Wireless Pers Commun (2015) 82:2363–2375 DOI 10.1007/s11277-015-2352-z DOA Estimation of Noncircular Signal Based on Sparse Representation Xuemin Yang · Guangjun Li · Zhi Zheng Published online: 3 February 2015 © Springer Science+Business Media New York 2015 Abstract In this paper, we propose a novel method employing subspace fitting principle for DOA estimation of noncircular signal based on the sparse representation technology. The proposed method combines the signal information contained in both the covariance and elliptic covariance matrix of the received data matrix. We use the eigenvalue decomposition of the extended covariance to obtain the signal eigenvectors, and represent the steering vector on overcomplete basis subject to sparse constraint in subspace fitting method. After casting multiple dimensional optimization problem of the classical subspace fitting method as a sparse reconstruction problem, we use L1-norm penalty for sparsity, and optimization by the second order cone programming framework to obtain the DOA estimates. The proposed method can be used in arbitrary array configuration. Compared with the existing algorithms, the simulation results show that the proposed method has better performance in low SNR. Compared with L1-SVD, the proposed method also own better resolution probability. Keywords DOA estimation · Noncircular signal · Sparse representation · Subspace fitting 1 Introduction In recent decades, the problem of estimating the direction-of-arrival (DOA) with high reso- lution has been widely concerned in military, sonar, radar and wireless communication. The traditional DOA estimators based on subspace method, like ROOT-MUSIC [1–3], ESPRIT X. Yang (B) · G. Li · Z. Zheng School of Communication and Information Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China e-mail: [email protected] G. Li e-mail: [email protected] Z. Zheng e-mail: [email protected] 123 2364 X. Yang et al. [4,5], and subspace fitting (SF) method [6], want to obtain high resolution probability, which all needs large quantity of measurements or high signal-to-noise ratio (SNR). Recently, sparse representation technique of signal has been applied in many areas, such as wireless channel estimation, cognitive radio and image processing, which also provides a new thought for DOA estimation. Several DOA estimation techniques based on sparse representation have been published in the literature [7–11]. In [7], a method called L1- SVD, which can effectively estimate DOA with single measurement. By using singular value decomposition (SVD) of received data matrix, it also works in multiple measurements case. Furthermore, the author presents an adaptive grid refinement method which is able to reduce the influence of the limitation of grid size. In [8], a mixed L2,0-norm-based joint sparse approximation technique is introduced into DOA estimation by Hyder and Mahata. In their work, the L0-norm constraint is approached by a set of convex functions, and a method called JLZA-DOA is proposed. Although JLZA-DOA method meets performance, global convergence is not guaranteed. To reduce the computational cost of the above methods, a low complexity DOA estimator based on sparse representation of array covariance matrix is proposed in [9]. The method combines Khatri–Rao product and signal sparse representation to estimate DOA by reconstructing single measurement vector, which means the computational complexity of the method is lower than L1-SVD. As the methods in [7–9] need to set the hyperparameters, Stoica et al. [10], present the sparse iterative covariance-based estimation (SPICE) approach, which can be obtained by the minimization of covariance matrix fitting criterion and can be used in both single and multiple measurements cases. It does not require selection of any hyperparameters, and it has global convergence properties. However, the algorithm based on power is limited by the grid size. In [11], Abeida et al. propose a series of user parameter-free iterative sparse asymptotic minimum variance (SAMV) approaches for array processing applications based on the asymptotically minimum variance (AMV) criterion. These approaches can effectively reduce the limitation of the grid size, and improve the performance of algorithm based on power. When the measurements are enough, SAMV is proved to coincide with the maximum likelihood (ML) estimator. Compared with the methods based on subspace, these approaches above mentioned own better performance especially in low SNR and limited measurements. Most of the researchers studying the DOA estimators are based on the assumption of complex circular signal. But, in practical communication channel there are a lot of complex noncircular signals, such as signal of BPSK, MSK, GMSK, PAM and UQPSK. Although the methods of noncircular signal take more computational cost for DOA estimation, as NC-MUSIC-like [12–16],and NC-ESPRIT [17], they can get better performance of resolu- tion probability and noise-robust. In [18], the authors present a method called noncircular covariance matrix sparse representation (NC-CMSR). NC-CMSR realizes DOA estimation by jointly representing the covariance and elliptic covariance matrices of the array output on overcomplete dictionaries subject to sparsity constraint. The noncircular signal DOA estimators can extend the array aperture, and improve the DOA estimation performance in assumption of complex circular. Thus, in this paper we study the noncircular signal DOA estimation. As the classical SF methods need multiple dimensional optimizations with a large com- putational complexity, we use the sparse representation technique to solve SF method, and present a new noncircular signal DOA estimator, called NC-SFSR. This method represents the steering vectors on overcomplete dictionaries. And the problem of multiple dimensional optimizations is casted as a sparse reconstruction problem. We use the perturbance between true signal eigenvectors and estimated signal eigenvectors to estimate the hyperparameters. Then, the sparse reconstruction problem can be optimized by SOCP framework, which can 123 DOA Estimation of Noncircular Signal 2365 reduce the computational complexity of the classical SF method. Later, we compare the existing algorithms with NC-SFSR, and present the simulation results. , , [•], [•],v (•), (•), , , , , In this paper, the notations *, T H E D ec tr 0 1 2 F Im denote conjugate, transpose, conjugate transpose, expectation, variance, vectorization, trace, L0-norm, L1-norm, L2-norm, Frobenius-norm, and m×m identity matrix, respectively. 2 Data Model We consider K far-field stationary and uncorrelated narrowband noncircular signals impinge on the array consisting of M sensors from direction angles θk, k = 1, 2, ..., K , which are corrupted by additive circular complex Gaussian white noise. The signals are irrelevant to the noise. The array output vector at time t can be expressed as: K x(t) = a(θk)sk (t) + n(t) k=1 = As(t) + n(t), t = 1, 2, ..., N (1) j2πd sin θ /λ j2πd − sin θ /λ T where x(t) is the receiving data of the array, a(θk) = 1, e 1 k , ..., e M 1 k is the M×1arraysteeringvector,dm is the distance between the mth sensor and the first sensor, λ is the wavelength of the carrier. s (t) is the kth zero-mean noncircular signal, k ◦ ◦ θk is the kth direction of the arrival, θk ∈[−90 , 90 ], n(t) is Gaussian white noise with 2 zero-mean and covariance matrix σ IM . N is the number of snapshots. Equation (1) can be written in matrix form X = AS + N (2) where A = [a(θ1), a(θ2), ..., a(θK )] (3) A is the M × K array manifold matrix, S = [s(1), s(2), ..., s(N)] (4) S is the K × N noncircular signal matrix, T s(t) = [s1(t), s2(t), ..., sK (t)] (5) s(t) is the K × 1 noncircular signal vector, N = [n(1), n(2), ..., n(N)] (6) N is the M × N noise matrix. For arbitrary noncircular signal, we have E SST = ρE SSH = 0 (7) where ρ = (ρ ,ρ , ..., ρ ), ≤ ρ ≤ diag 1 2 K 0 k 1(8) ◦ ◦ jϕ1 jϕ2 jϕK = diag e , e , ..., e ,ϕk ∈ 0 , 180 (9) 123 2366 X. Yang et al. and ρk is the kth noncircular rate of the noncircular signal, ϕ is the additional phase in the communication channel, called noncircular phase. Some special modulated signals, like AM, MASK and BPSK, whose noncircular rate equal 1, are the maximum noncircular rate signal. In this work, we pay attention to the maximum noncircular rate signal. According to the noncircularity of the received signal, we can combine the received data matrix and its complex conjugate counterpart into a matrix X Y = (10) X∗ and the extended data covariance matrix can be expressed as A A H R = E YYH = R + σ 2I Y A∗∗ s A∗∗ 2M H 2 B(θ; )Rs B(θ; ) + σ I2M (11) where H Rs = E SS (12) A(θ) B(θ; ) = (13) A(θ)∗∗ As SF method is based on the eigenvalue structure of covariance matrix, here the eigen- decomposition of RY is given by = H + H RY Es sEs En nEn (14) where the columns of Es are the eigenvectors corresponding to the K largest eigenvalues which constitute the diagonal matrix s , and the columns of En are the eigenvectors corre- sponding to the 2M − K smallest eigenvalues which constitute the diagonal matrix n. Es and B span the subspace, which are orthogonal to the noise subspace spanned by the matrix En. Then, there exists a nonsingular K × K matrix T satisfying the following relation Es = B(θ; )T (15) The estimation of the eigenvectors of the extended covariance is usually obtained by eigen- decomposing the sample covariance matrix as N 1 Rˆ = y(t)yH (t) Y N t=1 = ˆ ˆ ˆ H + ˆ ˆ ˆ H Es sEs En nEn (16) ˆ Because of the existence of noise, there are some perturbations between Es and Es ,andwe can get the following expression ˆ Es = B(θ; )T + ε (17) where ε is the perturbation matrix.