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IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 17, NO. 1, JANUARY 2018
An Asymmetric Mapping Method for the Synthesis of Sparse Planar Arrays Dingcheng Dai
, Minli Yao, Hongguang Ma, Wei Jin, and Fenggan Zhang
Abstract—This study proposes an asymmetric mapping method to design sparse planar arrays with multiple constraints including the aperture, number of elements, and minimum spacing between adjacent elements. Benefiting from two different mapping functions, which are established in this letter, this approach allows one to overcome the limitations of the existing matrix mapping method in terms of flexibility and performances. The numerical validation points out that the proposed method outperforms the matrix mapping method and modified real genetic algorithm in the design of array arrangements. Index Terms—Antenna arrays, constraint optimization, differential evolution (DE), sidelobe level (SLL), sparse planar arrays.
I. INTRODUCTION HE optimization design of an irregular array is a complex nonlinear problem due to the unequally spaced elements positions [1]–[4]. To suppress the peak sidelobe level (PSLL) and decrease the number of elements, many numerical and analytical techniques have been proposed [5]–[14]. Typically, large thinned arrays are used to meet the requirement of high directivity owing to their superior performance in terms of cost, weight, energy consumption, and hardware complexity. Several optimization approaches have been quite effective for the synthesis of thinned arrays with a given array aperture and a thinning factor [5]–[9]. For example, the matrix pencil methods reveal their high effectiveness in reducing the element number in reconfigurable linear antenna arrays [8], [9]. Moreover, some analytical techniques such as difference sets [10] and almost difference sets [11], [12] provide reliable and a priori predictable bounds for the PSLL of the synthesized array by exploiting the autocorrelation properties of binary sequences. In recent years, sparse array has increasingly attracted attention because it has more degrees-of-freedom (DOFs) than thinned arrays [4]. Usually, sparse arrays are limited to multiple constraints that include the aperture, number of elements, and minimum element spacing dc (usually dc = 0.5 λ). Many studies on sparse linear arrays have been reported [13], whereas there are few studies on sparse planar arrays. To search for the best solution, the convex optimization method [14] and Bayesian compressive sense [15] are introduced to synthesize a sparse array. However, these methods need to set a suitable reference pattern as a precondition.
T
Manuscript received October 10, 2017; accepted November 13, 2017. Date of publication November 16, 2017; date of current version January 10, 2018. This work was supported by the National Natural Science Foundation of China under Grant 61179004 and Grant 61179005. (Corresponding author: Fenggan Zhang.) The authors are with Xi’an Research Institute of High Technology, Xi’an 710025, China (e-mail:
[email protected];
[email protected]; mhg_xian@ 163.com;
[email protected];
[email protected]). Digital Object Identifier 10.1109/LAWP.2017.2774498
Fig. 1.
Array layout generated by matrix mapping method.
To overcome this drawback, the modified real genetic algorithm (MGA) [1] and matrix mapping method with differential evolution (DE) [2] have been proposed. The core concept in these two methods is transforming the minimum element spacing constraint to the Chebychev distance constraint. In this way, the diversity of element distribution and computational efficiency can be remarkably improved. However, despite the advantages mentioned above, these two methods have some limitations that could prevent further improvement of array performance and may lead to infeasible solutions. As shown in Fig. 1, the array layout is generated by matrix mapping method [2]. The results show that the number of elements in the ydirection is limited to 3, but the maximum number that can be placed is 5. Additionally, three pairs of elements could not meet the requirement of the minimum spacing constraint. The main reasons are as follows. 1) The dimension of element position matrices defined in [1] and [2] limit the DOFs of the element distribution. 2) The mapping functions in [1] and [2] cannot absolutely guarantee the minimum element spacing because the distances between diagonal adjacent elements have not been considered. This letter aims at overcoming the above limitations, and an asymmetric mapping method (AMM) is proposed. The innovations of the proposed method include. 1) Redefined sizes of elements position matrices. Both the number of elements that can be placed and distributable spacing are considered to improve the flexibility of element distribution. 2) When the number of elements defined in the position matrices is more than the actual number, a selection matrix is established to determine which elements in the position matrix should be turned off. 3) The minimum spacing constraints is transformed into two categories, and then two mapping functions are well designed to transform these problem into unconstrained problems. Hence, all the infeasible solutions can be avoided during the optimization procedure.
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DAI et al.: ASYMMETRIC MAPPING METHOD FOR THE SYNTHESIS OF SPARSE PLANAR ARRAYS
Fig. 2.
Fig. 3.
Geometry of a quarter of a symmetric planar array.
II. ANTENNA FORMULATION AND PROPOSED METHOD As shown in Fig. 2, 4N elements are randomly distributed on an axisymmetric planar array with an aperture of 2L × 2H. Each element is denoted by a pair of real numbers (xi , yi ) and the corresponding excitation current Ii is set to be identical (i.e., Ii = 1 for all elements). Since the other 3N elements can be symmetrically mapped by the positions of N elements in Fig. 2, the array factor can be characterized as 4N AF(θ, ϕ) = Ii ej k (x i sin θ cos ϕ+y i sin θ sin ϕ) i=1 N −1 (1) =4 cos(kxi u) cos(kyi v) + cos(kLu) cos(kHv)
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Sketch map of formulation (4).
As shown in Fig. 3, it can be easily proved that the elements position matrices X and Y could satisfy the element spacing constraint if they meet the requirement of the formulation (4). To satisfy the formulation (4), two mapping functions T 1 and T 2 are established. The mapping functions work as follows. First, two optimization variables matrices A and B are set as ⎤ ⎡ β1,1 · · · β1,Q ⎡ ⎤ α1,1 · · · α1,Q γ1 ⎢ . .. ⎥ .. ⎢ .. ⎢ .. ⎥ . . . . ⎥ . ⎥ ⎢ . . . A=⎣ . . . . ⎦ B=⎢ ⎥ ⎣βP ,1 · · · βP ,Q ⎦ αP ,1 · · · αP ,Q γP η1 ··· ηQ (5)
i=1
where k = 2π/λ, and λ is the wavelength. θ and ϕ are the azimuth and elevation angles, respectively. u = sin θ cos ϕ, v = sin θ sin ϕ. The element positions are transformed into two position matrices X ∈ RP ×Q and Y ∈ RP ×Q , and the sizes of X and Y are defined as ⎧
P × Q − N L − (Q − 0.5)dc H − (P − 0.5)dc ⎪ ⎪ max + + ⎪ ⎨ Pm × Qm L H + ⎪ P × Q ≥ N, P, Q ∈ Z ⎪ ⎪ ⎩s.t. P ≤ Pm , Q ≤ Qm (2) where Pm = H/dc , Qm = L/dc are the maximum numbers of placeable elements in the y- and x-directions, respectively. If P × Q = N , X and Y are full matrices. If P × Q > N , the position matrices should be thinned, and P × Q − N elements should be turned off. A selection matrix W ∈ RP ×Q is set up to determine which elements should be turned off. The element wi,j ∈ [0, 1] in W denotes the weight of the element determined by X and Y in ith row and jth column. W is randomly generated in the initial stage and would be taken as the optimization variable during the optimization process. Only N elements with larger weights will be reserved. Note that wi,j is not the excitation current. The constraint of minimum can be described as (xi,j − xk , l )2 + (yi, j − yk , l )2 ≥ dc ; i, k ∈ [1, P ]; j, l ∈ [1, Q].
(3)
To satisfy this requirement, we split the distance between elements into two classes. The one is the element spacing in the same row (or column), and the other is the distance between different rows (or columns). Then, the constraint in (3) can be transformed to i de = min (|xi,j +1 − xi,j |) ≥ dc . (4) dir = min(yi+1,j ) − max(yi,j ) ≥ dc
where αi,j , γi , βi,j , and ηj are randomly generated among the range of [0, 1]. Second, the remaining regions in x- and y-directions are calculated. Because of the symmetry of the array, the first row of Y and the first column of X should be no less than 0.5dc . Considering the minimum element spacing, the distributable spacings in the x- and y-directions are U x = L − (Q − 0.5)dc . (6) U y = H − (P − 0.5)dc Then, the average distributable spacing for each element in the x- and y-directions can be expressed as V x = U x/Q and V y = U y/P , respectively, assuming that V x > V y. Third, generate the element position matrices X and Y . Step 1: The element position coordinates in the x-direction can be mapped from variable matrix A ⎧ ⎨xi,j = jt=1 Δxi, t (7) Q ⎩Δxi, j = Δdxj + γi U x × αi, j t=1 αi, t where the Q-dimensional vector Δdx = [0.5dc , dc , dc , . . . , dc ], Δxi, j = xi, j − xi, j −1 represents the element spacing of ith row in the x-direction with the set mapping function (7) as X = T 1 (A). Step 2: Calculate the element position matrix Y . First, compute the distributable spacing matrix Sy according to variable matrix B: P βt, j . (8) Syi, j = ηj U y×βi, j t=1
Then, find the maximum value ymi in each row of Sy: ymi = max (Syi, 1 , Syi, 2 , . . . , Syi, Q ) , i ∈ [1, P ]. (9)
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Fig. 4.
IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 17, NO. 1, JANUARY 2018
Element spacing mapped by T 2 .
To guarantee the aperture, Sy should be reassigned when P i=1 ymi > U y. Sy = Sy × U y/
P
ymi .
Fig. 5.
Results of Mont Carlo simulations in Example A.
Fig. 6.
Layout of the best solution in Example A.
Fig. 7.
Results of Monte Carlo simulations in Example B.
(10)
i=1
As shown displayed in Fig. 4, the element spacing in ydirection can be expressed as Δdy1 + Sy1, j , i=1 Δyi,j = (11) Δdyt + ymi−1 − Syi−1, j + Syi, j , i ≥ 2 where Δdy ∈ RP ×1 has a similar structure with Δdx. The position matrix Y can be calculated by yi, j =
i
Δyt, j .
(12)
t=1
Set mapping functions (8)–(12) as Y = T 2 (B). Let X = T 2 (A) and Y = T 1 (B) when V x ≤ V y. Finally, place the element (xP ,Q , yP ,Q ) at (L, H) to satisfy the aperture constraint. To obtain a better performance of the sparse planar array, the fitness function can be defined as the sum of the maximum SLL in ϕ = 0 and ϕ = π/2 planes:
AF(θ, π/2) AF(θ, 0) . + max f (X, Y ) = max FFm ax FFm ax (13) This function can also be defined as the maximum SLL in all planes:
AF(θ, ϕ) (14) f (X, Y ) = max FFm ax where FFm ax is the peak of the main beam and the main beam is excluded from AF(θ, ϕ). In conclusion, an optimal model for the sparse planar array can be described as ⎧ f [T 1 (A), T 2 (B), W ] ⎨ min αi, j , βi, j , γi , ηj , wi,j ∈ [0, 1] ⎩ s.t. i ∈ [1, P ], j ∈ [1, Q]; xP , Q = L, yP , Q = H. (15) III. NUMERICAL RESULTS The sparse rectangular planar array with an aperture of 2L × 2H = 9.5 λ × 4.5 λ and the minimum element spacing dc = 0.5 λ are assumed. Two examples are studied to verify the effectiveness of the proposed method. In addition, 100 independent runs for each case are conducted.
The DE/best/1 strategy is applied to generate the donor individuals. All the parameters are set to be the same as those in [1] and [2]. A. Example 4N = 108 Equation (13) is selected as the fitness function. According to (2), we can calculate that P = 4 and Q = 9. Fig. 5 shows the final fitness obtained by the proposed method during 100 independent runs. The best fitness obtained by AMM is –61.454 dB (–34.597 dB in the ϕ = 0 plane and –26.143 dB in the ϕ = π/2 plane), which is lower than result using the matrix mapping method (–51.499 dB) [2] and MGA (–45.456 dB) [1]. Moreover, the mean fitness of the proposed method is –58.922 dB, which is also better than the best result of [1] and [2]. The deployment of the best solution is revealed in Fig. 6, and the corresponding coordinates are presented in Table I. B. Example 4N = 100 Equation (14) is set as the fitness function. The sizes of position matrices are P = 4 and Q = 9. Fig. 7 depicts the PSLL generated by the proposed method during 100 Mont Carlo simulations. The best result obtained by AMM is –21.886 dB in all plane, which is a 1.502 dB improvement compared with the best result (–20.384 dB) of matrix mapping method [2]. Moreover, the average PSLL of the 100 independent runs is –20.456 dB, which is also lower than the best results in [1] and [2]. The geometry of the best array is displayed in Fig. 8, and its corresponding coordinates are presented in Table II. According to Figs. 6 and 8, the array obtained by proposed method has more elements placed in the y-direction than those mapped by the matrix mapping method. It can be concluded that
DAI et al.: ASYMMETRIC MAPPING METHOD FOR THE SYNTHESIS OF SPARSE PLANAR ARRAYS
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TABLE I ELEMENTS POSITION COORDINATES OF THE BEST ARRAY OF EXAMPLE 4N = 108 Row 1 2 3 4
Column 1
Column 2
Column 3
Column 4
Column 5
Column 6
Column 7
Column 8
Column 9
0.250, 0.250 0.250, 0.750 0.250, 1.251 0.250, 1.751
0.750, 0.250 0.750, 0.751 0.750, 1.447 0.750, 2.069
1.250, 0.250 1.250, 0.750 1.250, 1.250 1.250, 1.750
1.750, 0.250 1.750, 0.788 1.750, 1.361 ——
2.250, 0.250 2.250, 0.753 2.257, 1.257 ——
2.757, 0.250 2.758, 0.751 2.802, 1.256 ——
3.302, 0.250 3.302, 0.750 —— ——
3.826, 0.250 3.826, 0.750 —— ——
4.326, 0.250 —— —— 4.750, 2.250
The first and second number denote x-, y -direction, in wavelength.
TABLE II ELEMENTS POSITION COORDINATES OF THE BEST ARRAY OF EXAMPLE 4N = 100 Row 1 2 3 4
Column 1
Column 2
Column 3
Column 4
Column 5
Column 6
Column 7
Column 8
Column 9
0.250, 0.250 0.250, 0.755 0.256, 1.518 ——
0.750, 0.251 0.750, 0.911 0.810, 1.444 0.758, 2.004
1.250, 0.257 1.251, 0.798 1.315, 1.431 ——
1.750, 0.253 1.752, 0.770 1.924, 1.409 ——
2.251, 0.255 2.305, 0.830 2.590, 1.654 ——
2.751, 0.374 2.905, 1.042 —— ——
3.273, 0.312 3.463, 0.889 3.645, 1.557 ——
4.002, 0.304 4.228, 1.069 —— ——
4.732, 0.349 —— —— 4.750, 2.250
The first and second number denote x-, y -direction, in wavelength.
Fig. 8.
Layout of the best solution in Example B.
the sizes of the position matrices play an important role in the optimization process. IV. CONCLUSION An AMM to synthesize sparse planar arrays with multiple constraints has been presented. The goal is to minimize the SLL with a given aperture and number of elements. In addition, the interelement spacing should not be less than the minimum element spacing to avoid mutual coupling effects. These constraints result in a complex and strong nonlinear optimization. The proposed method transforms the multiple constraints problem to an unconstrained problem by establishing two mapping functions. Then, the stochastic global optimization method, DE, is applied to search for the best solutions. According to the numerical results, the proposed method improves the performance of two given arrays by 7.37% and 19.33% over the matrix mapping method, respectively. REFERENCES [1] K. S. Chen, H. Chen, L. Wang, and H. Wu, “Synthesis of sparse planar array using modified real genetic algorithm,” IEEE Trans. Antennas Propag., vol. 55, no. 4, pp. 1067–1073, Apr. 2007. [2] H. Liu, H. Zhao, W. Li, and B. Liu, “Synthesis of sparse planar arrays using matrix mapping and differential evolution,” IEEE Antennas Wireless Propag. Lett., vol. 15, pp. 1905–1908, 2016.
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