The outline of the paper is as follow. In Section 2, the DOA estimation problem is nearly formulated. The new algorithm, called JSDOA, is proposed in Section 3. In section 4, the validity of the proposed algorithm is proved by a number of simulations. Finally, conclusions are presented in Section 5.2.?Problem Formulation2.1. DOA Estimation ProblemConsider a linear array consisting of M identical sensors and receiving signals from K narrowband signals s1(t), s2(t), , sK(t), which arrive at the array from directions 1, 2, K with respect to the line of array. The received signal ym(t) at the mth sensor can be written as:ym(t)=��k=1Ka(�ȡ�k)sk(t)+nm(t)(1)where Inhibitors,Modulators,Libraries a(�ȡ�k)k=1K denote steering vectors, nm(t) (m = 1, 2 , M) stand for the additive noise.
Let y(t) = [y1(t), y2(t), , yM(t)]T, n(t) = [n1(t), , nM(t)], Equation (1) can be written as:y(t)=A(�ȡ�)s(t)+n(t)(2)where the manifold matrix A () consists of the steering Inhibitors,Modulators,Libraries vectors a(�ȡ�k)k=1K:A(��)=[a(�ȡ�1),a(�ȡ�2),?,a(�ȡ�K)]Therefore, DOA estimation is to find K and k from T snapshots y(t)t=t1tT.2.2. Joint-Sparse Recovery for DOA Estimation ProblemBecause sources are sparse in space, DOA estimation with sensor arrays can be expressed as a joint-sparse recovery problem. Let ? denote the set of possible locations, ��nn=1N denotes a grid that covers ?. We assume that the grid is fine enough such that the true location parameters of the existing sources lie on the grid.
Let:x(t)=[x1(t), x2(t), ?,xN(t)]TThen the received signal ym(t) at the mth sensor can be written as:ym(t)=��n=1Na(��n)xn(t)+n(t)(3)where the nth element xn(t) of x(t) is nonzero only if ��n = k, k [1,2, , K] and Inhibitors,Modulators,Libraries in that case xn(t) = sk(t):Let �� = [a(��1), a(��2), , a(��N)], (3) can be expressed as:y(t)=��x(t)+n(t)(4)when the number of snapshots is denoted as T, Equation (4) can be written as:Y=��X+N(5)where Y = [y(t1), y(t2), , y(tT)], X = [x(t1), x(t2), , x(tT)]. As we know, X is row-sparse and only K rows have nonzero elements, so DOA estimation can be obtained by a joint-sparse recovery problem, which is also called a multiple measurement vectors (MMV) problem. Using lp,q norm to express joint sparsity, the MMV problem can be converted to lp,q norm minimization:{min��X��p,qs.t.Y=��X+N(6)where Inhibitors,Modulators,Libraries p and q are non-negative, and ��X��p,q is defined as:��X��p,q=��i=1M(��X(i,:)��p)q(7)Concretely, Eldar and Mishali [15] use p = 2, q = 1.
Chen and Huo [16] study for p = 1, q = 1. The above algorithms do not perform very well for ��X�� 2,1 and ��X�� 1,1 can��t Carfilzomib sufficiently reflect joint sparsity. Considering ��X�� 2,0 can reflect joint sparity sufficiently, we minimize ��X�� 2,0 norm to solve the MMV problem. However, ��X�� 2,0 norm minimization can hardly be solved directly. selleck compound Therefore, in this paper we approximate ��X�� 2,0 norm by an arctan function and estimate DOA by solving an approximate ��X�� 2,0 norm minimization problem.3.