IMPROVED BLIND 2D-DIRECTION OF ARRIVAL ESTI- MATION WITH L-SHAPED ARRAY USING SHIFT IN- VARIANCE PROPERTY

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1 J. of Electromagn. Waves and Appl., Vol. 23, , 2009 IMPROVED BLIND 2D-DIRECTION OF ARRIVAL ESTI- MATION WITH L-SHAPED ARRAY USING SHIFT IN- VARIANCE PROPERTY X. Zhang, X. Gao, and W. Chen Department of Electronic Engineering Nanjing University of Aeronautics & Astronautics Nanjing , China Abstract In this paper, we address the problem of bind 2D Direction Of Arrival (DOA) estimation with L-shaped array. Based on shift invariance property, an improved method for estimating 2D DOA is presented. The algorithm uses the array geometries to construct a matrix and then obtain the required signal subspace via the eigen decomposition of the constructed matrix. This algorithm works well without spectral peak searching. Our algorithm has much better 2D- DOA estimation performance than conventional ESPRIT algorithm, and it can identify more DOAs than conventional ESPRIT algorithm. 1. INTRODUCTION Antenna arrays have been used in many fields such as radar, sonar, communications, seismic data processing, etc. [1 16]. The directionof-arrival (DOA) estimation [17 27] of signals impinging on an array of sensors is a fundamental problem in array processing, and many methods have been proposed for its solution. Uniform linear arrays for estimation of wave arrival have been studied extensively, and the most popular techniques for the DOA estimations are the classical subspace method such as ESPRIT [26] and MUSIC [27] algorithms. Compared with uniform linear array, L-shaped array can identify 2D- DOA. 2D-DOA estimation with L-shaped array has been has received considerable attention in the field of array signal processing [28 36], and they contain ESPRIT algorithms [28, 29], MUSIC algorithm [30], matrix pencil methods [31, 32], propagator methods [23 35], high order cumulant method [36], etc. Corresponding author: X. Zhang (fei zxf@163.com).

2 594 Zhang, Gao, and Chen High order cumulant method requires the signal statistical properties, and needs a larger computation complexity. ESPRIT and MUSIC algorithms are based on signal subspace, and have better DOA estimation performance. MUSIC requires spectral peak searching, which is computationally expensive. The primary computational advantage of ESPRIT is that it eliminates the inherent search procedure. ESPRIT produces signal parameter estimates directly in terms of (generalized) eigenvalues. Propagator method has low complexity, but its 2D-DOA estimation performance is less than ESPRIT algorithm. Ref. [28] used ESPRIT method for 2D-DOA estimation with L-shaped array, and Ref. [29] proposed the improved ESPRIT algorithm for 2D-DOA estimation, which had better 2D- DOA estimation performance than Ref. [28]. An improved 2D-DOA estimation with L-shaped array is investigated in this paper. The algorithm uses full signal shift invariance property to estimate 2D- DOA, and has better DOA estimation than ESPRIT [29]. Furthermore it can identify more DOAs than conventional ESPRIT algorithm. This paper is structured as follows. Section 2 develops data models. Section 3 deals with algorithmic issues. Section 4 presents simulation results, and Section 5 summarizes our conclusions. Denote: We denote by (.) the complex conjugation, by (.) T the matrix transpose, and by (.) H the matrix conjugate transpose. The notation (.) + refers to the Moore-Penrose inverse (pseudo inverse). F stands for Forbenius norm. 2. DATA MODEL We consider an L-shaped array with sensors at 2M 1 different locations as shown in Fig. 1. A uniform linear array containing M elements is located in Y -axis, and the other uniform linear array containing M elements is located in X-axis. We suppose that there are Z d 12,,...,M 12,,...,M Y X Figure 1. The structure of an L-shaped array.

3 2D-DOA estimation with L-shaped array 595 K sources impinge the L-shaped array with (θ k, φ k ), k = 1, 2,..., K, where θ k, φ k are the elevation angle and the azimuth angle of the kth source, respectively. It is assumed that there are K sources impinging the array. The received signal of M elements in X-axis is shown X = A x S + N x (1) where S C K N is the source matrix, N x C M N is received noise, and A x C M K is 1 1 e j2πd cos θ 1 sin φ 1 /λ e j2πd cos θ 2 sin φ 2 /λ A x =.. e j2πd(m 1) cos θ 1 sin φ 1 /λ e j2πd(m 1) cos θ 2 sin φ 2 /λ 1 e j2πd cos θ K sin φ K /λ... (2). e j2πd(m 1) cos θ K sin φ K /λ where d is element spacing. The received signal of M elements in Y -axis is denoted as Y = A y S + N y (3) where N y is received noise, and A y C M K is 1 1 e j2πd sin θ 1 sin φ 1 /λ e j2πd sin θ 2 sin φ 2 /λ A y =.. e j2πd(m 1) sin θ 1 sin φ 1 /λ e j2πd(m 1) sin θ 2 sin φ 2 /λ 1 e j2πd sin θ K sin φ K /λ.... e j2πd(m 1) sin θ K sin φ K /λ A x and A y are Vandermonde matrices. The matrices X C M N, Y C M N, A x C M K, A y C M K are denoted as X1 x1 X = = (5) x M X 2 Y1 y1 Y = = (6) y M Y 2 (4)

4 596 Zhang, Gao, and Chen Ax1 ax1 A x = = (7) a xm A x2 Ay1 ay1 A y = = (8) a ym A y2 where x 1, x M are the first and last rows of matrix X, respectively. y 1, y M are the first and last rows of matrix Y, respectively. a x1, a xm are the first and last rows of matrix A x, respectively. a y1, a ym are the first and last rows of matrix A y, respectively. According to Eqs. (5) (8), we construct the following matrices, C 1 = E{X 1 Y H 1 } = A x1 R S A H y 1 + N 1 (9) C 2 = E{X 2 Y H 1 } = A x1 Φ x R S A H y 1 + N 2 (10) C 3 = E{X 1 Y H 2 } = A x1 R S Φ H y A H y 1 + N 3 (11) C 4 = E{X 2 Y H 2 } = A x1 Φ x R S Φ H y A H y 1 + N 4 (12) where Φ x = diag[e j(2π/λ)d cos θ 1 sin φ 1, e j(2π/λ)d cos θ 2 sin φ 2,..., e j(2π/λ)d cos θ K sin φ K ], N 1, N 2, N 3 and N 4 are noise matrices. Φ y = diag[e j(2π/λ)d sin θ 1 sin φ 1, e j(2π/λ)d sin θ 2 sin φ 2,..., e j(2π/λ)d sin θ K sin φ K ], Φ x, Φ y are rotation matrices, and we can use shift invariance properties to estimate 2D-DOA. E{.} is the expectation; R S = E{SS H } is the source correlation matrix with K K. For the independent sources, R S should be a diagonal matrix, and then the Eq. (11) and Eq. (12) can be expressed as C 3 = A x1 Φ H y R S A H y 1 + N 3 (13) C 4 = A x1 Φ x Φ H y R S A H y 1 + N 4 (14) 3. IMPROVED 2D-DOA ESTIMATION WITH SHIFT INVARIANCE PROPERTY According to Eqs. (9), (10), (13), (14), we form the following matrix, C 1 C C = 2 C (15) 3 C 4 For Eq. (15), R C = CC H. We denote the matrix containing the eigenvectors {f k } K k=1 associated with the K largest eigenvalues of R C

5 2D-DOA estimation with L-shaped array 597 by E. In the no-noise case, the matrix E can expressed as E 1 A x1 E E = 2 E = A x1 Φ x 3 A x1 Φ H y T (16) E 4 A x1 Φ x Φ H y where E 1 = A x1 T, E 2 = A x1 Φ x T, E 3 = A x1 Φ H y T, E 4 = A x1 Φ x Φ H y T, T is a K K full-rank matrix, Φ x, Φ y are the rotation matrices, and then we can use shift invariance property to estimate 2D-DOAs. According to Eq. (16), we get E3 E1 = T E 4 E 1 Φ H y T (17) 2 Define Ω y = T 1 Φ H E y T. Eq. (17) becomes 3 E = 1 Ω E 4 E y, and Least 2 squires solution for Ω y, ˆΩ y = [ ] + [ ] E1 E3 E 2 E 4 (18) Because ˆΩ y has the same eigenvalues as Φ H y, we use eigenvalue decomposition (EVD) for ˆΩ y to get e j(2π/λ)d sin θ k sin φ k, k = 1, 2,..., K, and attain u k, the estimated value of sin θ k sin φ k. According to Eq. (16), we get E2 E1 = T E 4 E 1 Φ x T (19) 3 Define Ω x = T 1 Φ T E x. Eq. (19) becomes 2 E = 1 Ω E 4 E x and Least 3 squires solution for Ω x, ˆΩ x = [ ] + [ ] E1 E2 E 3 E 4 (20) Because ˆΩ x has the same eigenvalues as Φ x, we use EVDfor ˆΩ x to get e j(2π/λ)d cos θ k sin φ k, k = 1, 2,..., K, and obtain v k, the estimated value of cos θ k sin φ k. Using the pairing method proposed by Dong [28], we get the pairs (u k, v k ), k = 1, 2,..., K. The 2D-DOAs are estimated via ( ) ˆφ k = sin 1 u 2 k + v2 k (21) ˆθ k = tan 1 (u k /v k ) (22)

6 598 Zhang, Gao, and Chen The steps of our proposed algorithm are shown as follows Step 1. For the received noisy signal, we construct the matrices C 1, C 2, C 3, C 4, and get the matrix C in Eq. (15). Step 2. Compute E by EVD for R C, where R C = CC H, and get E 1, E 2, E 3 and E 4. Step 3. Compute Ω y using Eq. (18), and employ EVD for Ω y to attain the estimated value of sin θ k sin φ k, k = 1, 2,..., K. Step 4. Compute Ω x using Eq. (20), and employ EVD for Ω x to attain the estimated value of cos θ k sin φ k, k = 1, 2,..., K. Step 5. After pairing, 2D-DOA are estimated via Eq. (21) and Eq. (22). ESPRIT algorithm in [29] used Ω x = E + 1 E 2 and Ω y = E + 2 E 4 for 2D-DOA estimation. ESPRIT can identify M 1 sources, and the maximum source number which our algorithm can estimate is 2M 2. So our algorithm can identify more DOA than conventional ESPRIT. In contrast to ESPRIT [29], our algorithms have a larger computational load, which is usually dominated by formation of the covariance matrix and calculation of EVD. For our algorithm, the computational complexity of formation of the covariance matrix is O(4 2 (M 1) 3 ); calculation of its eigendecomposition requires O(4 3 (M 1) 3 ); and eigenvalue decomposition for Ω x or Ω y requires O(K 3 ). So the major computational complexity of our algorithm is O(80(M 1) 3 + 2K 3 ), while ESPRIT requires O(36(M 1) 3 + 2K 3 ). 4. SIMULATION RESULTS We present Monte Carlo simulations that are to assess 2D-DOA estimation performance of the proposed algorithms. The number of Monte Carlo trials is There are two signals impinging on L- shaped array at (30, 30 ) and (40, 40 ), respectively. We supposed there is additive Gaussian white noise. We consider an L-shaped array with 2M 1 sensors, which is shown in Fig. 1. A half wavelength of the incoming signals is used for the spacing between the adjacent elements in each uniform linear array m=1 θ m θ 0 2, where θ m is the Define RM SE = estimated DOA of the mth simulation, θ 0 is the perfect DOA. Figure 2 shows 2D-DOA estimation of our proposed algorithm at SNR = 15 db, and Fig. 3 presents 2D-DOA estimation of our proposed algorithm at SNR = 25 db. The L-shaped array with 9 antennas, 40 Monte Carlo simulations and data length N = 500 are used in Fig. 2

7 2D-DOA estimation with L-shaped array azimuth angle/deg elevation angle/deg Figure 2. 2D-DOA estimation performance at SNR = 15 db azimuth angle/deg elevation angle/deg Figure 3. 2D-DOA estimation performance at SNR = 25 db. and Fig. 3. Form Fig. 2 and Fig. 3, we find that our proposed algorithm works well. We compare our algorithms against ESPRIT [29]. Their DOA estimation performance comparisons under different SNR are shown in Figs The L-shaped array with 9 antennas and data length N = 500 are used in Figs Fig. 4 shows the RMSE of the estimate of (30, 30 ) versus SNR, and Fig. 5 presents the DOA estimation performance comparison of the estimate of (40, 40 ). From Fig. 4 and Fig. 5 we find that our algorithm has much better DOA estimation

8 600 Zhang, Gao, and Chen elevation angle estimation ESPRIT Our algorithm azimuth angle estimation ESPRIT Our algorithm Figure 4. 2D-DOA (30, 30 ) estimation performance comparison. elevation angle estimation ESPRIT Our algorithm azimuth angle estimation ESPRIT Our algorithm Figure 5. 2D-DOA (40, 40 ) estimation performance comparison. performance than ESPRIT algorithm [29]. Our algorithm fully utilizes the signal shift invariance structure, and has better capability to suppress noise. Figures 6 and 7 present 2D-DOA estimation performance of our algorithm with different array configurations. The L-shaped array has different DOA estimation performance with different antennas. The data length with 500 samples is used in Figs From Fig. 6 and Fig. 7 we find 2D-DOA estimation performance of our algorithm is improved with the number of antennas increasing. When the number of antennas increases, our algorithm has higher receive diversity. As a result, our algorithm has better 2D-DOA estimation performance.

9 2D-DOA estimation with L-shaped array 601 elevation angle estimation L-shape array with 7 antennas L-shape array with 9 antennas L-shape array with 11 antennas L-shape array with 13 antennas 10 2 azimuth angle estimation L-shape array with 7 antennas L-shape array with 9 antennas L-shape array with 11 antennas L-shape array with 13 antennas Figure 6. 2D-DOA (30, 30 ) estimation performance comparison with different array. elevation angle estimation L-shape array with 7 antennas L-shape array with 9 antennas L-shape array with 11 antennas L-shape array with 13 antennas azimuth angle estimation L-shape array with 7 antennas L-shape array with 9 antennas L-shape array with 11 antennas L-shape array with 13 antennas Figure 7. 2D-DOA (40, 40 ) estimation performance comparison with different array. Figures 8 and 9 present 2D-DOA estimation performance of our algorithm with different data lengths. The L-shaped array with 9 antennas is used in Figs From Fig. 8 and Fig. 9, we find 2D-DOA estimation performance of our algorithm is improved with the number of snapshot increasing. When the number of snapshot increases, the constructed matrix C in Eq. (15) has better capability to suppress noise. As a result, our algorithm has better 2D-DOA estimation performance. As we mentioned above, ESPRIT algorithm only works well when K M 1. When K > M + 1, ESPRIT algorithm fails to

10 602 Zhang, Gao, and Chen work. Our proposed algorithm has no such a constrain. Suppose there are 6 signals impinging on L-shaped array with 9 (2M 1 = 9, then M = 5) sensors at (10, 15 ), (20, 25 ), (30, 35 ), (40, 45 ), (50, 55 ), (60, 65 ), respectively. Fig. 10 shows the DOA estimation at 35 db with 30 independent trials. From Fig. 10, we conclude that our proposed algorithms have better DOA estimation performance with larger source number. elevation angle estimation N=125 N=250 N=500 N= azimuth angle estimation N=125 N=250 N=500 N= Figure 8. 2D-DOA (30, 30 ) estimation performance comparison with different data length. elevation angle estimation N=125 N=250 N=500 N= azimuth angle estimation N=125 N=250 N=500 N= Figure 9. 2D-DOA (40, 40 ) estimation performance comparison with different data length.

11 2D-DOA estimation with L-shaped array azimuth angle/deg elevation angle/deg Figure 10. number. 2D-DOA estimation performance with larger source 5. CONCLUSION An improved method for estimating 2D DOA based on shift invariance property is presented. The algorithm uses the array geometries to construct a matrix and then obtain the required signal subspace via the eigen decomposition of the constructed matrix. Without spectral peak searching, this algorithm works well. Our algorithm has much better 2D-DOA estimation performance than conventional ESPRIT algorithm, and it can identify more DOAs than conventional ESPRIT algorithm. ACKNOWLEDGMENT This work is supported by NSF Grants ( ) and Ph.D. programs foundation of ministry of education of China (No ). The authors are grateful to the anonymous referees for their constructive comments and suggestions in improving the quality of this paper. REFERENCES 1. Liu, H.-X., H. Zhai, L. Li, and C.-H. Liang, Progressive numerical method combined with MON for a fast analysis of large waveguide slot antenna array, Journal of Electromagnetic Waves and Applications, Vol. 20, No. 2, , Zhang, X., X. Gao, and Z. Wang, Blind paralind multiuser detection for smart antenna cdma system over multipath fading

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