Concentric Ring Array Synthesis Using Taguchi

Concentric Ring Array Synthesis Using Taguchi Algorithm for MIMO Applications Elies Ghayoula

1,2

Najib Fadlallah3

and Ammar Bouallegue1

1

Sys’Com Laboratory, National Engineering School of Tunis, ENIT, El Manar University,Tunis, Tunisia. 2 Department of Electrical and Computer Engineering, Laval University Quebec, G1V0A8, Canada. [email protected]

Equipe RADICOM, Institut Universitaire de Technologie de Saida Universit´e de Lebanon, P.O.Box 813 36, Lebanon.

Abstract—In this paper, we study an electromagnetic optimization technique using Taguchi’s method and apply it to concentric ring antenna array design. Taguchi’s method was developped on the basis of the orthogonal array (OA) concept, which offers systematic and efficient characteristics. The newly proposed idea is the implementation of Taguchi optimization method for Concentric Circular Antenna Array (CCAA). The optimization procedure is then used to provide an optimum set of weights for different CCAAs. Obtained results show that the desired radiation pattern with optimum sidelobe level (SLL) reduction is successfully achieved. The numerically simulated patterns are obtained and compared with those of concentric circular isotropic arrays (12, 18, 24, 30 and 36 elements). Compared to traditional optimization techniques and well-known algorithms (Evolutionary Programming (EP) algorithm and Firefly Algorithm (FA)), Taguchi’s method is easy to implement and efficient to reach the optimum solutions. Keywords—Array synthesis, concentric circular antenna array (CCAA), optimization method, sidelobe level (SLL), Taguchi method.

I.

Ridha Ghayoula2,4 and Amor Smida4

3

I NTRODUCTION

Concentric Circular Antenna Arrays (CCAA) has an interesting features over other array configurations such as linear one-dimensional or two-dimensional arrays [1]. Circular antenna arrays have considerable interest in a variety of applications including radar, sonar, and mobile and commercial satellite communication systems [2]. Concentric ring arrays have been optimized previously in literature for different purposes [1], [3]. A first metaheuristic approach towards the design of circular arrays is found in [4] where a real-coded genetic algorithm (GA) has been applied for designing circular arrays with maximal sidelobe level reduction coupled with the constraint of a fixed beam width. Shihab et al. in [5] applied the particle swarm optimization (PSO) algorithm to the same problem and achieved better results as compared to those reported in [4]. Recently, Panduro et al. [6] compared three powerful population based optimization algorithms PSO, GA, and differential evolution (DE) on the design problem of scanned circular arrays. Ovidio et al. in [7] exploits a deterministic two dimensional density taper approach for fast design of uniform amplitude pencil beams arrays. Biller et al. used steepest descent iterative process to find out element

4

Unit of Research in High Frequency Electronic Circuits and Systems, Faculty of Mathematical, Physical and Natural Sciences of Tunis, Tunis El Manar University – 2092, Tunis, Tunisia.

weights and ring spacing to get lower sidelobe levels and control over beam width [8]. Huebner reduced the sidelobe levels for small concentric ring array by adjusting the ring radii using optimization technique [9]. Kumar et al. also proposed optimum ring radii for generating lower sidelobes [10]. In [11] Hammami et al. compared sidelobe reduction of a planar array antenna by complex weight control using SQP algorithm and Tchebychev method. A number of methods have been proposed that define the excitation coefficients of an array antenna in order to shape the radiation pattern. The methods are usually based on beam steering or sidelobe suppression. We propose the use of Taguchi algorithm to determine the excitation coefficients needed yielded any desired beam pattern. A simulation, which assumed an 12, 18, 24, 30 and 36 elements with CCAA geometry. The paper is organized as follows: In section II, concentric circular antenna array CCAA geometry and mathematical formulation of the array factor is developed. A brief description of Taguchi algorithm is given in section III. Implementation procedure of Taguchi’s method and numerical results are provided and compared to the results obtained using other optimization methods in section IV. Validation using CST Microwave Studio for concentric ring arrays with 18 antennas (N1 = 2, N2 = 4 and N3 = 6) is presented in section IV. Finally, section V makes conclusion. II.

PROBLEM FORMULATION FOR CONCENTRIC RING ANTENNA ARRAYS SYNTHESIS

Fig. 1 shows the general configuration of CCAA with M concentric circular rings, where the mth (m = 1, 2, ..., M ) ring has a radius rm and the corresponding number of elements is Nm . Assuming that all the elements (in all the rings) are isotopic sources, then the radiation pattern of this array can be written in terms of its array factor only. For the concentric ring array with M rings and Nm elements in the corresponding mth ring, the array factor is given as [12], [13]: AF (φ, w)

=

M N m P P

wmn exp [j (krm sin θ (cos (φ − φmn ))

m=1 n=1

+ αmn )] (1)

Step1. Initializing the Problem: The optimization procedure begins with the problem initialization, which includes the selection of a proper OA, and an appropriate design of the fitness function. The selection of an OA depends on the number of input parameters of the optimization problem, and the number of levels of each parameter. Usually three levels are necessary for each input parameter to describe the nonlinear effect.

Fig. 1: Multiple concentric circular ring arrays of isotropic antennas in x-y plane.

where k is the wave number, λ is the signal wavelength, dm rm = Nm is the radius of the mth ring, dm is the inter 2π is element arc spacing of the mth ring, φmn = 2π (n−1) Nm the angular position of the nth element of the mth ring, wmn is the current excitation of the nth element of the mth ring, φ and θ are the azimuth and zenith angle respectively, αmn = −krm cos (φ0 − φmn ) is the residual phase and φ0 is the value of φ where main beam is to be directed. From equation (1), we can conclude that three parameters are controlling the AF are the amplitudes, the phases, and the positions of the elements. In the next section, Taguchi’s optimization method is used to design concentric circular arrays (CCA) by optimizing these parameters individually [17], [18]. Normalized absolute power pattern P (φ, w) in dB can be expressed as follows [1]:  P (φ, w) = 20 log10

AF (φ, w) AF (φ, w)M ax

 (2)

The objective is to improve radiation pattern performance of this concentric ring antennas arrays in (1) by optimizing weigth of amplitude (αn ) using Taguchi optimization algorithm. Taguchi’s method [19], [20], [21] was developed based on the concept of the orthogonal array (OA) [22], which can effectively reduce the number of tests required in a design procedure. Orthogonal arrays (OAs) play an essential role in Taguchi’s method. They provide an efficient and systematic way to determine control parameters so that the optimal result can be found with only a few experimental runs. Taguchi method provides an efficient way to choose the design parameters in an optimization problem. This paper shows that the proposed method is straight forward and easy to implement and can quickly converge to the optimum designs. III.

TAGUCHI ’ S OPTIMIZATION ALGORITHM

Taguchi’s optimization method will be briefly described here. The interested reader may consult [14] for more details. The steps taken in Taguchi’s optimization can be summarized as follows:

Step2. Designing Input Parameters Using an OA: In this step, the input parameters are selected to guide the experiments (i.e., the evaluation of the fitness function). For an OA with s = 3, the value of level 2 for each parameter is chosen at the center of the optimization range corresponding to that parameter. Then, the values of the other levels (1 and 3) are respectively evaluated by subtracting and adding a specific level difference (LD) to the value of level 2. The equation which determines the level difference in the first iteration is taken as [21]: (max − min) (3) s+1 where max and min are the upper and lower bounds of the optimization range, respectively. LD1 =

Step3. Conducting Experiments: After converting the OA entries to proper input values, the fitness function for each experiment can be calculated analytically or through numerical simulations. The fitness value is used to calculate the corresponding S/N ratio () in Taguchi’s method through the following formula: η = −20 log(F itness ) (dB)

(4)

After conducting all the experiments and finding the fitness values and the corresponding S/N ratio, a response table is built by averaging the S/N for each parameter n and level m using: X 1 ηi (5) η¯(m, n) = N i,OA(i,n)=m

Step4. Identifying Optimal Level Values and Conducting Confirmation Experiment: When the response table is established, the optimal level for each parameter can be identified by finding the largest (S/N) ratio. Next, a confirmation experiment is carried out by
using the combination of the optimal level values wn |opt . i Step5. Reducing the Optimization Range: If the termination criteria are not satisfied, the optimal level for the current iteration will be the center of the next iteration. wn 2i+1 = wn opt (6) i Also, in equation (7) the optimization range for the next iteration is minimized by multiplying the current level difference by the reducing rate (rr). rr can be set between 0.5 and 1 according to the problem. So, for the (i + 1)th iteration. LDi+1 = RR(i) × LD1 = rri × LD1

(7)

i

where RR(i) = rr is called the reduced function. Step6. Checking the Termination Criteria: Each time the number of iterations increases, the LD of each element

decreases. So, the level values are near to each other, and the fitness value of next iteration is close to the fitness value of the current iteration. The next equation can be used as a termination criterion for the optimization procedure [23]. LDi+1 ≤ converged value LD1

(8)

Usually, the converged value can be set between 0.001 to 0.01 depending on the problem. If the design targets are achieved or equation (8) is satisfied, the optimization process will finish. Finally, the aforementioned steps are repeated until a specific termination criterion is achieved or a specific number of iterations are reached. Taguchi’s method is used in the synthesis of circular antenna array to minimize the maximum SLL by controlling only amplitude parameter. In the next section more details and results are given regarding the application of Taguchi algorithm. IV.

I MPLEMENTATION PROCEDURE OF TAGUCHI ’ S METHOD FOR CONCENTRIC RING ANTENNA ARRAY

This section addresses the application of the proposed Taguchi algorithm to different concentric ring array designs CCAA. The number of rings, their radius and the number of elements in each ring are some of parameters under designer’s control that need to be optimized. Each CCAA maintains a fixed optimal inter-element spacing between the elements in each ring. The limits of the radius of a particular ring of CCAA are decided by the product of number of elements in the ring and the inequality constraint for the inter-element spacing d . For all the cases, 0◦ is considered so that the peak of the main lobe starts from the origin. Taguchi’s optimization method will be applied on 12, 18, 24, 30 and 36 elements CCAA (N1 = 2, N2 = 4 and N3 = 6), (N1 = 4, N2 = 6 and N3 = 8), (N1 = 6, N2 = 8 and N3 = 10), (N1 = 8, N2 = 10 and N3 = 12) and CCAA (N1 = 6, N2 = 8, N3 = 10 and N4 = 12). A. Example 1: CCAA with (N1 = 2, N2 = 4 and N3 = 6) To minimize the levels of the secondary lobes, the fitness function is selected according to the optimization goal: f itness = min (max {20 log |AF (θ)|})

(9)

In equation (1), there are six parameters that should be optimized in this case. Thus, the orthogonal experimental design selected must have five columns (k = 6)to represent these parameters. To characterize the non-linear effect, three levels (s = 3) are considered adequate for each parameter. Usually, an orthogonal experimental design with 2 strength (t = 2) is effective for most problems. In summary, an orthogonal experimental design with 6 columns, 3 levels, and 2 strength is required. Having access to the online database OA [22], an orthogonal experimental design (orthogonal arrays), OA (27, 6, 3, 2) is available. The objective function (fitness) (9) is selected depending on the optimization goal. The input parameters should be selected for experiments . When the orthogonal experimental design is selected, the numerical values

TABLE I: The OA(27, 10, 3, 2), Fitness values, and S/N ratios in the first iteration. Experiences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

w1 0.25 0.5 0.75 0.25 0.5 0.75 0.25 0.5 0.75 0.25 0.5 0.75 0.25 0.5 0.75 0.25 0.5 0.75 0.25 0.5 0.75 0.25 0.5 0.75 0.25 0.5 0.75

w2 0.25 0.25 0.25 0.5 0.5 0.5 0.75 0.75 0.75 0.25 0.25 0.25 0.5 0.5 0.5 0.75 0.75 0.75 0.25 0.25 0.25 0.5 0.5 0.5 0.75 0.75 0.75

Elements w3 w4 0.25 0.25 0.5 0.5 0.75 0.75 0.25 0.5 0.5 0.75 0.75 0.25 0.25 0.75 0.5 0.25 0.75 0.5 0.5 0.25 0.75 0.5 0.25 0.75 0.5 0.5 0.75 0.75 0.25 0.25 0.5 0.75 0.75 0.25 0.25 0.5 0.75 0.25 0.25 0.5 0.5 0.75 0.75 0.5 0.25 0.75 0.5 0.25 0.75 0.75 0.25 0.25 0.5 0.5

Fitness R(S/N)dB w5 0.25 0.5 0.75 0.5 0.75 0.25 0.75 0.25 0.5 0.5 0.75 0.25 0.75 0.25 0.5 0.25 0.5 0.75 0.75 0.25 0.5 0.25 0.5 0.75 0.5 0.75 0.25

w6 0.25 0.75 0.5 0.5 0.25 0.75 0.75 0.5 0.25 0.5 0.25 0.75 0.75 0.5 0.25 0.25 0.75 0.5 0.75 0.5 0.25 0.25 0.75 0.5 0.5 0.25 0.75

-10.087 -10.055 -15.006 -7.616 -12.581 -9.948 -6.737 -7.785 -12.268 -9.706 -15.412 -6.776 -9.284 -13.083 -7.247 -10.44 -9.420 -6.497 -6.399 -7.859 13.200 -13.200 -6.993 -6.993 -6.993 -6.993 -6.993

-20.076 -20.047 -23.525 -17.635 -21.994 -19.955 -16.569 -17.825 -21.775 -19.741 -23.757 -16.620 -19.355 -22.334 -17.203 -20.375 -19.481 -16.254 -16.122 -17.907 -22.411 -26.022 -16.894 -19.027 -22.212 -16.632 16.8536

corresponding to the three levels of each input parameter must be determined in the first iteration. The value for level 2 is selected in the middle of the range optimization. The values of level 1 and 3, are calculated after that. After determining the input parameters, the objective function (fitness) for each experiment can be calculated. For example, the fitness value for the experience number 1 (the first line of Table I) is calculated using equation (9), the result is −10.087 . Then the value of fitness using Taguchi method is converted into a signal to noise ratio (S/N ), denoted (η) using the formula ( 4). The corresponding fitness values and ratios (S/N ) are listed in Table I. These results are then used to construct a response table (Table II) of the first iteration , by making the average of the ratios (S/N ) for each parameter through equation (5). To identify the optimal value for each parameter, we have to find the greatest ratio (S/N ) in each column of Table II; as it is indicated with a yellow color in Table III . When the optimal levels are identified, a confirmation test is performed using the corresponding numerical values for optimal levels identified in the Table III. If the termination criteria are not satisfied; which are examined using equation (8), and after determining the values of optimal levels of the current iteration that are used as central values for the next iteration (6) (case in our values). The

process optimization is repeated in the next iteration with an optimization interval reduction using equation (7). TABLE II: Response table in the first iteration .

Level 1 Level 2 Level 3

1 -19,79 -19,65 -19,29

2 -20,02 -20,04 -18,66

Elements 3 4 -17,31 -18,45 -19,73 -19,95 -21,68 -20,32

5 -19.77 -19,71 -19,24

6 -21,13 -19,60 -17,98

TABLE III: Optimized excitation magnitudes of the CCAA (N1 = 2, N2 = 4 and N3 = 6).

(a) SLL reduction for CCAA with 12, 18, 24, 30 and 36 elements.

Elements 1 2 3 4 5 6 Optimized 1.0000 0.6303 0.9210 0.9818 0.6175 0.2294 excitations

In the example 1, Taguchi’s optimization method was applied on 12 elements of Concentric Circular Antenna Array CCAA (N1 = 2, N2 = 4 and N3 = 6). Table III (amplitude values) hold the optimum values of the amplitude obtained using Taguchi’s method (after 60 iterations) with rr = 0.7. Pattern synthesis calculations were performed using MATLAB R2014a on a desktop PC with a Core i7 processor running at 3.2 GHz. (b) Convergence of Fitness function for 10 elements.

Fig. 3: Simulation results for CCAA with 12, 18, 24, 30 and 36 elements.

TABLE IV: Comparison results of CCAA.

Fig. 2: SLL reduction for CCAA with 12 elements (N1 = 2, N2 = 4 and N3 = 6). The minimum Fitness values are plotted against the number of iteration cycles to get the convergence profiles as shown in Fig. 3(b). The Taguchi technique yield convergence to the minimum SLL in less than 40 iterations. To overcome the above mentioned contrasting results of parameters (Table IV), this paper proposes a modified concentric circular array configuration with number of rings M = 4. As shown in Table IV , it is evident that by varying the number of elements in each ring rather than increasing the number of rings have shown improvements in performance in terms of number of sidelobes, reduced HPBW and desired main lobe

CCAA synthesis HPBW Main lobe to using Taguchi Method (deg.) sidelobe level (dB) (N1 = 2, N2 = 4 and N3 = 6 ) 14.5 -11.1418 (N1 = 4, N2 = 6 and N3 = 8 ) 12.4 -17.6870 (N1 = 6, N2 = 8 and N3 = 10 ) 11 -21.2004 (N1 = 8, N2 = 10 and N3 = 12) 9 -25.1911 (N1 = 6, N2 = 8, N3 = 10 and N4 = 12 ) 8.3 -27.5831

to sidelobe ratio. For the result, as shown in Table IV, there are 3 and 4 rings in each case. The number of elements, however, varies in each case. The first row of Table IV shows number of elements 2, 4 and 6 for 3 rings respectively. Similarly final row shows the elements 6, 8, 10 and 12. It can seen that for CCAA, the HPBW is reduced to 8.3 with 36 antennas. V.

VALIDATION USING CST M ICROWAVE S TUDIO FOR CONCENTRIC RING ANTENNNA ARRAY

The proposed antenna shown in Fig. 4 is printed on a plexiglass substrate with relative permittivity of 2.5, loss tangent of 0.02, and thickness of 4mm for IEEE 802.11 MIMO

application [24]. The overall dimensions are L = 36 mm and W = 36 mm for 2.45 GHz.

TABLE V: Optimum amplitude values found by different algorithm of CCAA with (N1 = 4, N2 = 6 and N3 = 8). CCAA

N1

N2

Fig. 4: Geometry of the proposed antenna. It was found that the antenna resonates in the desired frequency band as shown in Fig. 5(a). Indeed, for |S11| < −10 dB, we have band ranges from 1.5 to 3.5 GHz with a resonant frequency 2.45 GHz. The 3D radiation pattern of our antenna is given by Fig. 5(b)The bandwidth is 105 MHz which is used for WLANs based on IEEE 802.11 MIMO applications.

N3

Number of elements 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Uniform 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

EP [16] 0.3416 0.0496 0.3242 0.0283 0.5321 0.2114 0.1923 0.4901 0.1876 0.1994 0.1204 0.2555 0.3527 0.2450 0.1229 0.2294 0.3449 0.2400

FA [15] 0.7025 0.1410 0.6770 0.1215 0.9999 0.4349 0.4084 0.9999 0.4076 0.4305 0.2352 0.4789 0.7366 0.4831 0.2542 0.4790 0.7172 0.4730

Taguchi 0.5417 0.4623 0.5417 0.4623 0.4931 1.0000 0.8540 0.4931 1.0000 0.8540 0.5184 0.8743 0.5000 0.1893 0.5184 0.8743 0.5000 0.1893

show that the synthesized pattern of CCAA with Taguchi method is better than those presented in comparison in terms of sidelobe level and main lobe directivity. VI. (a) Reflection coefficient S11.

(b) 3D Radiation pattern at 2.45 GHz.

Fig. 5: Simulation results of the proposed patch antenna. The optimal design of an excited CCAA with uniform inter-element spacing and without central element feeding has been described using optimization techniques as Taguchi method, Firefly Algorithm FA and Evolutionary Programming technique EP. Taguchi technique proves to be faster and robust technique; yields optimal excitations and global maximum directivity for all sets of CCAA designs. Five examples are presented in the paper for comparison. It is evident from the results that use of Taguchi amplitude excitations in the CCAA reduces the SLL more effectively. A. CCAA with (N1 = 4, N2 = 6 and N3 = 8) Fig.6(a) shows the structure of our concentric antenna array, using CST and excitations values shown in the Table V, The obtained optimized excitations by Taguchi was compared to that obtained using other well-known optimization techniques (FA, EP and Uniform) we have as a result Figure 6(b), 6(c), 6(d) and Figure 6(e). We can conclude that Using Taguchi method improve the synthesis of radiation patterns more than other algorithm. Obtained results by Taguchi optimization for different structures of CCAA are shown in Table Vand its comparison to other well-known optimization techniques (FA, EP and Uniform). Using CST Microwave Studio, different results clearly

C ONCLUSION

In this paper, a global optimization technique based on Taguchi’s method is used for designing and synthesizing a three and four-ring of concentric circular antenna array (CCAA). The implementation procedure is described in detail, and five CCAA examples are discussed to demonstrate its validity. Optimized results show that SLLs is reduced to more than −27.5 dB with narrow beamwidth about 8◦ for CCAA with 36 antennas. The optimized array factor was compared to that obtained using other well-known optimization techniques (Uniform, EP and FA). Array factor patterns obtained from Taguchi results outperform those presented in the literature. CST Microwave Studio is used to design and validate this results for CAA with 18 patch antennas that resonate at 2.45 Ghz for IEEE 802.11 MIMO applications. Results clearly show a very good agreement between the desired and synthesized specifications. R EFERENCES [1]

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(a) Array structure CCAA (N1 = 4, N2 = 6 and N3 = 8 ).

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(d) Firefly Algorithm (FA).

(c) Evolutionary Programming (EP) algorithm.

(e) Taguchi algorithm.

Fig. 6: 3D radiation pattern synthesis for CCAA (N1 = 4, N2 = 6 and N3 = 8 ) using different algorithm at 2.45 GHz.

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