Optimization of Sparse Planar Arrays with Minimum Spacing and Geographic Constraints in Smart Ocean Applications

Sparse arrays can fix array aperture with a reduced number of elements to maintain resolution while reducing cost. However, grating lobe suppression, high peak side-lobe level reduction (PSLL), and constraints on the location of the array elements in the practical deployment of arrays are challenging problems. Based on simulated annealing, the element locations of a sparse planar array in smart ocean applications with minimum spacing and geographic constraints are optimized in this paper by minimizing the sum of PSLL. The robustness of the deployment-optimized spare planar array with mis-calibration is further considered. Numerical simulations show the effectiveness of the proposed solution.


Introduction
Sensor and antenna arrays play an important role in fields such as radar and sonar due to their higher processing gain and angular resolution; moreover, they are often employed in smart ocean applications for collecting information, positioning and communication. Recently, large-aperture arrays have been used widely for long-range acoustic communication in deep water [1], continuous monitoring of fish populations [2], as well as in detection, localization and classification of mechanized ocean vessels [3,4], etc. However, large-aperture arrays always require many elements with higher cost, and the deployment, impact on the ecological environment, energy supply, and maintenance of a large number of elements in the ocean are challenging problems. Therefore, sparse arrays with fewer elements are increasingly attracting interest in many applications such as target localization, ocean acoustic remote sensing [5], and passive underwater imaging [6]. In particular, sparse arrays are expected to be a potential scheme for seafloor observatory networks in oceanic engineering, radar and sonar systems with distributed arrays, etc. Furthermore, sparse arrays and optimization can also be used in the node deployment of large-scale underwater acoustic sensor networks [7][8][9] and the optimal experimental design [10,11].
A sparse array can fix array aperture with a smaller number of elements to maintain angular resolution while reducing cost. However, sparse arrays face challenges such as grating lobe suppression, high peak side-lobe level (PSLL) reduction, and even some constraints on the location of elements in the practical deployment of the array. The optimization of a sparse array is, thus, element. Without loss of generality, we let (x 0 , y 0 ) = (0, 0). The phase difference of arrival ∆φ n (θ, ϕ) between the reference element and the nth element is Ref. [34]: λ (x n sin θ cos ϕ + y n sin θ sin ϕ), n = 0, 1, · · · , N − 1 (1) where λ is the signal wavelength, and θ and ϕ are the elevation and azimuth angles, respectively. If an array in the 3-D space is considered, we have [34]: λ (x n sin θ cos ϕ + y n sin θ sin ϕ + z n cos θ), where z n denotes the nth element location on the z-axis. Every Sonar or Radar in a distributed array system can be treated as a subarray of a sparse array, where all element locations of every subarray can be clustered and described by x n + ∆x nj , y n + ∆y nj , z n + ∆z nj . ∆x nj , ∆y nj and ∆z nj are the distance between the reference element and the jth element of the nth subarray in the x-, y-, and z-axes, respectively, and are generally known. Thus, only (x n , y n , z n ) need to be optimized in the deployment of the distributed array system.

Constraints
In practical applications, a sparse array is often chosen for high angular resolution with a smaller number of elements, where the minimum spacing between two elements is generally larger than λ/2. In fact, there may be no alternative in many scenarios, e.g., mutual coupling [13], element size larger than λ/2, etc. The constraint can be mathematically written as: (x n 1 − x n 2 ) 2 + (y n 1 − y n 2 ) 2 ≥ ∆d, n 1 , n 2 = 0, 1, · · · , N − 1, n 1 = n 2 , where ∆d is the minimum spacing between two elements. For convenience of optimization implementation, the minimum spacing constraint shown in Equation (7) is rewritten as: max(|x n 1 − x n 2 |, |y n 1 − y n 2 |) ≥ ∆d, n 1 , n 2 = 0, 1, · · · , N − 1, n 1 = n 2 , which is the Chebyshev distance [27][28][29] between two elements. It is easy to prove that the minimum spacing constraint in Equation (7) is true when the requirement in Equation (8) is met, then the adjustment of the element locations in the optimization with the minimum spacing constraint can be done in only one direction (x-or y-axis), which would improve the diversity of element distribution and computational efficiency. There may be hills, rivers, boulder, oceanic ridges and trenches, etc. in the given area, which are obstacles for array deployment; thus, geographic constraints must be considered for the optimization of a large-sized sparse planar array. In this paper, geographic obstacles are approximated in shape by their bounding rectangles, and mathematically written as: where L is the number of obstacles in the given area of D X × D Y . The lth obstacle is described mathematically by G l x1 , G l x2 ∩ G l y1 , G l y2 . G l x1 , G l x2 and G l y1 , G l y2 are its ranges along the xand y-axes, respectively, and ∪ and ∩ denote union and intersection, respectively, in set theory. The detailed mathematical description of the obstacles with other more complex shapes is shown in Ref. [35].

Implementation of Constraints in Optimization
The minimum spacing and geographic constraints in optimization would be independently implemented in the direction of the y-axis and x-axis, respectively.
Suppose the adjustment of element locations with the minimum spacing constraint given in Equation (8) is done in the direction of the y-axis. At the kth iteration in optimization, the element locations in the y-axis y(k) = [y 0 (k), · · · , y N−1 (k)] T are expressed as: where y n (k) denotes the nominal location in the y-direction of the nth elements andỹ(k) = are generated and then sorted in the ascending order as the initialized values y n (0), n = 1, · · · , N − 2. Thus, At the kth iteration in optimization, y n (k), n = 1, · · · , N − 2, is always updated in the interval (0, D Y − (N − 1)∆d), i.e.,: (11) (N − 1)∆d is reserved for N elements and then the minimum spacing ∆d for every two adjacent element is imposed in the direction of the y-axis as in Equation (10). Therefore, y n (k), n = 1, · · · , N − 2, is certainly in the interval (0, D Y ). Meanwhile: and any two elements are separated by at least ∆d. Equations (10)-(13) make y n (k), n = 0, · · · , N − 1, could be always deployed in [0, D Y ]. But the optimization is based on y n (k), which ensure the minimum spacing constraint is always satisfied but the implementation of optimization is simple and flexible.
At the kth iteration in optimization with the geographic constraint, similarly, the element locations in the x-direction x(k) = [x 0 (k), · · · , x N−1 (k)] T are first expressed as: where At the kth iteration in optimization, the implementation of the geographic constraint is shown in Algorithm 1, where x n (k), n = 1, · · · , N − 2, is always updated in the interval (0, D X ].

Algorithm 1. Implementation of Geographic Constraint
The geographic constraint is independently implemented in the direction of the x-axis. Thus, no matter how the elements move in the direction of the x-axis, the implementation of the geographic constraint in optimization does not conflict with that of the minimum spacing constraint. The elements cannot be deployed in G l x1 , G l x2 ∩ G l y1 , G l y2 , l = 1, · · · , L, which is the mathematical description of the lth obstacle. With the if-then mode in Table. 1, the nth element could be moved from the area of the lth obstacle.

Optimization via Simulated Annealing
Simulated annealing [17,36] is a probabilistic technique often used for approximating the global optimum of a given function depending on many parameters, e.g., array element locations in this paper. The optimization of a sparse planar array with minimum spacing and geographic constraints via simulated annealing is summarized in Algorithm 2.
Two sparse planar arrays with random and optimized deployment are shown in Figure 1a. The elements of both sparse planar arrays are not located in areas corresponding to geographic constraints, and the minimum spacing between any two adjacent elements is greater than ∆d. The beampattern of the optimized sparse planar array when the peak of the main-lobe is at (30 • , 0 • ) is given in Figure 1b. Figure 1c shows the beampatterns in ϕ = 0 • , ϕ = 60 • and ϕ = 120 • planes whose PSLLs are −12.9 dB, −9.69 dB and −11.76 dB, respectively. It is shown the optimization of sparse planar arrays with minimum spacing and geographic constraints is successfully implemented in Figure 1a-c. The good convergences of the fitness value ( f in Equation (5) [12,13,24], random sparse planar array (RSPA) without optimization, and uniform planar array (UPA) in = 60° and = 120° planes. The UPA has 275 elements and is also deployed in the same area. It is shown that their main-lobe beam widths are almost at the same level, which means that the spare planar array can have a larger array aperture with a smaller number of elements for high angular resolution. The PSLLs of beampatterns versus different beam shifting directions for the UPA, the RSPA, and the OSPA optimized by the proposed solution and PSO in = 60° and = 120° planes are illustrated in Figure 2c,d. In Figure 2, the sidelobe level of the OSPA is higher than that of the UPA but significantly lower than that of the RSPA. Generally, the low side-lobe level is very important for interference suppression in target localization.   [12,13,24], random sparse planar array (RSPA) without optimization, and uniform planar array (UPA) in ϕ = 60 • and ϕ = 120 • planes. The UPA has 275 elements and is also deployed in the same area. It is shown that their main-lobe beam widths are almost at the same level, which means that the spare planar array can have a larger array aperture with a smaller number of elements for high angular resolution. The PSLLs of beampatterns versus different beam shifting directions for the UPA, the RSPA, and the OSPA optimized by the proposed solution and PSO in ϕ = 60 • and ϕ = 120 • planes are illustrated in Figure 2c,d. In Figure 2, the side-lobe level of the OSPA is higher than that of the UPA but significantly lower than that of the RSPA. Generally, the low side-lobe level is very important for interference suppression in target localization.

Robustness of a Deployment-Optimized SPA
The optimization of a sparse planar array with minimum spacing and geographic constraints yields a distribution of elements optimized for high angular resolution and low side-lobe level, which means that the performance of the optimized sparse planar array in practical applications may be sensitive to mismatching, e.g., mis-calibration in hydrophone arrays or distributed underwater acoustic sensor networks, in which case the array deployment would have extremely strict accuracy requirements for the location of elements. Thus, the robustness of the deployment-optimized sparse planar array against mismatching is further considered in this paper.
Assume that the observation X( ) received by the sparse planar array in the presence of additive white Gaussian noise (AWGN) can be expressed by: where A is the steering matrix, and S( ) and N( ) correspond to the signals and noise, respectively. The standard Capon beamformer (SCB) and robust Capon beamformer (RCB) are always taken together for performance comparison when the mismatching occurs. Generally, the RCB belongs to the class of diagonal loading techniques [37]. The SCB has the optimal weight w 0 = R −1 a 0 a 0 R −1 a 0 ⁄ , and the weight vector of RCB is generally given by: where R X is the covariance matrix estimate of X( ) and R X = ∑ X( )X ( ) =1 ⁄ , is the number of snapshots. in Equation (16) is the diagonal loading value and can be often effectively determined as follows [38]:

Robustness of a Deployment-Optimized SPA
The optimization of a sparse planar array with minimum spacing and geographic constraints yields a distribution of elements optimized for high angular resolution and low side-lobe level, which means that the performance of the optimized sparse planar array in practical applications may be sensitive to mismatching, e.g., mis-calibration in hydrophone arrays or distributed underwater acoustic sensor networks, in which case the array deployment would have extremely strict accuracy requirements for the location of elements. Thus, the robustness of the deployment-optimized sparse planar array against mismatching is further considered in this paper.
Assume that the observation X(t) received by the sparse planar array in the presence of additive white Gaussian noise (AWGN) can be expressed by: where A is the steering matrix, and S(t) and N(t) correspond to the signals and noise, respectively. The standard Capon beamformer (SCB) and robust Capon beamformer (RCB) are always taken together for performance comparison when the mismatching occurs. Generally, the RCB belongs to the class of diagonal loading techniques [37]. The SCB has the optimal weight w 0 =R −1 X a 0 /a H 0R −1 X a 0 , and the weight vector of RCB is generally given by: whereR X is the covariance matrix estimate of X(t) andR X ∑ N t t=1 X(t)X H (t)/N t , N t is the number of snapshots. λ in Equation (16) is the diagonal loading value and can be often effectively determined as follows [38]: where std(·) means the standard deviation and diag(·) means the diagonal elements of a matrix. In this paper, we let λ = std diag R X . The performance comparison is given in Figure 3a,b, where standard Capon beamforming (SCB) without mis-calibration, SCB with mis-calibration, and robust Capon beamforming (RCB) with mis-calibration are presented. The mis-calibration in simulations is assumed to have a Gaussian distribution with a standard deviation of ∆d/2. The elements are illustrated in Figure 3c. It is shown the OSPA with miscalibration still has good performance by taking the robust adaptive beamforming. Figure 3d shows the beampatterns of RCB with mis-calibration through 10 trials. where std(•) means the standard deviation and diag(•) means the diagonal elements of a matrix. In this paper, we let = std(diag(R X )).
The performance comparison is given in Figure 3a,b, where standard Capon beamforming (SCB) without mis-calibration, SCB with mis-calibration, and robust Capon beamforming (RCB) with miscalibration are presented. The mis-calibration in simulations is assumed to have a Gaussian distribution with a standard deviation of ⁄ . The elements are illustrated in Figure 3c. It is shown the OSPA with miscalibration still has good performance by taking the robust adaptive beamforming. Figure 3d shows the beampatterns of RCB with mis-calibration through 10 trials.

Conclusions
In this paper, the optimization of SPAs in smart ocean applications with minimum spacing and geographic constraints has been presented, where the cost function is defined as the sum of PSLL for different beam shifting directions and simulated annealing has been used for the optimization. The implementation of these constraints in optimization is also given. Furthermore, the robustness of the deployment-optimized SPA against mis-calibration of the array in practical applications is also verified. Simulations show that the optimized SPA is effective for the DOA estimation.

Conflicts of Interest:
The authors declare no conflicts of interest.