IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 13, 2014
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Pair-Matching Method by Signal Covariance Matrices for 2D-DOA Estimation Yinsheng Wei, Member, IEEE, and Xiaojiang Guo
Abstract—The two-dimensional direction-of-arrival (2D-DOA) estimation for the L-shaped array is usually divided into two one-dimensional DOA (1D-DOA) estimation for the two linear subarrays, and then the elevation and azimuth angles are obtained from the two correctly matched 1D-DOAs. This letter presents a novel method for pair-matching of the two 1D-DOAs. The key points of the proposed method are: 1) construct two signal covariance matrices corresponding to the two estimated 1D-DOAs; 2) introduce an objective function, defined over the two signal source covariance matrices, to find the matched 1D-DOAs; 3) by introducing a permutation matrix for the signal covariance matrices, the optimal pair-matching is obtained with an efficient method. Simulations under different scenarios, with low signal-to-noise ratio (SNR), snapshot deficiency, and uncorrelated or coherent signal settings, show the robustness of the proposed method. Index Terms—L-shaped array, pair matching, signal covariance matrices, two-dimensional direction-of-arrival (2D-DOA) estimation.
I. INTRODUCTION ECENTLY, two-dimensional direction-of-arrival (2D-DOA) estimation of incident signals impinging on array antennas has received increasing attention in a broad area of signal processing such as radar, sonar, and wireless communication. Compared to many other simple structured arrays, it seems that the L-shaped array provides higher accuracy potential for 2D-DOA estimation [1]. The L-shaped array is usually divided into two linear subarrays, and two one-dimensional DOAs (1D-DOAs) and (functions of the elevation angle and the azimuth angle ) can be separately estimated from each subarray. By this process, it can avoid two-dimensional spectral peak searching and easily decorrelate coherent incident signals [2]. However, it will also result in the pair-matching problem because the two estimated 1D-DOAs and may not be correctly matched. Therefore, conventional process of 2D-DOA estimation for the L-shaped array usually consists of two parts: 1) 1D-DOA estimation from each subarray; and 2) pair-matching of the two estimated 1D-DOAs. In a practical viewpoint, major studies on 2D-DOA estimation for L-shaped arrays have been focused on improving estimation accuracy, reducing the computational cost, and
R
Manuscript received April 15, 2014; revised May 23, 2014; accepted June 12, 2014. Date of publication June 17, 2014; date of current version June 30, 2014. This work was supported by the National Natural Science Foundation Key Project of China under Grant 61032011. The authors are with the School of Electronics and Information Engineering, Harbin Institute of Technology, Harbin 150001, China (e-mail:
[email protected];
[email protected]). Digital Object Identifier 10.1109/LAWP.2014.2331076
dealing with complex processing environments. For instance, Gu et al. [2] employed matrix decomposition technique using cross-correlation matrix to decorrelate coherent signals and to improve estimation accuracy. Tayem and Kwon [3] proposed modified propagator methods to reduce the computational cost. Liang proposed methods to solve estimation failure problem for high elevation angles between 70 and 90 or low elevation angles between 0 and 20 in [4] and estimate 2D-DOA in the presence of mutual coupling in [5]. As for pair-matching of the two estimated 1D-DOAs, methods proposed by [2] and [3] expressed the azimuth angles by using the elevation angles and tried to obtain the corresponding azimuth angles by eigenvalue decomposition. However, the alignment of these eigenvalues still needs to be paired (see, e.g., [6]). Al-Jazzar et al. [7] presented a singular value decomposition (SVD)-based 2D-DOA estimation algorithm, which performs automatic pair-matching by solving a constrained nonlinear functions. Liang et al. [8], [9] proposed to match angles pairs by comparing the difference between the cosine values of the two estimated 1D-DOAs. Wang and Xin [10] searched the elevation and azimuth angle pairs by minimizing a cost function based on the asymptotically orthogonal properties of signal subspace and noise subspace. Liu and Mendel [11] obtained the matched angles based on the array manifolds of the two linear arrays corresponding to the projecting columns of eigenvectors. Even though the above methods in [8]–[11] work well for independent signal sources, their performance would be compromised when coherent signal sources exist, such as in multipath-propagation scenarios. To address such a problem, Palanisamy and Kalyanasundaramb [12] offered closed-form automatically paired 2D-DOA estimation methods for an L-shaped array with acoustic vector sensors. Kikuchi et al. [13] developed an automatic pair-matching method using the cross-correlation matrix methods, however the method is not robust due to the inferior accuracy of the two 1D-DOAs cosine difference using the Toeplitz method and just applies to L-shaped arrays with uniform linear subarrays. This letter proposes an effective method for pair-matching of elevation and azimuth angles under scenarios of uncorrelated and coherent signals for L-shaped arrays with arbitrary linear subarrays. First, the proposed method obtains maximum likelihood estimation (MLE) of incident signals. Then, we construct two covariance matrices using the estimated incident signals: One corresponds to 1D-DOAs impinging on the - or -axis subarray, and the other to 1D-DOAs impinging on both subarrays. Finally, the matched angle pairs are obtained by minimizing a cost function, which is built based on the equivalence relation of the two signal covariance matrices. Simulations show that the proposed method can provide higher detection
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IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 13, 2014
and . This letter will not repeat how to estimate the 1D-DOAs here. A. Two Signal Covariance Matrices Under the AGWN environment, the MLE of array manifold and the complex envelope of incident signals can be obtained by solving the nonlinear least squares problem [15] (3) where is the number of snapshots. If we construct array manifold using and substitute it for in (3), the complex envelope estimation of incident signals can be written as
Fig. 1. L-shaped array with two subarrays along the - and -axis.
probability of successful pair-matching than the CCM method in [13], especially under scenarios of low signal-to-noise ratio (SNR), high correlation, snapshot deficiency, and small angular separation. II. SIGNAL MODEL As shown in Fig. 1, the L-shaped antenna array consists of two linear orthogonal arrays in the -plane. The subarray along the -axis consists of omnidirectional sensors, and the subarray along the -axis consists of omnidirectional sensors. The element placed at the origin is common for referencing purpose. Suppose far-field narrowband signals, with carrier wavelength , impinge on the antenna array. Denote by the th incident signal, its elevation angle, and its azimuth angle, ( , , ). Then, the impinging angles along the -axis subarray along -axis subarray are given via the equations and and . The steering vectors of the subarray along the -axis can be expressed as
(1) is the distance from the th sensor to the origin along where the -axis. Accordingly, the sampled received signals along the -axis are given by
(4) where the superscript “ ” denotes the Moore–Penrose inverse. The signal source covariance matrix can be expressed as (5) is the covariance matrix of the received signals at where the subarray along the -axis
(6) is a full-column-rank matrix and is less than When , we have approximately . Thus, can be written as
(7) In the same way, we can obtain another estimation of the incident signals from the -axis subarray (8) corresponds to , we can If the alignment of construct the second signal source covariance matrix
(2) where is time index, is the array manifold, is the signal vector, and is the vectors of additive white Gaussian noise (AGWN). Similarly, we have the array manifold , sampled received signals , and vectors of AGWN along the -axis.
(9) where
is the last
low of
,
.
B. Alignment of the Two Estimated 1D-DOAs
III. PAIR-MATCHING METHOD
In this section, we present an efficient algorithm to obtain the correctly matched 1D-DOAs and . Define a permutation matrix with the size of . If is the th row and the th column element of , we have , , , . Suppose the
Much research has been done on how to estimate the two 1D-DOAs and for the linear subarrays, such as [2], [3], [13], and [14]. The aim of pair-matching is to find the corresponding relationship of the estimated 1D-DOAs
order of corresponds to , i.e., . We try to find the corresponding angle pairs by rearranging the using the permutation matrix . Then, alignment of corresponding array manifolds of the subarray along the -axis
WEI AND GUO: PAIR-MATCHING METHOD BY SIGNAL COVARIANCE MATRICES FOR 2D-DOA ESTIMATION
can be written as , i.e., can be written as
, we have
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.
(10) Thus, we combine the two signal source-covariance matrices in (7) and (9) via . Clearly, if the signals are incoherent, i.e., , we have . Then, and will be corresponding angle pairs. However, when the signals are coherent, i.e., , is irreversible. We propose to solve by minimizing the following cost function. (11) where PSCM denotes Pair-matching method by Signal Covari” denotes the Frobenius norm. If ance Matrices and “ is the th row of and is the th row of , respectively, we have
Fig. 2. Detection probability versus SNR for two independent signal sources.
(12) It is clear that (11) has less computational cost than (12). C. Algorithm Steps Suppose that the number of incident signals is assumed a priori or has been estimated. Estimate 1D-DOAs and (in this letter, we used the ESPRIT algorithm [14]). a) Calculate the two source-covariance matrices by (5) and (9). b) Obtain the permutation matrix by optimizing (11) and (12). c) Obtain the corresponding angles pairs , by .
Fig. 3. Detection probability versus SNR for two coherent signal sources.
IV. SIMULATIONS In this section, we evaluated the performance of the proposed pair-matching method in 2D-DOA estimation by several simulations. The proposed method was investigated versus SNR in independent sources (Fig. 2) and coherent sources (Fig. 3), versus correlation coefficients (Fig. 4), versus number of snapshots (Fig. 5), and versus angular separation (Fig. 6). Consider an L-shaped array, consisting of two uniform linear orthogonal subarrays with equal number of elements, i.e., . The sensors are separated by a half-wavelength in each uniform linear array (ULA), i.e., element spacing . Results on each of the simulations were analyzed over 500 independent trials. Two signals impinge on the L-shaped array along the angle pairs and with equal amplitude but different phases. The proposed method “PSCM” is compared to the method “CCM” proposed by [13] and the method “Eigenvector” proposed by [11]. The two 1D-DOAs and were estimated independently by using the method proposed in [14]. Detection probability of successful pair-matching means the number of correctly matched trials divides the total number of trials. First, we evaluated the performance of the proposed method versus SNR for independent signal sources (Fig. 2) and coherent signal sources (Fig. 3), and versus correlation coefficients at dB (Fig. 4). The number of snapshots
Fig. 4. Detection probability versus correlation coefficients.
was set at . The detection probability of successful pair-matching is shown in Figs. 2–4. It is observed that the proposed method outperformed CCM in low SNR. The CCM method utilized the cosine difference of the two 1D-DOAs to achieve pair-matching, however accuracy of the cosine difference estimated by Toeplitz matrix was inferior, especially in low SNR, and the two signal sources highly correlated. In addition, the proposed method still performed well under scenarios of coherent signal sources and with the increase of the correlation coefficients. The performance of method “Eigenvector” in [11] was bad under scenarios of coherent signal sources because the
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IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 13, 2014
the method proposed in [11] performed badly for coherent signals and only applied to L-shaped arrays with uniform linear subarrays. V. CONCLUSION
Fig. 5. Detection probability versus number of snapshots.
In this letter, we have proposed a novel method for pairmatching of elevation and azimuth angles in 2D-DOA estimation with the L-shaped array. The proposed method utilizes two estimated signal source covariance matrices constructed by the received signals and the two estimated 1D-DOAs to exploit the characteristics of the corresponding angle pairs, and then obtains the angle pairs by minimizing a cost function built by the two covariance matrices. Results on extensive simulations showed the robustness of the proposed method, under a wide range of scenarios, from low SNR, high correlation, and snapshot deficiency to small angular separation. Furthermore, the proposed method is also applicable to array geometries with two linear subarrays, e.g., cross array and T-shaped array. The linear subarrays along - and -axes can even be unequal and sparse line arrays, while methods proposed by [11] and [13] just apply to subarrays of equal element space. REFERENCES
Fig. 6. Detection probability versus angular separation.
projecting eigenvectors used for pair-matching were bad for coherent signal sources. Second, we evaluated the performance of the proposed method versus the number of snapshots for independent signal sources. The SNR was set at 5 dB, and the number of snapshots varied from 20 to 400. The detection probability of successful pair-matching is shown in Fig. 5. Again, the proposed methods PSCM show better performance than CCM. Furthermore, we evaluated the performance of the proposed method versus angular separation. The SNR was set at 5 dB, and the number of snapshots was set at . One signal was still at , and the other changed into . varied from 2 to 10 . The detection probability of successful pair-matching is shown in Fig. 6. It is clearl that the proposed PSCM performed encouragingly in small angular separation. In summary, the proposed method outperformed the CCM method proposed by [13], under scenarios of low SNR, high correlation, snapshot deficiency, and small angular separation. Because the proposed method takes full advantage of all received signal data of the array and 1D-DOAs estimated, each step of the pair-matching method has high accuracy (such as the MLE of incident signals). However, the CCM method utilizes Toeplitz matrix in order to obtain the cosine difference of the two 1D-DOAs, which has poor accuracy. Moreover, performance of the proposed method was similar to the Eigenvector method in [11] for independent signal sources. However,
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