Consider a linear array consisting of M identical sensors and receiving signals from K narrowband signals s₁(t), s₂(t), ⋯, s_K(t), which arrive at the array from directions θ̄₁, θ̄₂, ⋯ θ̄_K with respect to the line of array. The received signal y_m(t) at the m^th sensor can be written as: (1) y m ( t ) = ∑ k = 1 K a ( θ ¯ k ) s k ( t ) + n m ( t )where { a ( θ ¯ k ) } k = 1 K denote steering vectors, n_m(t) (m = 1, 2 ⋯, M) stand for the additive noise.

Let y(t) = [y₁(t), y₂(t), ⋯, y_M(t)]^T, n(t) = [n₁(t), ⋯, n_M(t)], Equation (1) can be written as: (2) y ( t ) = A ( θ ¯ ) s ( t ) + n ( t )where the manifold matrix A (θ̄) consists of the steering vectors { a ( θ ¯ k ) } k = 1 K: A ( θ ) = [ a ( θ ¯ 1 ) , a ( θ ¯ 2 ) , ⋯ , a ( θ ¯ K ) ]

Therefore, DOA estimation is to find K and θ̄_k from T snapshots { y ( t ) } t = t 1 t T.

Because sources are sparse in space, DOA estimation with sensor arrays can be expressed as a joint-sparse recovery problem. Let Ω denote the set of possible locations, { θ n } n = 1 N denotes a grid that covers Ω. 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 ) = [ x 1 ( t ) ,   x 2 ( t ) ,   ⋯ , x N ( t ) ] T

Then the received signal y_m(t) at the m^th sensor can be written as: (3) y m ( t ) = ∑ n = 1 N a ( θ n ) x n ( t ) + n ( t )where the n^th element x_n(t) of x(t) is nonzero only if θ_n = θ̄_k, k ∈ [1,2, ⋯, K] and in that case x_n(t) = s_k(t):

Let Φ = [a(θ₁), a(θ₂), ⋯, a(θ_N)], (3) can be expressed as: (4) y ( t ) = Φ x ( t ) + n ( t )when the number of snapshots is denoted as T, Equation (4) can be written as: (5) Y = Φ X + Nwhere Y = [y(t₁), y(t₂), ⋯, y(t_T)], X = [x(t₁), x(t₂), ⋯, x(t_T)]. 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 l_p,q norm to express joint sparsity, the MMV problem can be converted to l_p,q norm minimization: (6) { min ‖ X ‖ p , q s . t . Y = Φ X + Nwhere p and q are non-negative, and ‖X‖_p,q is defined as: (7) ‖ X ‖ p , q = ∑ i = 1 M ( ‖ X ( i , : ) ‖ p ) q

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 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. 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.

Let ξ_i = ‖X(i,:)‖₂, i = 1, 2, ⋯, M, we have: (8) ‖ X ‖ 2 , 0 = ∑ i = 1 M ( ‖ X ( i , : ) ‖ 2 ) 0 = ‖ ξ ‖ 0where ξ = (ξ₁, ⋯, ξ_M). Considering the following arctan function: (9) f δ ( s ) = 2 π arc   tan ( s 2 2 δ 2 )where δ is a positive parameter and s is a variable parameter. Then f_δ(s) has the following property: (10) lim δ → 0 f δ ( s ) = { 1 s ≠ 0 0 s = 0

Let F δ ( ξ ) = ∑ i = 1 N f δ ( ξ i )

From Equation (10), we have: (11) lim δ → 0 F δ ( ξ ) = ‖ ξ ‖ 0

From Equations (8) and (10), then: lim δ → 0 F δ ( X ) = ‖ X ‖ 2 , 0

So DOA can be obtained by solving the approximate ‖X‖_2,0 norm minimization: (12) { min F δ ( X ) s . t . Y = Φ X + Nwhere δ is a small positive constant. Because noise is unknown, we synchronously hope to minimize F_δ(X) and ‖ Φ X − Y ‖ F 2. Then Equation (12) is converted into a multiple objective optimization: (13) { min X F δ ( X ) min X ‖ Φ X − Y ‖ F 2

Using the linear weighting method, Equation (13) can be written as: (14) min x       L δ , λ ( X ) = F δ ( X ) + λ ‖ Φ X − Y ‖ F 2where ‖•‖_F denotes the Frobenious norm and the parameter λ will be discussed in Section 3.2. When the parameter δ is very small, the objective function F_δ(X) is highly unsmoothed and contains a lot of local minimization, so its global minimization is not easy. On the other hand, if the parameter δ is larger, the objective function F_δ(X) is smoother and contains less local minimization. In order to obtain global minimization, we select a decreasing sequence for δ, denoted as: δ = [ δ 1 , δ 2 , ⋯ , δ J ] ,     δ j + 1 < δ jwhere δ₁ is a relatively large value, and δ_J is a small value. For δ = δ_J−1, the solution of Equation (14) is denoted as x_δj–1, and x_δj–1 is used as the initial value for δ = δ_j, thus we hope the proposed algorithm can escape from getting trapped into a local minimum and reach the global minimization for a small value δ = δ_J.

For some fixed value δ = δ_j, minimization problem Equation (14) is solved by the quasi-Newton method in this paper. One of the most successful quasi-Newton methods is the BFGS algorithm, which is second order convergent and has good numerical stability. Therefore, we use the BFGS algorithm to solve Equation (14) for some fixed value δ = δ_j.

The conjugate gradient for a matrix variable is defined as: ∂ L δ , λ ( X ) ∂ X * = 1 2 ( ∂ L δ , λ ( X ) ∂ X R + i ∂ L δ , λ ( X ) ∂ X I )where X_R and X_l denote the real part and the imaginary part respectively, X^*denotes the conjugate of X. Then the conjugate gradient of L_δ,λ(X) (for its derivation see Appendix I) can be expressed as: (15) ∂ L δ , λ ( X ) ∂ X * = 1 2 δ 2 Λ X − λ Φ * ( Y − Φ X )where Λ = diag { δ 2 / ( ξ 1 4 + 4 δ 4 ) ,   δ 2 / ( ξ 2 4 + 4 δ 4 ) , ⋯ , δ 2 / ( ξ n 4 + 4 δ 4 ) }

Solving by the BFGS algorithm, the main iterative steps are as follows:

Search step length t_k, satisfy: t k = min t     L δ j , λ   ( x − t H k ∇ L δ j , λ ( X ) )

Based on the above idea, JSDOA can be described as shown in Table 1.

Remark:

(1) Select parameter λ: in this paper, we select the parameter λ by the α– method [17]. Set f₁(X) = F_δ(X), f 2 ( X ) = ‖ AX − y ‖ F 2, and minimizing f₁(X) and f₂(X), we have; f i ( X ( i ) ) = min x     f i ( X )   i = 1 , 2

It is easy to know that X⁽ⁱ⁾(i = 1,2) is 0 and A^T(AA^T)⁻¹Y respectively. Then we have: f ij = f i ( X ( j ) )   i , j = 1 , 2

Set λ = λ₂/λ₁ and introduce auxiliary parameter α, we can have: { ∑ i = 1 2 f ij λ i = α ∑ i = 1 2 λ i = 1

Setting the coefficient matrix f(i,j) = f_ij, from the above equations, the solution is: (17) { [ λ 1 , λ 2 ] = e T f − 1 e T f − 1 e α = 1 e T f − 1 ewhere e = [1,1]^T, and λ = λ₂/λ₁.

(2) Select parameter δ: In this paper, we set δ_j = γδ_j–1, j = 2, ⋯, J, γ ∈ (0.5,1). Let x ˜ = max i | x i 0 |, we hope parameter δ₁ satisfies: f δ ( x ˜ ) = 2 π arc   tan   ( x ˜ 2 2 δ 1 2 ) ≤ 1 2 ⇒ δ 1 ≥ x ˜ 2

In order to save computation cost, we set δ 1 = max i | x i 0 | / 2. When δ_J → 0, F_δJ(x) → ‖x‖₀. However, if δ_J is too small, F_δJ(x) will be sensitive to noise, so δ_J shouldn’t be too small.

(3) For coherent sources: Many of popular methods, such as MUSIC and ESPRIT require the assumption that sources are uncorrelated, because the source covariance matrix remains nonsingular so long as none of these sources are coherent. However, the algorithm proposed in this paper doesn’t need to compute the source covariance matrix, and joint sparsity is used to estimate DOA, so the proposed algorithm can handle highly coherent sources as well.

(4) Limitation: The algorithm proposed in this paper suffers from two problems. First, there exist no clear-cut guidelines for the selection of δ_J, especially when knowledge of noise is unknown. Second, the proposed algorithm can’t perform well when the SNR is relatively low.

In this simulation, we compare spatial spectrum of different algorithms under various snapshots. We consider two uncorrelated sources located at 13° and 20° with SNR = 10. For JSDOA, we set δ 1 = max i | x i 0 | / 2, δ_j = γδ_j–1, γ = 0.5, J = 7. In Figures 1(a,b), we set T = 10 and T = 5, respectively. It can be seen from Figure 1 that JSDOA and L1-SVD can resolve the two sources as T = 10. However, only JSDOA can resolve the two sources as T = 5.

In this simulation, we compare the spatial spectrum of different algorithms with more than two uncorrelated sources. Set SNR = 10, T = 20, δ 1 = max i | x i 0 | / 2, δ_j = γδ_j–1, γ = 0.5, J = 7. In Figure 2(a), there are three sources located at 10, 20 and 30. In Figure 2(b), there are five sources located at 0, 10, 20, 30 and 40. It is shown from Figure 1 that JSDOA and L1-SVD can resolve the three sources, but only JSDOA can resolve the five sources.

In this simulation, we compare probability of resolution for two uncorrelated sources. JSDOA, MUSIC and L1-SVD are used for comparison and we set T = 5, δ_j = γδ_j–1, γ = 0.5, J = 7. For each Δθ or SNR, 200 independent simulation are carried out. In Figure 3(a), one source is fixed at 10°, the other source is located at 10° + Δθ. vary from 0 to 30. In Figure 3(b), two sources are located at 10° and 20°. SNR vary from −5 to 20. It is shown from Figure 3 that JSDOA has the higher probability of resolution than MUSIC and L1-SVD.

In this simulation, we compare spatial spectrum for completely coherent sources. Set SNR = 10, T = 20, δ 1 = max i | x i 0 | / 2, δ_j = γδ_j–1, γ = 0.5, J = 7. In Figure 4(a) two coherent sources, located at 10° and 20°, are considered.

In Figure 4(b) three coherent sources, located at 10° and 20°, are considered. It is shown from Figure 4 that JSDOA and L1-SVD can be used to estimate DOA for coherent sources and that JSDOA has higher resolution than L1-SVD.