2-D DOA Estimation of Multiple Signals Based on Sparse L-shaped Array Zhi Zheng, Yuxuan Yang, Wen-qin Wang, Jiao Yang and Yan Ge School of Communication and Information Engineering, University of Electronic Science and Technology of China, Chengdu, 611731, China; Email: [email protected] Abstract—This paper proposes a novel method for two- dimensional (2-D) direction-of-arrival (DOA) estimation of mul- tiple signals employing a sparse L-shaped array structured by a sparse linear array (SLA), a sparse uniform linear array (SULA) and an auxiliary sensor. In the proposed method, the elevation angles are estimated using the SLA and an improved MUSIC method. The azimuth angles are estimated by two stages. Firstly, the rough azimuth estimates are obtained by a cross- correlation matrix (CCM) of the array received data and the elevation estimates. Secondly, the fine azimuth estimates can be achieved using the SULA and the rough azimuth estimates. The proposed method can achieve automatic pairing of the 2-D DOA estimates. Simulation results show that our approach outperforms the existing methods based on L-shaped array. Index Terms—Sparse L-shaped array, 2-D DOA estimation, cross-correlation matrix (CCM), automatic pairing. 1. Introduction The two-dimensional (2-D) direction-of-arrival (DOA) es- timation of multiple signals has received considerable attention in radar, sonar and wireless communications, and other fields. In the past decades, lots of 2-D DOA estimation algorithms have been proposed with L-shaped array since it can provide a larger array aperture. However, the algorithms can not achieve automatic pairing for 2-D DOA estimation. Aiming at the problem, some methods with automatic pairing, e.g., JSVD [1], CCM-ESPRIT-AP [2] and J-2D [3] was proposed. However, the methods [1]–[3] are not very high in estimation accuracy. In this paper, we present a 2-D DOA estimation method based on a sparse L-shaped array. Our approach can achieve higher estimation accuracy and make the 2-D DOA estimates pairing automatically. Simulation results demonstrate the effectiveness of the proposed method. 2. Data Model Consider an L-shaped array geometry lying in the - plane as shown in Fig. 1. It consists of two subarrays: Subarray 1 and Subarray 2. Subarray 1 includes sensor elements and consists of an SULA and an auxiliary sensor, in which the inter-element spacing of the SULA is and the distance between the auxiliary sensor and reference sensor is . Subarray 2 is a SLA with sensor elements and inter-element spacings . In order to avoid phase ambiguity, the minimum inter-sensor spacing of SLA is set to . Assume that there are uncorrelated narrowband signals with wavelength im- pinging on the L-shaped array from distinct directions , , where and denote, respectively, the elevation and azimuth angles of the th signal. Fig. 1. Sparse L-shaped array for 2-D DOA estimation. The output vectors of subarrays 1 and 2 can be written as (1) where denotes the signal vector. and are the additive noise vectors. is the array response matrix of subarray 1, where (2) is the steering vector of subarray 1 for the th signal and . is the array response matrix of subarray 2, where (3) is the steering vector of subarray 2 for the th signal and . Moreover, define as the received vector of the SULA in subarray 1. Correspond- ingly, is the array response matrix of the SULA, where (4) and denotes the phase difference between two adjacent sensors in the SULA. 3. Proposed Method The covariance matrix of can be expressed as (5) Proceedings of ISAP2016, Okinawa, Japan Copyright ©2016 by IEICE 4C1-4 1014
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2-D DOA Estimation of Multiple Signals Based onSparse L-shaped Array
Zhi Zheng, Yuxuan Yang, Wen-qin Wang, Jiao Yang and Yan GeSchool of Communication and Information Engineering, University of Electronic Science and Technology of China,
Chengdu, 611731, China; Email: [email protected]
Abstract—This paper proposes a novel method for two-dimensional (2-D) direction-of-arrival (DOA) estimation of mul-tiple signals employing a sparse L-shaped array structured bya sparse linear array (SLA), a sparse uniform linear array(SULA) and an auxiliary sensor. In the proposed method, theelevation angles are estimated using the SLA and an improvedMUSIC method. The azimuth angles are estimated by two stages.Firstly, the rough azimuth estimates are obtained by a cross-correlation matrix (CCM) of the array received data and theelevation estimates. Secondly, the fine azimuth estimates can beachieved using the SULA and the rough azimuth estimates. Theproposed method can achieve automatic pairing of the 2-D DOAestimates. Simulation results show that our approach outperformsthe existing methods based on L-shaped array.
Index Terms—Sparse L-shaped array, 2-D DOA estimation,cross-correlation matrix (CCM), automatic pairing.
1. IntroductionThe two-dimensional (2-D) direction-of-arrival (DOA) es-
timation of multiple signals has received considerable attentionin radar, sonar and wireless communications, and other fields.In the past decades, lots of 2-D DOA estimation algorithmshave been proposed with L-shaped array since it can provide alarger array aperture. However, the algorithms can not achieveautomatic pairing for 2-D DOA estimation. Aiming at theproblem, some methods with automatic pairing, e.g., JSVD [1],CCM-ESPRIT-AP [2] and J-2D [3] was proposed. However,the methods [1]–[3] are not very high in estimation accuracy.In this paper, we present a 2-D DOA estimation method basedon a sparse L-shaped array. Our approach can achieve higherestimation accuracy and make the 2-D DOA estimates pairingautomatically. Simulation results demonstrate the effectivenessof the proposed method.
2. Data ModelConsider an L-shaped array geometry lying in the �-� plane
as shown in Fig. 1. It consists of two subarrays: Subarray 1and Subarray 2. Subarray 1 includes � � � sensor elementsand consists of an SULA and an auxiliary sensor, in whichthe inter-element spacing of the SULA is �� � � and thedistance between the auxiliary sensor and reference sensoris � � ���. Subarray 2 is a SLA with � sensor elementsand inter-element spacings �� � ��� �� � �� �� �� � ��.In order to avoid phase ambiguity, the minimum inter-sensorspacing of SLA is set to ���� � ���. Assume that thereare uncorrelated narrowband signals with wavelength � im-pinging on the L-shaped array from distinct directions ��� ,���� � �� �� ��, where �� and �� denote, respectively, theelevation and azimuth angles of the th signal.
Fig. 1. Sparse L-shaped array for 2-D DOA estimation.
The output vectors of subarrays 1 and 2 can be written as
���� � ������ � ������ ���� � ������ � ����� (1)
where ���� denotes the signal vector. ����� and ����� are theadditive noise vectors. �� � �������� ������� � � � � ������� isthe �� � ��� array response matrix of subarray 1, where
������ � ��� ���� � ����� � � � � � ��������� ��
(2)
is the steering vector of subarray 1 for the th signal and� � �� ������. �� � �������� ������� � � � � ������� is the� � array response matrix of subarray 2, where
������ � ��� ����� � � � � � ������
���
���
�� (3)
is the steering vector of subarray 2 for the th signal and � � �� �� ����.
Moreover, define ����� � ������� ������ � � � � �������� as
the received vector of the SULA in subarray 1. Correspond-ingly, ��� � ��������� � � � � �������� is the � � arrayresponse matrix of the SULA, where
������� � ��� ��� � � � � � ������� ��
(4)
and �� � �������� ���� � �� ���� denotes the phasedifference between two adjacent sensors in the SULA.
3. Proposed MethodThe covariance matrix of ���� can be expressed as
��� � �������� ���� � ��� ��� � ��� (5)
Proceedings of ISAP2016, Okinawa, Japan
Copyright ©2016 by IEICE
4C1-4
1014
In practice, the covariance matrix ��� is estimated by
���� ��
�
����
���� � �� ��� (6)
where � denotes the number of snapshots.The eigenvalue decomposition (EVD) of ���� yields
���� � � �� ����
��� (7)
where �� and � are the noise and signal subspace.Furthermore, we can construct the MUSIC spectrum
����� ��
��� �������������
(8)
Using (8), we can estimate the elevation angles by twosteps: 1) Create a rough grid with searching interval �� andfind peaks to get rough estimates of the elevation angles.2) Create a refined grid with searching interval �� around thelocations of the peaks in the first search and perform searchagain to obtain more accurate estimates of elevation angles.
Next, we will estimate azimuth angles. Let ����� �
������� ������ � � � � � ����� , and then we can obtain a cross-
correlation matrix between ���� and �����
���� � ������������� � ��� �
��� (9)
where ��� � ���� � �� ��.
Substituting the elevation estimates �� into ��, we canobtain ���� � ����� � �� ��. Under the AGWN environment,we can estimate ��� by solving the following least square(LS) problem
���� � ��� �������
������� ���� �����
����� (10)
The solution of (10) is���� � ����� ����� �
�.
According to (5) and (7), we have ��� ��� �
� �� , then the signal covariance matrix can be estimated
by
�� � �����
�� �
���� �
� (11)
Therefore, we can obtain the estimation of ��
��� � ����� ����� �
�������
�� � ��
�� �
����
(12)
Furthermore, we can obtain ���� through ���.To solve ambiguity, we firstly obtain rough estimates of
the azimuth angles by
��� � ����
����
������������������
����������������
��
����
�
� (13)
where �������� � �� ���� �� ������
and �������� �
����� �� ���� ����� �� ������
.According to the rough azimuth estimates above, determine
the range where each azimuth angle may be, then fineestimates of the azimuth angles can be obtained by
��� �
��� ���
���������������
�� ��� ��� ����
�������
������� ��� ����� �����
���������������
�� ��� ������ �����
(14)
where
��� � ���
����������������
��������������
�� � �� � � � � (15)
where ������� and ������� denote the first and last ��� elementsof ������ that denotes the th column of ���� .
4. Simulation ResultsWe consider an L-shaped array as shown in Fig. 1, and
assume that � � �, the inter-sensor spacings of Subarray 2are �� � ���� �� � ����� �� � ����� �� � �� �� � �. For J-2D, the inter-element spacings of the SLA along the �-axis are�� � ���� �� � �� �� � ����� �� � �� �� � �. The searchingrange of the elevation angle is ��Æ� ���Æ� with searching inter-vals �� � �Æ and �� � ��Æ. Two uncorrelated sources with 2-DDOAs ���Æ� ��Æ� and ���Æ� ��Æ� are considered. The snapshotsnumber is set to � � ���, and 1000 independent independenttrials are performed. Figs. 2 indicates that the proposed methodachieves better performance than the JSVD [1], CCM-ESPRIT-AP [2] and J-2D [3] methods, especially at low SNR.
0 5 10 15 20 250
0.5
1
1.5
SNR (dB)
RM
SE
(deg
ree)
ProposedJ−2DCCM−ESPRIT−APJSVD
Fig. 2. RMSE of 2-D DOA estimates versus the SNR.
5. ConclusionsA 2-D DOA estimation method based on sparse L-shaped
array has been proposed in this paper. The proposed methoddoes not require an extra pairing process and can achieveautomatic pairing for 2-D DOA estimation. Simulation resultsshow that our approach exhibit better performance than someexisting methods with automatic pairing based on L-shapedarray, such as JSVD, CCM-ESPRIT-AP and J-2D.
References[1] J.-F. Gu and P. Wei, ”Joint SVD of two cross-correlation matrices to
achieve automatic pairing in 2-D angle estimation problems,” IEEEAntennas Wireless Propag. Lett., vol. 6, pp. 553–556, 2007.
[2] J. Gu, P. Wei, and H.-M. Tai, ”DOA estimation using cross-correlationmatrix,” in Proc. IEEE Int. Symp. Phased Array Syst. Technol. (ARRAY),Waltham, USA, pp. 593–598, Oct. 2010.
[3] J.-F. Gu, W.-P. Zhu, and M. Swamy, ”Joint 2-D DOA estimation viasparse L-shaped array,” IEEE Trans. Signal Process., vol. 63, no. 5, pp.1171–1182, Mar. 2015.
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