Two-Dimensional DOA Estimation for Uniform Rectangular Array Using Reduced-Dimension Propagator Method

A novel algorithm is proposed for two-dimensional direction of arrival (2D-DOA) estimation with uniform rectangular array using reduced-dimension propagator method (RD-PM). The proposed algorithm requires no eigenvalue decomposition of the covariance matrix of the receive data and simplifies two-dimensional global searching in two-dimensional PM (2D-PM) to onedimensional local searching. The complexity of the proposed algorithm is much lower than that of 2D-PM. The angle estimation performance of the proposed algorithm is better than that of estimation of signal parameters via rotational invariance techniques (ESPRIT) algorithm and conventional PMalgorithms, also very close to 2D-PM.The angle estimation error andCramér-Rao bound (CRB) are derived in this paper. Furthermore, the proposed algorithm can achieve automatically paired 2D-DOA estimation. The simulation results verify the effectiveness of the algorithm.


Introduction
Direction-of-arrival (DOA) estimation is a fundamental problem in array signal processing and has been widely used in many fields [1][2][3][4][5]. Till now, many algorithms have been proposed for DOA estimation with uniform linear array. Among them, multiple signal classification (MUSIC) algorithm [6] and estimation of signal parameters via rotational invariance technique (ESPRIT) algorithm [7] are widely used superresolution methods. The two-dimensional DOA (2D-DOA) estimation problem, which plays an important role in array signal processing, has received more attention. This problem is usually considered with rectangular array, two parallel uniform linear arrays, L-shape array, and so forth. Also, many algorithms have been considered for 2D-DOA estimation, which include 2D-MUSIC algorithm [8], 2D Unitary ESPRIT algorithm [9], modified 2D-ESPRIT algorithm [10], matrix pencil methods [11,12], maximum likelihood method [13], parallel factor (PARAFAC) algorithm [14], and high order cumulant method [15].
Propagator method, which is known as a low complexity method without eigenvalue decomposition (EVD) of the covariance matrix of the received data, has been proposed for DOA estimation through peak searching [17][18][19][20]. Due to its low complexity, PM algorithms are widely used for 2D-DOA estimation. In [21], a PM-based DOA estimation algorithm is proposed for two parallel uniform linear arrays via the rotational invariance property of propagator matrix, which requires extra pairing match. Reference [22] presents an efficient 2D-DOA estimation algorithm with two parallel uniform linear arrays using PM. In [23], an improved PM algorithm is proposed for 2D-DOA estimation, which has better angle estimation performance than the algorithms in [21,22]. The above-mentioned PM algorithms can be extended to the rectangular array for 2D-DOA estimation; however, they only employ the rotational invariance property of propagator matrices, and as a result of that, the accuracy of angle estimation performance is not sufficient in some cases, especially with low signal to noise ratio (SNR). Twodimensional PM (2D-PM) algorithm through peak searching can be extended for 2D-DOA estimation; however, the high computational complexity caused by 2D peak searching makes it inefficient.

International Journal of Antennas and Propagation
In this paper, we derive a reduced-dimension PM (RD-PM) algorithm, which reduces the high complexity for 2D-DOA estimation with uniform rectangular array compared with 2D-PM algorithm. The proposed algorithm applies the rotational invariance property of propagator matrix to get the initial angle estimation and then employs onedimensional local searching to get more accurate angle and finally obtains the other angle via least square (LS) method and estimate azimuth and elevation angles. The proposed algorithm has the following advantages: (1) the proposed algorithm has lower computational complexity than 2D-PM algorithm since it requires no EVD of the covariance matrix of the receive data and it only requires one-dimensional local searching; (2) the angle estimation performance of the proposed algorithm is better than that of ESPRIT algorithm and the PM algorithms in [21,23], also very close to that of 2D-PM algorithm; (3) it can obtain automatically paired parameters estimation.
The remainder of this paper is structured as follows: Section 2 shows the data model for uniform rectangular array, while Section 3 proposes the RD-PM algorithm for 2D-DOA estimation. The angle estimation error is derived as well as CRB in Section 4. In Section 5, the simulation results verify the feasibility and effectiveness of the proposed algorithm, and the conclusions are showed in Section 6.

Data Model
As illustrated in Figure 1, consider a uniform rectangular array having × sensors; and are the numbers of sensors in -axis and -axis, respectively. The distance between the two adjacent elements is in both -axis andaxis. Assume that there are uncorrelated sources, and , are the elevation and azimuth angles of the th source.
Similarly, the received signal of the th subarray x ( ) in the rectangular array is also denoted as where Φ = diag( − 2 sin 1 sin 1 / , . . . , − 2 sin sin / ) and n ( ) is the additive white Gaussian noise of the th subarray. So the integrated received signal of the rectangular array is shown as follows: where x ( ) ( = 1, . . . , ) stands for the received signal of the th subarray. Equation (3) is also denoted as where  (4), we collect snapshots to get the covariance matrixR aŝ

Two-Dimensional DOA Estimation Algorithm
In this section, we give a brief introduction of 2D-PM algorithm. Partition the matrix A as where International Journal of Antennas and Propagation 3 where P ∈ C ×( − ) is the propagator matrix. We define the matrix Q as According to (6) and (7), we have Then, we construct the function of the 2D-PM spatial spectrum as By searching and , the largest peaks of the costing function in (10) correspond to the angles information. However, the high computational cost caused by the 2D peak searching makes this algorithm inefficient. For estimating the propagator matrix P, we partition the covariance matrixR asR whereĜ ∈ C × ,Ĥ ∈ C ×( − ) . The propagator matrix P meets the minimization problem csm (P) = ‖Ĥ −ĜP‖ 2 , and the estimation ofP isP

2D-DOA Estimation.
Define Equation (18) can also be denoted as where Equation (19) is the problem of quadratic optimization. We consider the constraint e 1 a (V) = 1, where e 1 = [1, 0, . . . , 0] ∈ R ×1 . The optimization problem can be reconstructed via linearly constrained minimum variance as [24] The costing function is where is a constant. We have According to (22), we obtain a (V) = C −1 ( )e 1 , where is also a constant. As e 1 a (V) = 1, = 1/{e 1 C −1 ( )e 1 }. Then, a (V) can be expressed as Combining (20) and (23), the estimation of iŝ = arg min 1 Equation (24) also can be present aŝ We find the largest peaks of the (1, , and 1 is the estimation of V . The solution to c is Eĝ .
Finally, we get the estimation of , aŝ Till now, we have achieved the proposed algorithm for 2D-DOA estimation. The major steps and complexity of the algorithm are shown as follows: (1) construct the covariance matrixR via (5);. . ., { 2 2 }; (2) partition theR to get P; then construct Q, P , P 1 , P 2 and obtain the initial estimation of̂i ni from the EVD of P + 1 P 2 ;. . ., { 2 2 + 2 + 2 2 ( − 1) + 2 3 };  Remark 1. In this paper, we assume that the target number is a prior known parameter. If not, we can estimate by applying the method in [25,26].
Remark 2. The proposed algorithm gets the estimation oft hrough searching ∈ [̂i ni − Δ ,̂i ni + Δ ] and then obtains the correspondingâ (V ) and estimates correspondingV . SincêandV are correspondence, the proposed algorithm can obtain automatically paired parameters estimation.
Remark 3. The proposed algorithm requires , V ( = 1, . . . , ) being different. If the sources have the same or V, then the direction matrix A = A ∘ A is a matrix of rank loss according to the property of Khatri-Rao product [27]. That will lead many algorithms invalid, such as ESPRIT, PMs in [21,23], and our algorithm.
The complexity comparison with different parameters is shown in Figures 2 and 3. From Figures 2 and 3, we can find that the proposed algorithm has much lower complexity than 2D-PM algorithm since the proposed algorithm only requires a one-dimensional local searching.

Performance Analysis
This section aims at analyzing the theoretic estimation error of the proposed algorithm and deriving the Cramér-Rao bound (CRB).
The estimation of propagator matrix P can be denoted aŝ

P = [(G + ΔG) (G + ΔG) ] −1 (G + ΔG) (H + ΔH) . (33)
With a first-order expansion in ΔG and ΔH,P is rewritten aŝ where ΔP = (GG ) −1 G(ΔH − ΔG P). We also obtain where SinceĈ =QQ = C + ΔC, using first-order expansion we obtain The proposed algorithm has the same cost function as 2D-PM algorithm, which is a reduced-dimension form of 2D-PM International Journal of Antennas and Propagation  After the Taylor expansion around (use instead of ( , V) for simplification), where is a vector of some value on the line segment joininĝ and . Define Under large sample, (̂− ) is asymptotically zero-mean Gaussian distributed [17][18][19], so the associated asymptotic covariance matrix is given by According to (36) and (41) and taking into account that a Q = 0, we can determine where (2,2), and [ V ] = Φ (2, 1). According to (29), the mean square error (MSE) of azimuth and elevation can be derived as The comparison of theoretical estimation error and simulation results will be shown in Section 5.

Simulation Results.
We present 1000 Monte Carlo simulations to assess the angle estimation performance of the RD-PM algorithm. Define root mean square error (RMSE): adjacent elements is = /2. Note that and are the numbers of sensors in -axis and -axis, respectively. is the number of snapshots. Simulation 2. Figures 5 and 6 present angle estimation performance comparison among the proposed algorithm, 2D-PM algorithm, the PM algorithm in [21], the PM algorithm in [23], ESPRIT algorithm, RD-MUSIC algorithm, CRB, and the theoretic estimation error of RD-PM. We can see that the estimation performance of the proposed algorithm is better than ESPRIT algorithm and the PM algorithms in [21,23] and is very close to 2D-PM algorithm and RD-MUSIC algorithm. Compared with 2D-PM algorithm, the proposed algorithm has the same costing function and simplifies two-dimensional global searching to one-dimensional local searching. As a result of that, our algorithm has very close angle estimation performance to 2D-PM algorithm and greatly reduces complexity. Furthermore, Figures 5 and 6 also show that the RMSE of the proposed algorithm is almost the same as the theoretic estimation error.
Simulation 3. Figure 7 presents angle estimation performance of the proposed algorithm with = 8, = 8, = 3, and different values of . We can find that the angle estimation performance is improved when increases. When increases, we get more samples to estimate more accurate propagator matrix P, and the angle estimation performance is enhanced.    angle estimation performance of the proposed algorithm is affected.
Simulation 5. Figures 9 and 10 investigate angle estimation performance of the proposed algorithm with = 3, = 100, and different values of or ,respectively. It is indicated that the performance of the proposed algorithm is improved when or increases. Multiple sensors enhance the aperture of the array as well as diversity gain, so the angle estimation performance is improved. Simulation 6. Figure 11 depicts angle estimation performance of the proposed algorithm when the two sources are closely spaced with = 8, = 8, = 100, and SNR = 20 dB. It is shown in Figure 11 that our algorithm works well for the closely spaced sources.

Conclusions
In this paper, we have presented a novel algorithm for 2D-DOA estimation for uniform rectangular array using RD-PM algorithm. The proposed algorithm, which only requires one-dimensional local searching and avoids the EVD of the covariance matrix of the receive data, has a much lower complexity than 2D-PM algorithm. Simulation results show that the angle estimation performance of the proposed algorithm is better than that of ESPRIT algorithm and conventional PM algorithms, also very close to that of 2D-PM algorithm. Furthermore, the proposed algorithm can achieve paired 2D-DOA estimation automatically.
International Journal of Antennas and Propagation