Abstract
Twodimensional (2D) directionofarrival (DOA) estimation has played an important role in array signal processing. In this article, we address a problem of bind 2DDOA estimation with Lshaped array. This article links the 2DDOA estimation problem to the trilinear model. To exploit this link, we derive a trilinear decompositionbased 2DDOA estimation algorithm in Lshaped array. Without spectral peak searching and pairing, the proposed algorithm employs well. Moreover, our algorithm has much better 2DDOA estimation performance than the estimation of signal parameters via rotational invariance technique algorithms and propagator method. Simulation results illustrate validity of the algorithm.
Keywords:
array antennas; directionofarrival estimation; Lshaped array1. Introduction
Antenna arrays have been used in many fields, such as radar, sonar, communications, seismic data processing, and so on. The directionofarrival (DOA) estimation of signals impinging on an array of sensors is a fundamental problem in array processing, and many DOA estimation methods have been proposed for its solution [110]. Uniform linear arrays for estimation of wave arrival have extensively been studied. Compared with uniform linear array, Lshaped array can identify twodimensional (2D) DOA. 2DDOA estimation with Lshaped array has been received considerable attention in the field of array signal processing [513], and it contains estimation of signal parameters via rotational invariance techniques (ESPRIT) algorithms [57], multiple signal classification (MUSIC) algorithm [8], matrix pencil methods [9,10], propagator methods [1113], and highorder cumulant method [14].
Highorder cumulant method requires the signal statistical properties, and it needs a heavy computation load. MUSIC algorithm is based on the noise subspace, and has a good DOA estimation performance. However, MUSIC requires spectral peak searching, which is computationally expensive. Propagator method has low complexity, but its 2DDOA estimation performance is less than ESPRIT algorithm. ESPRIT produces signal parameter estimates directly in terms of (generalized) eigenvalues, and the primary computational advantage of ESPRIT is that it eliminates the search procedure inherent. Authors of [5,6] used ESPRIT method for 2DDOA estimation with Lshaped array, and Zhang et al. [7] proposed the improved ESPRIT algorithm for 2DDOA estimation, which had better 2DDOA estimation performance than that of [5,6]. The algorithms in [57] require an extra paring within 2DDOA estimation. Paring usually fails to work in the condition of low signaltonoise ratio (SNR) and the large number of sources.
This study links 2DDOA estimation problem of Lshaped array to trilinear model, and derives a novel blind 2DDOA algorithm whose performance is better DOA estimation than ESPRIT algorithms and propagator method. Furthermore, our algorithm employs well without spectral peak searching and pairing. Bro et al. [15] proposed a 2DDOA algorithm for uniform squares array using trilinear decomposition. There are some differences between this study and that of [15] in some aspects. First, Bro et al. [15] proposed a 2DDOA algorithm for uniform squares array, while this study is to estimate 2DDOA for Lshaped array. Second, the received signal of uniform squares array can be modeled directly with trilinear model, and then that of [15] proposed joint azimuthelevation estimation using trilinear decomposition in uniform squares array. This article is to estimate 2DDOA estimation in Lshaped array, and the received signal of Lshaped array cannot be modeled directly with trilinear model. We use the cross correlation of received signal for constructing the trilinear model.
The rest of the article is structured as follows. Section 2 develops a data model. Section 3 deals with algorithmic issues. Section 4 presents simulation results, and Section 5 provides conclusions.
2. Data model
We consider an Lshaped array with 2M  1 sensors at different locations as shown in Figure 1. A uniform linear array containing M elements is located in yaxis, and the other uniform linear array containing M elements is located in xaxis. We suppose that there are K sources impinge on the Lshaped array with (θ_{k},ϕ_{k}), k = 1,2,...,K, where θ_{k},ϕ_{k }are the elevation and the azimuth angles of the kth source, respectively. The received signal of M elements in xaxis is
Figure 1. The structure of Lshaped array.
where
where α_{k }= 2πd cosθ_{k }sin ϕ_{k }/ λ (k = 1, ..., K), d is the element spacing, and λ is the wavelength. d ≤ λ/2 is required in the array.
The received signal of M elements in yaxis is denoted as
where n_{y}(t) is an M × 1 Gaussian white noise vector of zeros mean and covariance matrix σ^{2}I_{M}, and
where β_{k }= 2πd sinθ_{k }sin ϕ_{k }/ λ, k = 1, ..., K. A_{x }and A_{y }are Vandermonde matrices.
where x_{1 }and x_{M }are first and last rows of x(t), respectively. y_{1 }and y_{M }are first and last rows of the y(t), respectively. a_{x1 }and a_{xM }are first and last rows of the matrix A_{x}, respectively. a_{y1 }and a_{yM }are first and last rows of the matrix A_{y}, respectively.
According to Equations 58, we construct the following matrices
where
We define the matrix Ω as
Equations 912 can be denoted by
where D_{l}(.) is to extract the lth row of its matrix and construct a diagonal matrix out of it. Now, the noiseless signal in (14) can be denoted as a trilinear model [1620], which is shown as
where a_{m, k }is the (m,k) element of the matrix A_{x1}, h_{l, k }stands for the (l,k) element of the matrix Ω, b_{n, k }represents the (n,k) element of the matrix
3. Blind 2D DOA estimation
In this section, we utilize the trilinear decomposition for blind 2DDOA estimation in Lshaped array, where the received signal has been reconstructed with trilinear model. We use trilinear decomposition for obtaining the direction matrices
3.1 Trilinear decomposition
Since trilinear alternating LS (TALS) algorithm is a common data detection method for trilinear model [19], it can be discussed in detail as follows. According to (14), we construct the following matrix in this form
where ⊙ stands for KhatriRao product. LS fitting is given by
LS update for A_{y1 }can be shown as
Similarly, from the second way of slicing, we have
and the LS update for Ω is
where
and the LS update for A_{x1 }is
where
According to (20), (22), and (24), the matrices A_{y1}, Ω, and A_{x1 }are continually updated with conditional LSs, respectively, until convergence. TALS algorithm has several advantages: it is quite easy to implement, guarantee to converge, and comparatively simple to be expanded to the higherorder data. In this article, we use the complexvalued parallel factor analysis model (COMFAC) algorithm [17] for trilinear decomposition. COMFAC algorithm is essentially a fast implementation of TALS, and it can speed up the LS fitting.
For the blind 2DDOA estimation algorithm that we have investigated, trilinear decomposition has been adopted for obtaining the estimated matrices, and then 2DDOA estimation is correspondingly shown.
3.2 Identifiablity
In this section, we discuss the sufficient and necessary conditions for uniqueness of trilinear decomposition.
Theorem 1 [19]: Considering
where k_{Ω }is the kth rank [18] of the matrix Ω, the matrices A_{y1}, Ω, and A_{x1 }are unique up to permutation and scaling of columns.
When the matrix
If K ≥ 4, then min(4,K) = 4 and hence, the identifiability is K ≤ M. If K ≤ 4, then min(4,K) = K and hence, the identifiability in practice becomes K ≤ 2M  4.
For the received noisy signal, we use trilinear decomposition for obtaining the estimated matrices
where ∏ is a permutation matrix, Δ_{1}, Δ_{2}, Δ_{3 }are diagonal scaling matrices satisfying Δ_{1}, Δ_{2}, Δ_{3 }= I_{K}, V_{1}, V_{2}, and V_{3 }are estimation error matrices. Within trilinear decomposition, permutation and scale ambiguities are inherent. Notably, the scale ambiguity can be resolved by means of normalization.
3.3 DOA estimation for Lshaped array
The direction matrices
and then the following vector is obtained by
where angle(.) is get the phase angles, for each element of complex array. Thereafter, LS principle is adopted for estimating sin ϕ_{k }cos θ_{k}. The estimated array steer vector
w = [w_{0},w_{x}]^{T}, in which w_{x }is the estimated value of sin ϕ_{k }cos θ_{k}, and w_{0 }is the other estimation parameter. The LS solution to w is
Similarly,
Up to now, as deducted above, we have proposed the trilinear decompositionbased 2DDOA estimation for Lshaped array in this section. The algorithmic steps in detail are shown as follows:
Step 1. We collect L snapshots to construct the matrices C_{i}, i = 1,2,...,4.
Step 2. According to the symmetry characteristics of trilinear model, we obtain Y_{m}, m = 1,...,M  1, and Z_{n }n = 1, ...,M1.
Step 3. Initialize randomly for the matrices A_{y1}, Ω and A_{x1}.
Step 4. LS update for the source matrix A_{y1 }according to (20).
Step 5. LS update for the source matrix Ω according to (22).
Step 6. LS update for the channel matrix A_{x1 }according to (24).
Step 7. Repeat Steps 46 until convergence.
Step 8. Estimate 2DDOA according to the estimated matrices and LSs principle.
It is noted that our algorithm can obtain automatically paired 2DDOA estimation. In our algorithm, we employ trilinear decomposition for obtaining the estimated direction matrices
It is also noted that for the coherent source, spatial smoothing technique is used for attaining fullrank source matrix, followed by our algorithm to estimate coherent DOA. However, the spatial smoothing decreases the array aperture and the identifiable number of targets.
3.4 Complexity analysis and CramerRao lower bounds (CRLB)
In contrast to ESPRIT algorithms in [6,7], our algorithm has a heavy computational load. For our algorithm, the complexity of each TALS iteration is O(3K^{3 }+ 12(M  1)^{2}K) [16], only a few iterations of this algorithm with COMFAC are usually required to achieve convergence. The total complexity of our algorithm is O{4L(M  1)^{2 }+ n(3K^{3 }+ 12(M  1)^{2}K)}, where L is the number of snapshots, and n is the number of iterations. The algorithm in [6] requires O(4L(M  1)^{2 }+ 36(M  1)^{3 }+ 2K^{3}), and the ESPRIT algorithm in [7] needs O(4L(M  1)^{2 }+ 80(M  1)^{3 }+ 2K^{3}).
We define the matrix A
which is also denoted by A = [a_{1 }a_{2 }... a_{K}], where a_{K }is the kth column of the matrix A. According to [21], we derive the CRLB for angle estimation in Lshaped array,
where ⊕ stands for Hadamard product.
4. Simulation results
We present Monte Carlo simulations that are to assess 2DDOA estimation performance of the proposed algorithm. The number of Monte Carlo trials is 1000. There are two signals impinging on Lshaped array with (30°, 30°) and (40°, 40°), respectively. We consider the Lshaped array with 2M  1 sensors, and a half wavelength of the incoming signals is used for the spacing between the adjacent elements in each uniform linear array. L = 300 snapshots are used in the simulations.
Let
We first investigate the convergence performance of our proposed algorithm in this simulation. The sum of squared residuals (SSR) in the trilinear fitting is defined as
where
Figure 2. Algorithmic convergence performance.
Figure 3 shows 2DDOA estimation of the proposed algorithm at SNR = 15 dB, and Figure 4 shows 2DDOA estimation of our algorithm at SNR = 24 dB. The Lshaped array with 13 antennas is used in Figures 3 and 4. From Figures 3 and 4, we find that our proposed algorithm employs well.
Figure 3. 2DDOA estimation performance at SNR = 15 dB.
Figure 4. 2DDOA estimation performance at SNR = 24 dB.
We compare our algorithm against ESPRIT algorithms [6,7], propagator method, and CRLB. Their DOA estimation performance comparison is shown in Figure 5, where the Lshaped array with 13 antennas is used. From Figure 5, we find that our algorithm has much better DOA estimation performance than ESPRIT algorithms and propagator method.
Figure 5. Angle estimation performance comparison.
Figure 6 shows 2DDOA estimation performance of our algorithm with different array configurations. It is seen from Figure 6 that 2DDOA estimation performance of our algorithm is improved with the number of antennas increasing. When the number of antennas increases, our algorithm has higher received diversity.
Figure 6. Angle estimation performance with different array.
5. Conclusion
This article links the Lshaped array 2DDOA estimation problem to the trilinear model. To exploit this link, we have proposed trilinear decompositionbased DOA estimation in Lshaped array. Without spectral peak searching and pairing, the proposed algorithm employs well. Furthermore, the proposed algorithm has much better 2DDOA estimation performance than conventional ESPRIT algorithms and propagator method.
Notations
Bold symbols denote matrices or vectors. Operators (.)*, (.)^{T}, (.)^{H}, (.)^{1}, (.)^{+ }, and ._{F }denote the complex conjugation, transpose, conjugatetranspose, inverse, pseudoinverse, and Forbenius norm, respectively. I_{P }denotes a P × P identity matrix. 1_{N×1 }is an N × 1 vector of ones. diag(v) stands for diagonal matrix whose diagonal is the vector v. ⊙ and ⊕ stand for KhatriRao and Hadamard product, respectively. E{.} denotes statistical expectation.
Competing interests
The authors declare that they have no competing interests.
Acknowledgements
This study was supported by the China NSF Grants (60801052), Aeronautical Science Foundation of China (2009ZC52036), Nanjing University of Aeronautics & Astronautics Research Funding (NS2010114, NP2011036) and the Graduate Innovative Base Open Funding of Nanjing University of Aeronautics & Astronautics.
References

X Zhang, Theory and application of array signal processing (National Defense Industry Press, Beijing, 2010)

X Zhang, D Xu, Improved coherent DOA estimation algorithm for uniform linear arrays. Int J Electron 96(2), 213–222 (2009). Publisher Full Text

H Chen, B Huang, Y Wang, Directionofarrival estimation based on direct data domain (D3) method. J Syst Eng Electron 20(3), 512–518 (2009)

X Zhang, X Gao, D Xu, Multiinvariance ESPRITbased blind DOA estimation for MCCDMA with an antenna array. IEEE Trans Veh Technol 58(8), 4686–4690 (2009)

Y Dong, Y Wu, G Liao, A novel method for estimating 2D DOA. J Xidian Univ 30(5), 369–373 (2003)

J Chen, S Wang, X Wei, New method for estimating twodimensional direction of arrival based on Lshape array. J Jilin Univ (Eng Technol Edition) 36(4), 590–593 (2006)

X Zhang, X Gao, W Chen, Improved blind 2ddirection of arrival estimation with Lshaped array using shift invariance property. J Electromag Waves Appl 23(5), 593–606 (2009). Publisher Full Text

Y Hua, A pencilMUSIC algorithm for finding twodimensional angles and polarizations using crossed dipoles. IEEE Trans Antennas Propag 41(3), 370–376 (1993). Publisher Full Text

R'ıo JE Fern'andez del, MF C'atedraP'erez, The matrix pencil method for twodimensional direction of arrival estimation employing an Lshaped array. IEEE Trans Antennas Propag 45(11), 1693–1694 (1997). Publisher Full Text

P Krekel, E Deprettre, A two dimensional version of matrix pencil method to solve the DOA problem. Proceedings of European Conference on Circuit Theory and Design, 435–439 (1989)

N Tayem, HM Kwon, Lshape2D arrival angle estimation with propagator method. IEEE Trans Antennas Propag 53(5), 1622–1630 (2005)

P Li, B Yu, J Sun, A new method for twodimensional array signal processing in unknown noise environments. Signal Process 47(3), 319–327 (1995). Publisher Full Text

Y Wu, G Liao, HC So, A fast algorithm for 2D directionofarrival estimation. Signal Process 83(8), 1827–1831 (2003). Publisher Full Text

B Tang, X Xiao, T Shi, A novel method for estimating spatial 2D direction of arrival. Acta Electonica Sinica 27(3), 104–106 (1999)

R Bro, ND Sidiropoulos, GB Giannakis, Optimal joint azimuthelevation and signalarray response estimation using parallel factor analysis. Proceedings of 32nd Asilomar Conference Signals, System, and Computer, 1594–1598 (1998)

SA Vorobyov, Y Rong, ND Sidiropoulos, Robust iterative fitting of multilinear models. IEEE Trans Signal Process 53(8), 2678–2689 (2005)

R Bro, ND Sidiropoulos, GB Giannakis, A fast least squares algorithm for separating trilinear mixtures. Proceedings of International Workshop ICA and BSS (France, 1999), pp. 289–294

ND Sidiropoulos, GB Giannakis, R Bro, Blind PARAFAC receivers for DSCDMA systems. IEEE Trans Signal Process 48(3), 810–823 (2000). Publisher Full Text

ND Sidiropoulos, X Liu, Identifiability results for blind beamforming in incoherent multipath with small delay spread. IEEE Trans Signal Process 49(1), 228–236 (2001). Publisher Full Text

X Zhang, G Feng, J Yu, Anglefrequency estimation using trilinear decomposition of the oversampled output. Wireless Pers Commun 51, 365–373 (2009). Publisher Full Text

P Stoica, A Nehorai, Performance study of conditional and unconditional directionofarrival estimation. IEEE Trans Signal Process 38, 1783–1795 (1990). Publisher Full Text