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Fast synthesis of large planar arrays using active element pattern method and fine-grained parallel micro-genetic algorithm.

1. INTRODUCTION

Planar antenna arrays for mobile, remote sensing, satellite and radar communications systems can provide a good control of the radiation patterns. Typically, the elements of planar arrays are arranged in a rectangular, triangular grid or concentric circular ring. The performance of these systems is closely related with the antenna array design. To gain high performance of the planar array, many synthesis methods that permit the calculation of the array excitation distribution are applied [1-10]. In [1], a method based on invasive weed optimization (IWO) algorithm and iterative Fourier technique (IFT) was proposed for the synthesis of large planar thinned arrays. Recently, global optimization algorithms have shown great potential in the solution of planar array synthesis problems, such as genetic algorithm (GA) [2-4], particle swarm optimization [5,6], differential evolution strategy (DES) [7,8], and ant colony optimization (ACO) [9]. In [10], a PGAPack parallel GA was used for the synthesis of planar phased-array antennas that can take a significantly greater amount of time to converge to the desired solution. However, these approaches all neglect the mutual coupling effects.

Full-wave electromagnetic simulations based on the finite-difference time-domain (FDTD) method [11,12], finite element method (FEM) [13-15] and method of moment (MoM) [16-18] are applied to analyze the mutual coupling between antenna arrays. However, the calculation of most planar array antennas requires extremely long computation time, which makes it impossible to obtain the optimal excitation amplitudes and phases of array elements. Due to the computer memory limitations, full-wave methods are difficult to simulate the whole planar arrays, when the number of array elements is large.

Active element pattern (AEP) method which can rigorously include the effects of mutual coupling is an effective way to compute the far-field patterns of a fully excited array [19,20]. The AEP is defined as the antenna pattern of a single excited element in the array, and all other elements terminated in matched loads. The AEP of each element is different from each other. The difference is mainly due to the different relative position of each element and mutual couplings between elements. In [21], AEP method is used to design linear microstrip arrays with taylor synthesis method. The AEP method has been applied for planar antenna array with asymmetrical elements [22]. For a circular array, array beamforming is achieved based on the AEP method and GA [23]. The AEP method has been widely used for analyzing conformal antenna arrays [24-27]. At present, the AEP method has not been applied to the triangular grid planar array and multiple concentric circular ring array.

In this paper, the AEP method is used to calculate the patterns of rectangular grid, triangular grid and multiple concentric circular ring E-shaped patch antenna arrays. The AEPs are calculated by CST Microwave Studio. Through optimizing excitation amplitudes or phases based on the fine-grained parallel micro-genetic algorithm (FGPMGA), the desired radiation patterns are obtained. Our research shows that the FGPMGA significantly outperforms microgenetic algorithm (MGA), in terms of both the convergence rate and exploration ability [28].

This paper is organized as follows. Section 2 describes the FGPMGA, a linear array pattern synthesis with low side lobe and a wide null is provided to demonstrate the advantages of FGPMGA. Section 3 gives the formulations of AEP method, and the correctness of this method is validated by using CST Microwave Studio. In Section 4 the 256-element rectangular grid, 200-element triangular grid and 4-circle 60-element concentric circular ring E-shaped patch antenna arrays are optimized by FGPMGA. Conclusions are given in Section 5.

2. FINE-GRAINED PARALLEL MICRO-GENETIC ALGORITHM

The fine-grained parallel model is also called neighborhood model. Each processor is allocated only one individual and selects parents for recombination from local neighborhood by considering neighbors at different distances. For the FGPMGA, a 4-n neighborhood is used, as shown in Fig. 1. In this case, each node represents a single individual. The FGPMGA is implemented using C++ on a message passing interface (MPI) environment.

In each neighborhood, different crossover and mutation probability, as well as different methods of mutation and crossover. The procedure of the FGPMGA is described in [28].

A linear array with 2N elements is symmetrically placed on the X-axis, and the array factor for nonuniform excitation amplitude and uniform excitation phase can be written as

AF(I, [theta]) = 2 [N.summation over (n=1)] [I.sub.n] COS [(2N -1/2)kdsin[theta]] (1)

where, d (d = 0.5[lambda]) is the separation between the elements; In is the excitation amplitude of nth element; k is the propagation constant; [theta] is the elevation angle.

To validate the effectiveness of FGPMGA, we synthesize a 32-element (N = 16) linear array with a wide null located at 45[degrees] ~ 50[degrees]. 5 independent runs of the FGPMGA are repeated. The desired sidelobe level DSLL = -20 dB and the null depth level DNULL = -40 dB [29]. The population size and fitness function are the same settings as in [29].

Figure 2 shows the optimal excitation amplitudes optimized by FGPMGA. The average convergence rate of the parallel algorithm is shown in Fig. 3. The normalized radiation pattern is given in Fig. 4. Table 1 shows the comparative results obtained by FGPMGA and improved differential evolution (IDE) algorithm proposed in [29]. It can be seen that the SLL and null depth level found by FGPMGA are smaller than those obtained by the IDE, as well as FGPMGA needs less number of iterations.

3. ACTIVE ELEMENT PATTERN METHOD FOR PLANAR ANTENNA ARRAY

3.1. Rectangular Grid Planar Array

We construct a uniform 256-element rectangular grid planar E-shaped patch antenna array, as shown in Fig. 5. The distances between elements in the x and y direction are 45 mm (0.79 [lambda] at the operating frequency 5.25 GHz). The E-shaped patch antenna element was optimized in [30]. We change the size of ground to 40 mm x 40 mm. As the excitation phases are 0, the radiation pattern of xz-plane can be expressed using active element patterns as

E([theta], [phi] = 0) = [[N.sub.x].summation over (m=1)] [[N.sub.y].summation over (n=1)] [I.sub.mn] [E.sub.mn][([theta], [phi] = 0).sub.e][jkmd.sub.x] sin [theta] (2)

and yz-plane is

E([theta], [phi] = 90[degrees]) = [[N.sub.x].summation over (m=1)] [[N.sub.y].summation over (n=1)] [I.sub.mn] [E.sub.mn][([theta], [phi] = [90[degrees]).sub.e][jknd.sub.y] sin [theta] (3)

where, [d.sub.x] = [d.sub.y] = 45 mm are the inter-element spacings; [N.sub.x] = [N.sub.y] = 16 are the number of elements in x and y directions; k is the freespace propagation constant; [theta] denotes the azimuth angle; [I.sub.mn] are the excitation amplitudes for m = 1, ..., [N.sub.x] and n = 1, ..., [N.sub.y]; [E.sub.mn] are the active element patterns for m = 1, ..., [N.sub.x] and n = 1, ..., [N.sub.y].

The patterns of the principal planes can be deduced from two small Ns-element (Ns is odd and less than [N.sub.x], [N.sub.y]) linear arrays. The procedure can be described as follows [31]:

1. Calculate the AEPs of [N.sub.s]-element linear antenna arrays arranged along both the x and y axis. The AEPs of i-th element in x and y axis are set as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] respectively.

2. In X-axis, the n-th AEP [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of [N.sub.y]-element linear array is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

In y-axis, the n-th AEP [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of [N.sub.y]-element linear array is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

3. When [phi] = 0, the AEPs of the planar array in L-th line are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

when [phi] = 90[degrees], the AEPs of the planar array in C-th column are,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

4. Using Eqs. (2) and (3), we can obtain the radiation patterns in the principal planes.

The amplitude [I.sub.mn] is set as 1. As shown in Fig. 6, for [N.sub.s] = 5, the radiation patterns of the 256-element rectangular grid planar array in the xz and yz planes computing using Eqs. (2) and (3) are compared to the patterns full-wave simulated by CST. We can see that the results computed using AEP method are consistent with the CST simulations when the planar array is fully excited. The maximum gain is 32.9 dBi.

3.2. Triangular Grid Planar Array

A uniform triangular grid planar array composed of 200 E-shaped patch antennas (see Fig. 7). The excitation phases are set as 0. So the far field of xz-plane is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

and radiation pattern of yz-plane is expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

where, [d.sub.x] = [d.sub.y] = 45 mm, [N.sub.x] = [N.sub.y] = 10, and k is the free-space propagation constant; [theta], [E.sub.mn] and [I.sub.mn] are the same meanings as in Eqs. (2) and (3).

The analysis problem of triangular grid planar array can also be divided into small linear array problems just like rectangular grid planar array. Using two 5-element linear arrays, we can obtain all AEPs of this array. All excitation amplitudes are the same. From Eqs. (8) and (9), the radiation patterns of the principal planes have been computed, and the results given by the AEP method and CST simulation are shown in Fig. 8. It is shown that the results are in good agreement. The maximum gain is 31.3 dBi.

3.3. Multiple Concentric Circular Ring Array

A concentric circular ring E-shaped patch antenna array with four rings is shown in Fig. 9. The number of elements in the ring is 6, 12, 18, and 24, respectively. The far field pattern is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

where, [[phi].sub.a] = 2[pi](a-1)/6, [[phi].sub.b] = 2[pi](b-1)/12, [[phi].sub.c] = 2[pi](c-1)/18, [[phi].sub.d] = 2[pi](d-1)/24; [I.sub.1], [I.sub.2], [I.sub.3], [I.sub.4] are excitation amplitudes; [[phi].sub.1], [[phi].sub.2], [[phi].sub.3], [[phi].sub.4] are the excitation phases; [r.sub.1], [r.sub.2], [r.sub.3], [r.sub.4], are the radii of the four rings, and [r.sub.1] = 55 mm (0.96 [lambda]), [r.sub.2] = 110mm. [r.sub.3] = 165mm, [r.sub.4] = 220 mm.

We only need to calculate the AEPs of 10 elements in the dashed box (see Fig. 9). The others can be given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

where, [E.sub.m] ([theta], [phi]) is the m-th unknown AEP, [E.sub.i] ([theta], [phi]) is the i-th known AEP, and [[phi].sub.m] is the included angle between the m-th unknown element and i-th known element.

An analysis is given of the 4-ring array with excitation amplitudes equal to 1 and phases equal to 0. The radiation patterns are obtained from the Eq. (10). Fig. 10 shows the far field patterns by using the AEP method and CST simulation. Note that the results of the two methods are in good agreement.

4. OPTIMIZATION RESULTS OF PLANAR ANTENNA ARRAYS

The planar E-shaped patch antenna arrays are optimized by the FGPMGA on the Dawning 5000 computer system (belong to Institute of Applied Physics and Computational Mathematics).

4.1. Optimization of Rectangular Grid Planar Array

In the first example, a rectangular grid planar antenna array as shown in Fig. 5 is optimized. The excitation phases are set as 0. The optimized parameters are the excitation amplitude of each element and the objective function is to generate the minimum side lobe level (SLL) and keep the maximum gain same as the patterns in Section 2.1. The fitness function is given by

[F([bar.x]) = [w.sub.1] x SLL + [w.sub.2] x ([Gain.sub.max] - 32.9 dBi) (12)

where, [bar.x] = [I.sub.1,1], ..., [I.sub.16,16]; SLL indicates side lobe level, and [Gain.sub.max] is the maximum gain of planar array; [W.sub.1] and [W.sub.2] are weight factors. The population size is set to 400 and the total number of iterations is 2000.

Figure 11 shows the optimized patterns. It can be seen that the SLL of xz-plane is reduced from -13.25 dB to -20.53 dB, the SLL of yz-plane is reduced from -13. 35 dB to -18. 25 dB, and the maximum gain is 31.9 dBi. The optimization time by using the FGPMGA with 400 processors is about two hours in contrast to five days by using MGA on the personal computer.

4.2. Optimization of Triangular Grid Planar Array

In this example, amplitude-only pattern synthesis is used for triangular grid planar array (see Fig. 7). Our objectives are to meet the minimum SLL, and the same maximum gain as the array with equal amplitudes in Section 2.2. The fitness function is given by

F ([bar.x]) = [w.sub.1] x SLL + [w.sub.2] x ([Gain.sub.max] - 31.3 dBi) (13)

where, [bar.x] = [I.sub.1,1], ..., [I.sub.1,1]; SLL, [Gain.sub.max], [w.sub.1] and [w.sub.2] are the same meanings as in Eq. (12). The following FGPMGA parameter values are selected: the number of individuals and generations is 400 and 2000, respectively.

Shown in Fig. 12 is the optimized radiation patterns of the principal planes. It is found that the SLLs of xz-plane and yz-plane obtained by the FGPMGA are -15.94dB and -16.54 dB, respectively, in contrast to -12.98dB and -13.28dB of the array with equal amplitudes. The maximum gain is 30.5 dBi. The FGPMGA with 400 processors only needs 1.3 hours for computing, while MGA on the personal computer needs about three days.

4.3. Optimization of Multiple Concentric Circular Ring Array

In the last example, amplitude-phase pattern synthesis is used to make the power patterns of 4-ring array (see Fig. 9) be well matching with the desired patterns. The objective function is defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

where, [bar.x] = [I.sub.1], [I.sub.2], [I.sub.3], [I.sub.4], [[phi].sub.1], [[phi].sub.2], [[phi].sub.3], [[phi].sub.4], 0 [less than or equal to] [I.sub.i] [less than or equal to] 1, 0 [less than or equal to] [[phi].sub.i] [less than or equal to] 2[pi]; [omega].sub.1]= 10 and [w.sub.2] = 1 are the weight factors; [E.sub.max] is the maximum radiation electric field; [E.sub.main] ([theta], [phi]) indicates the radiation electric field of main lobe, and [E.sub.else]([theta], [phi]) indicates the radiation electric field in other regions. In the FGPMGA parameters, the initial population is set to 100, and the maximum number of iterations is 200.

First, we aim to produce a conical beam covering the sector 5[degrees] [less than or equal to] [absolute value of [theta]] [less than or equal to] 20[degrees]. The resulting beam patterns are shown in Fig. 13. The patterns of the principal planes are close to the desired pattern.

Next, a conical beam covering the sector 15[degrees] [less than or equal to] [absolute value of [theta]] [less than or equal to] 30[degrees] is generated. Fig. 14 shows the optimized results. It can be seen that the resulting beam pattern closely resembles to the desired pattern.

5. CONCLUSION

In this paper, we propose a technique based on AEP method and FGPMGA for large planar antenna array design. The AEP algorithm is used for computing the far field patterns of rectangular, triangular grid and multiple concentric circular ring planar arrays. The patterns obtained by the AEP method and CST Microwave Studio are in good agreements. The studies show that the AEP method is valid. For the three types of planar array synthesis as design examples, the FGPMGA achieves satisfactory results for planar arrays with many design variables to be determined, and effectively reduces the optimization time.

ACKNOWLEDGMENT

This work was supported by the National Basic Research Program of China (973 Program, Grant No. 2013CB328904), the NSAF of China (Grant No. 11076022), and the Research Fund of Key Laboratory of HPM Technology (2012-LHWJJ.006).

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Ling-Lu Chen (1), Cheng Liao (1), *, Lei Chang (1), Hai-Jing Zhou (2), and Han-Yu Li (2)

(1) Institute of Electromagnetics, Southwest Jiaotong University, Chengdu, Sichuan 610031, China

(2) Institute of Applied Physics and Computational Mathematics, Beijing 100094, China

Received 20 May 2013, Accepted 9 July 2013, Scheduled 17 July 2013

* Corresponding author: Cheng Liao (c.liao@swjtu.edu.cn).

Table 1. Comparisons of the optimized results for FGPMGA and
algorithm proposed in [29].

                                                   Number of
                       SLL      Null depth level   iterations

FGPMGA              -31.64 dB   less than -50 dB      150
algorithm in [29]    -22 dB     less than -40 dB   about 170
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Author:Chen, Ling-Lu; Liao, Cheng; Chang, Lei; Zhou, Hai-Jing; Li, Han-Yu
Publication:Progress In Electromagnetics Research C
Article Type:Abstract
Geographic Code:9CHIN
Date:Sep 1, 2013
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