Extreme Sparse-Array Synthesis via Iterative Convex Optimization and Simulated-Annealing Expanded Array

Abstract

Sparse-array synthesis can considerably reduce the number of sensor elements while optimizing the beam-pattern response performance. The sparsity of an array is related to the degrees of freedom of the array elements. A sparse-array method based on iterative convex optimization and a simulated-annealing expanded array is proposed in this paper. This method transforms the sparse-array problem into a minimum l₁ norm problem and obtains the sparse array by solving the convex optimization problem using the primal-dual algorithm. Meanwhile, to improve the degree of freedom, array elements are expanded using stochastic perturbation. According to the simulated-annealing algorithm, the closed array elements are reopened with a specific probability, which is iteratively thinned and expanded. The results show that the proposed method can obtain an extremely sparse array, which is better than that obtained using the existing methods.

1. Introduction

Sparse-array synthesis optimizes the spatial response performance of beam patterns, such as the mainlobe width and sidelobe peak (SLP), deletes redundant elements of the uniform sensor array, assigns weight coefficients to the remaining elements, and obtains a sparse array [1]. The sparse-array synthesis reduces the number of sensor array elements considerably and ensures the spatial response performance of the beam pattern. It is widely used in sonar, radar, and medical imaging applications [2,3,4].

Sparse-array methods mainly comprise two categories: (1) stochastic optimization algorithms and (2) deterministic optimization algorithms. Stochastic optimization algorithms include genetic [5], simulated-annealing (SA) [6,7], and particle-swarm-optimization (PSO) algorithms [8]. In recent years, based on the theory of compressed sensing (CS), deterministic optimization algorithms have been developed, including the Bayesian CS (BCS) algorithm [9,10], Focal Underdetermined System Solver (FOCUSS) algorithm [11,12], and the convex optimization algorithm [13,14,15].

The stochastic optimization algorithm is sensitive to the initial value, and the experimental results are stochastic, thereby requiring several experiments to obtain the global optimal solution. Deterministic optimization algorithms can perform global sparse optimization by solving the analytical formula. However, the deterministic optimization algorithm has proven difficult to effectively solve. In recent years, convex optimization algorithms have achieved satisfactory results in array-sparse technology. However, some challenges still exist with the array-sparse method, which is based on the convex optimization algorithm. First, the perturbed convex optimization algorithm using the first-order Taylor approximation improves the sparse degrees of freedom, thereby increasing the approximation error [16]. Further, increasing the number of initial elements and degrees of freedom may lead to a sparse problem that may not satisfy the restricted isometry property (RIP), thereby hampering the primal-dual algorithm from converging on the convex optimization problem. Consequently, an optimal solution may not be obtained [17].

To solve the above problems, this study proposes an array-sparse method based on iterative convex optimization and SA expanded array. This method combines the advantages of the stochastic optimization algorithm (which makes it easy to add variables to improve the degrees of freedom) and deterministic optimization algorithm (which can obtain an analytical solution). This method transforms the sparse-array problem into a minimum l₀ norm problem and obtains the sparse array by solving the convex optimization problem using the primal-dual algorithm. Simultaneously, to improve the sparse degree of freedom, the array is expanded based on the sparse array. According to the SA algorithm, the closed array elements are restarted using Boltzmann probability and by iteratively thinning the expanded array.

This study is further organized as follows: In Section 2, a sparse-array method based on iterative convex optimization and a simulated-annealing expanded array are proposed and analyzed. In Section 3, several sparse-array synthesis methods are described to evaluate the efficiency of the proposed method. In Section 4, experimental results are discussed, while the conclusions are drawn in Section 5.

2. Methods

Array sparsity should principally constrain the beam pattern and further consider the far-field beam pattern of the plane array. For an M uniform linear array, the array-element spacing is d, the array-element index number is m (1 ≤ m ≤ M), and the sensor element position x_m is:

$$x_m=(m-\frac{M+1}{2})d$$

(1)

As shown in Figure 1, when the target distance r meets r > πD²/4λ [18], where D is the array aperture, the target can be regarded as being in the far field. Here, the echo signal in the same direction can be regarded as a parallel beam. The direct method (DM) frequency domain beamforming algorithm expression is shown in Equation (2).

$$BP(u)=\left|{\displaystyle \sum _{m=1}^{M}w(m)}{S}_{(m)}(f_0)\exp (-j(\frac{2\pi }{\lambda })(x_m\sin {\theta }_a))\right|$$

(2)

where S_(m)(f₀) is the frequency domain echo signal of the receiving array at frequency f₀ and w(m) the weight coefficient of the array element. u is the steering direction equal to sin θ_a (which is evenly divided into P directions), and $\exp (-j(\frac{2\pi }{\lambda })(x_m\sin {\theta }_a))$ is the phase shift parameter. After the steering direction is normalized, the beamforming expression is represented as:

$$B(u)=\left|{\displaystyle \sum _{m=1}^{M}w(m)}\exp (-j(\frac{2\pi }{\lambda })(u{x}_m))\right|$$

(3)

The sparse-array problem includes minimizing the number of array elements whose weight coefficients are not zero when the beam pattern meets the specified constraints (solving the minimum l₀ norm problem). Array sparsity can be expressed as:

$$\min {\Vert w\Vert }_0 \mathrm{s}.\mathrm{t}. \{\begin{array}{l}B(0)=1\\ \left|B(ml)\right|\le 1\\ \left|B(sl)\right|\le \mathrm{SLP}\end{array}$$

(4)

where ‖w‖₀ is the l₀ norm of w (the number of nonzero elements in w), B(0) is the beam strength in the incident direction, ml is the main direction, sl is the sidelobe direction, and SLP is the sidelobe peak constraint. Solution w of the optimization problem in Equation (4) is the sparsest array obtained when the beam-pattern constraint is satisfied.

However, the minimum l₀ norm problem shown in Equation (4) is a nondeterministic NP-hard problem that is difficult to solve directly using mathematical methods. The sparse optimization problem (4) can be approximated as the minimum l₁ norm problem expressed below:

$$\underset{w}{\min }{\Vert w\Vert }_1 \mathrm{s}.\mathrm{t}. \{\begin{array}{l}B(0)=1\\ \left|B(ml)\right|\le 1\\ \left|B(sl)\right|\le \mathrm{SLP}\end{array}$$

(5)

where ‖w‖₁ represents the l₁ norm of w (the sum of the absolute values of the elements in w). The approximate Equation (5) is a convex optimization problem. In the minimum l₁ norm, the weight coefficient of the element is generally not equal to zero. Hence, an element with a weight coefficient of less than 10⁻⁸ is regarded as a zero element (a closed array element).

In the l₀ norm, non-zero elements are equivalent, whereas in the l₁ norm, elements with larger values have a greater impact on the objective function. The solution obtained using the minimum l₁ norm problem may not be the sparsest solution. To better solve the sparse problem, the minimum l₀ norm problem can be equivalent to the optimization problem of Equation (6) as expressed below:

$$\begin{array}{c}\underset{w^i}{\min }{\Vert w^i\cdot {\omega }^i\Vert }_1 \mathrm{s}.\mathrm{t}. \{\begin{array}{l}B(0)=1\\ \left|B(ml)\right|\le 1\\ \left|B(sl)\right|\le \mathrm{SLP}\end{array}\\ {\omega }^i=1/(w^{i-1}+\epsilon )\end{array}$$

(6)

where i is the number of iterations (i > 1) and wⁱ · ωⁱ is the Hadamard inner product of wⁱ and ωⁱ. parameter ωⁱ makes the variable w greater than wⁱ · ω. After i weighting, the non-zero elements are close to 1, while the zero elements remain close to 0. The value of parameter ε is slightly lower than the minimum weight coefficient of non-zero elements. Parameter ε ensures that the closed array elements still have a certain probability of opening in the next iteration. Compared to the minimum l₁ norm problem, solving the iteratively weighted minimum l₁ norm problem can yield sparser solutions which are closer to the minimum l₀ norm. The iteratively weighted minimum l₁ norm problem is solved as follows: In the first iteration, optimization problem (5) is solved to determine w¹. When i > 1, optimization problem (6) is solved until the sparse rate converges to obtain the final sparse result. The aforementioned iterative optimization problems are convex optimization problems that can be effectively solved using the MATLAB tool CVX [19] via the primal-dual algorithm.

For a sparse array obtained using the above method, the array element can only be at the initial M positions, while the array element at the off-grid position cannot be selected. This algorithm greatly limits the degrees of freedom of candidate array elements, thereby resulting in the poor position of the sparse array. To increase the degrees of freedom of the candidate positions of the array elements, the initial array can use a dense array with a smaller array-element spacing. However, this greatly increases the computational complexity of the sparse process and reduces the efficiency and performance of the sparse algorithm. Further, the use of dense arrays may not satisfy the RIP, thereby hampering the primal-dual algorithm from converging on the convex optimization problem. Consequently, an optimal solution may not be obtained. Additionally, the first-order Taylor expansion is used to introduce the element position perturbation parameter. However, this approximation method is not accurate, thereby increasing the optimization error.

After a sparse array is obtained solving the minimum l₁ norm problem, the array is expanded based on its element’s positions. Simultaneously, according to the SA algorithm, the closed array element was reopened using Boltzmann probability. This sparse process was repeated for the expanded array.

To maintain the size of the array aperture, the position of elements at two ends are fixed in thinning. After thinning and expanding as in Figure 2, the black, gray, and hollow circles represent open, reopened, and closed array elements, respectively, while the dots represent extended array elements. The SA algorithm starts from an initial temperature T₀ and gradually cools based on certain rules. At any temperature state T_i, each element is stochastically traversed, and some elements are reopened with the Equation (7) Boltzmann probability P_B. The energy function of the proposed SA method is the number of active array elements.

$$P_B=\exp ((M-N_s)/K_B{T}_i)$$

(7)

where K_B is the Boltzmann constant. The positions of the open and reopened array elements are L(i), where i is the sequence number of the array element. The total number of active elements is I. Q sub-elements are expanded for each active element, while the expanded array-element position L₁(i, q) is within the range of the array-element position (−d/2 to d/2), as expressed below:

$$L_1(i,q)=L(i)+(q-(Q+1)/2)d/Q+r(q)$$

(8)

where r(q) is the random perturbation added to the array-element spacing based on d/Q, |r(q)| ≤ d/2Q. The beam pattern of the expanded array is expressed in Equation (9). The expanded array element was used as the initial array, while the iterative weighted l₁ norm was used to obtain a sparse array iteratively. The method ended when the results remained unchanged 10 consecutive times or the temperature T_i dropped to the lowest value. The flowchart of the algorithm for this process is illustrated in Figure 3.

$$B(u)=\left|{\displaystyle \sum _{i=1}^I{\displaystyle \sum _{q=1}^{Q}w(m)}\exp (-j(\frac{2\pi }{\lambda })(u{L}_1(i,q)))}\right|$$

(9)

3. Results

To evaluate the effectiveness of the proposed sparse-array synthesis method, a sparse linear array with 50λ apertures was compared to the sparse results in References [7,16], and two sparse arrays with different configurations were compared to the sparse results in literature [14]. The experiments described in this section were simulated using MATLAB. The experimental platform processor is an Intel i7-10870 and the amount of memory is 32 GB.

The same array configurations as in References [7,16] were adopted. The steering directions u were set within −1 and 1 and were divided equally into 400 beams. The mainlobe directions were restricted within 0.022, while the SLP constraint was set to −23.5 dB. The initial array comprised 501 elements with a spacing of λ/10. To maintain the size of the array aperture, the position of elements at two ends (elements at 0 and 50λ) were fixed in thinning. The expanded number of elements Q was set to 10, initial temperature T₀ was 1000, cooling coefficient was 0.85, and parameter ε was 0.004.

Through the method, a sparse array containing 43 elements was obtained. The computation time was 536.9 s. The SLP was −23.80, while the angular resolution at −3 dB was 1.22°.

Table 1. Comparison of sparse array between the proposed method and [7,16].

Method	Number of Elements	SLP (dB)	Resolution (°)
[7]	58	−22	1.28
[16]	45	−23.67	1.22
Proposed method	43	−23.80	1.22

presents a comparison of the sparse results of the proposed method with those in References [7,16]. Further, to increase the degree of freedom, the initial array was simply expanded to 50,000 array elements with a spacing of λ/1000. Under the above conditions, the number of elements obtained using the convex optimization algorithm was 62. Compared to other sparse-array methods, the proposed method obtained a sparse array with fewer elements and a lower SLP.

The positions and weights of the optimized active elements are shown in Figure 4a and

Table 2. Positions and weight values of the sparse array with 43 elements using the proposed method.

Position (λ)	Weight	Position (λ)	Weight	Position (λ)	Weight
0	0.0218	18.3197	0.0259	32.8365	0.0241
3.9013	0.0216	19.2132	0.0278	33.8144	0.0203
4.7085	0.0247	20.2201	0.0280	34.8189	0.0191
6.7014	0.0152	21.2256	0.0251	35.7119	0.0213
7.6218	0.0184	22.1111	0.0285	36.7129	0.0175
8.6156	0.0231	23.1059	0.0257	37.7382	0.0142
9.5462	0.0181	24.1048	0.0290	38.6273	0.0220
10.5215	0.0178	25.1059	0.0320	40.6161	0.0226
11.4992	0.0195	26.0212	0.0261	41.5016	0.0199
12.4452	0.0203	27.0154	0.0277	42.4244	0.0230
13.4490	0.0206	27.9943	0.0306	43.4312	0.0216
14.4119	0.0267	28.9131	0.0262	44.3340	0.0243
15.3956	0.0255	29.9401	0.0301	50	0.0128
16.3129	0.0199	30.8915	0.0280	-	-
17.3104	0.0272	31.8464	0.0263	-	-

, respectively. The beam pattern of the optimized sparse array is shown in Figure 4b.

To verify beampatterns in different incident directions, beampatterns with incident directions of 0, 15, and 30 degrees are shown in Figure 5. Experiments show that beampattern response performance is consistent at different incident directions.

To simulate the application of the sparse array, we used a uniform line array sonar system with 50λ apertures and 100 elements shooting a pool scene in Sanya, Hainan Province, China [20]. The pool size is 10 × 12 × 10 m. Since the sparse array in this paper is a non-uniform array, echo signals need to be deduced according to Equation (2). As shown in Figure 6, the beampattern of the sparse array keeps the image quality close to that of uniform array.

Two sparse arrays were compared with [14]. In one experiment, the mainlobe directions were restricted within 0.01982, while the SLP constraint was set to −20 dB with 50λ apertures. In the other experiment, the mainlobe directions were restricted within 0.01, while the SLP constraint was set to −37 dB with 162λ apertures. Parameter ε is 0.004 and 0.001, respectively, and other configurations are almost the same as the previous experiment.

Through the method, sparse arrays containing 39 and 144 elements were obtained. The computation times were 602.3 and 2846.8 s, respectively.

Table 3. Comparison of sparse array between the proposed method and [14].

Array Configuration	Number of Elements
	Proposed Method	Method in [14]
50λ, SLP ≤ −20 dB, |uml| ≤ 0.01982	39	42
162λ, SLP ≤ −37 dB, |uml| ≤ 0.01	144	149

presents comparisons of the sparse results of the proposed method with those in literature [14]. Experiments show that the proposed method obtained a sparse array with fewer elements.

The positions and weights of the optimized active elements with 50λ apertures and 39 elements are shown in Figure 7a and

Table 4. Positions and weight values of the sparse array with 39 elements using the proposed method.

Position (λ)	Weight	Position (λ)	Weight	Position (λ)	Weight
0	0.0239	19.1971	0.0217	33.6252	0.0293
6.6953	0.0259	20.1119	0.0266	34.6094	0.0282
7.6274	0.0347	21.1058	0.0296	35.5084	0.0211
8.5922	0.0239	22.1034	0.0301	36.5089	0.0209
9.4932	0.0357	23.0033	0.0294	37.5194	0.0177
10.5118	0.0268	23.9946	0.0402	38.4983	0.0227
11.4087	0.0147	24.8996	0.0344	39.4030	0.0232
12.4058	0.0107	26.9099	0.0260	40.3132	0.0194
14.3688	0.0329	27.8997	0.0416	41.3192	0.0288
15.3156	0.0253	28.8034	0.0188	42.2122	0.0169
16.3037	0.0123	29.8068	0.0299	44.2037	0.0229
17.2972	0.0221	30.7049	0.0256	46.1132	0.0318
18.2135	0.0287	32.7045	0.0291	50	0.0161

, respectively. The beam pattern of the optimized sparse array is shown in Figure 7b.

The positions and weights of the optimized active elements with 50λ apertures and 39 elements are shown in Figure 8a. The beam pattern of the optimized sparse array is shown in Figure 8b.

4. Discussion

The proposed method based on convex optimization and the SA algorithm used the deterministic optimization algorithm (convex optimization) to obtain the global optimal analytical solution and stochastic optimization algorithm (SA algorithm) to improve the degree of freedom. It combined the advantages of both types of algorithms, thus obtaining better sparse results. Additionally, the initial array was expanded to 50,000 array elements with a spacing of λ/1000. The number of sparse-array elements obtained using convex optimization algorithm under the above conditions was 62. This shows that an increase in the number of initial elements may lead to worse sparse results as the sparse problem may not satisfy the RIP. Hence, it will be difficult to converge the primal-dual algorithm.

5. Conclusions

In this study, a sparse-array method based on iterative convex optimization and a simulated-annealing expanded array is proposed. This method solves the minimum l₁ norm problem to obtain the global optimal sparse array. Further, it iteratively expands and considers the array according to the SA algorithm. This method effectively improved the degrees of freedom of the array elements while obtaining a global optimal analytical solution. The experimental results showed that the proposed method obtained an extremely sparse array, which was better than that obtained using the existing methods.