increases, the diversity gain of the receiving antenna increases. It is illustrated clearly that the increased number of sensors leads to improved DOA estimation performance. Figure 8 exhibits the RMSE comparison of the proposed    Figure 8 that the proposed algorithm and AF-MUSIC algorithm, benefiting from the utilization of the received data of the entire CPA, outperform the decomposition-based RD-MUSIC, 2D-PSS, and 2D-ROOT algorithms. Furthermore, the proposed algorithm yields superior estimation performance to the AF-MUSIC algorithm, as the proposed algorithm directly performs two root-finding operations based on the spectral function (11), while the cascading process in the AF-MUSIC algorithm may result in performance degradation.

Conclusion
In this paper, we propose a computationally efficient 2D DOA estimation algorithm for CPA by exploiting the RD polynomial root-finding technique. e proposed algorithm first maps CPA into FCPA and exploits the received data of two subarrays jointly, where the mutual information loss is avoided and simultaneously the improved estimation performance as well as enhanced DOFs can be achieved. In particular, the FCPA we constructed is an ambiguity-free array with high extraction efficiency. Furthermore, we convert the 2D total spectral search into one 1D polynomial root-finding process via reduced-  Mathematical Problems in Engineering dimension transformation, which significantly reduces the computational cost and simultaneously preserves the estimation accuracy. Simulations demonstrates the superiority of the proposed approach in regard to complexity, achievable DOFs and DOA estimation performance.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.
Acknowledgments is work was supported by the China NSF (Grant nos. 61631020, 61971218, 61601167, and 61371169), Fund of Sonar Technology Key Laboratory (Research on the theory and algorithm of signal processing for two-dimensional underwater acoustics coprime array), and Fund of SONAR  Mathematical Problems in Engineering 9 Technology Key Laboratory (Range estimation and location technology of passive target via multiple array combination).