Synthesis of Non-Uniformly Spaced Linear Array of ...

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Synthesis of Non-Uniformly Spaced Linear Array of Unequal Length Parallel Dipole Antennas for Impedance Matching using QPSO Hemant Patidar*1, Gautam Kumar Mahanti2 and R Muralidharan3 1,2

Department of Electronics and Communication Engineering, National Institute of Technology, Durgapur, India Email: [email protected], [email protected] 3 Department of Electrical and Computer Engineering, Caledonian College of Engineering, Sultanate of Oman Email: [email protected]

Abstract- This work presents a method based on quantum particle swarm optimization (QPSO) for the synthesis of a linear array of non-uniformly spaced parallel unequal length very thin dipole antennas for impedance matching of all the antenna elements in the array. This technique is used along with the generation of the radiation pattern with low side lobe level and main lobe tilting including uniform null filling (NF) which is generally useful for broadcasting applications. Mutual coupling effect exists between the parallel dipole antennas and it is analyzed using induced electro-motive force (EMF) method, assuming the distribution of the current on each and every dipole to be sinusoidal. To obtain the desired results, current excitation, length of the antenna and spacing between the antennas are taken as the optimizing parameters and QPSO as the optimizing algorithm. Two examples are presented to show the effectiveness of the proposed method. The performance of this algorithm for impedance matching is validated by duly comparing it with a standard benchmark algorithm. Index Terms- Antennas array, FEKO, Induced EMF method, Quantum particle swarm optimization, main lobe tilting, null filling, Side lobe level.

I. INTRODUCTION Linear antenna arrays finds themselves as one of the most vital components in communications applications, in which they are used to generate sufficient narrow beam patterns as per the requirements. The shape of these patterns can be changed by altering the geometrical

configuration and antenna parameters like inter element spacing between the elements, excitation amplitude and phase, relative pattern of the individual elements [1]. Recent researches [2-3] on antenna arrays in the area of broadcasting are very much interesting. Many researchers have developed methods for the design of necessary base station antenna arrays in order to meet the essential demands [4-9]. Few of the demands which are generally fulfilled by a broadcasting antenna array are: (1) since transmitting antennas are usually located quite far away from the usage area, it becomes necessary to generate a very narrow beam. (2) When the broadcasting base stations are in high altitudes compared to the service area, the main lobe needs to be tilted from the horizontal plane. (3) Impedance matching condition becomes necessary for the purpose of reducing the power, etc. Moreover consideration of the mutual coupling constraints for a nonuniformly spaced linear array is quite a difficult task. Evolutionary algorithms have given considerable support in the past in the generation of these sort of requirements. Non-uniformly spaced linear antenna array synthesis using Taguchi initialized invasive weed optimization for broadcasting application is presented in [9]. A method for side lobe level control for nonuniformly spaced linear array with coupling considerations is stated in [10]. Non-uniformly spaced linear antenna array design using firefly Algorithm is also detailed in [11].

Still, the urge for better solutions is always a demand. Therefore, a powerful optimization

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technique is required to solve these kind of problems dealing with unequal height antenna elements. In this paper, we propose a technique based on quantum particle swarm optimization algorithm (QPSO) [12-13] for optimizing the excitations and geometry of the individual array elements. Coupling effect is compensated by minimizing the real and imaginary parts of the input impedance of the antenna elements of the linear array to the value near to the specified value. The reason for the choice of QPSO is that the lliterature reports [14-16] the validity of QPSO and its different versions over other algorithms. This paper also presents a validation of obtained results from simulation using FEKO Software. FEKO [17] is a comprehensive electromagnetic simulation software tool. It is used for the electromagnetic field analysis of 3D structures. The software is based on the Method of Moments (MoM). Moreover, this algorithm is compared with a well-known algorithm in the past, namely Particle swarm optimization (PSO) [18-19], which has been widely found in the field of antenna arrays.

FP( , )  AF(  , )  ELP( , ) 

 N  ...(1)   2In cos[kdn sin( )(sin( )  sin(0 ))  ELP( , ) n  1 

where n=element number, dn= distance from origin to center of the n th dipole, k=2π/λ = the wave number, λ=wavelength, θ is the polar angle of far-field measured from z-axis (-90o to +90o),  is the tilting angle, ϕ is the azimuth angle 0

measured from x-axis (for vertical plane ϕ=90o), In = amplitude of the excitation current of the n th element, N is the total number of elements from one side of the origin. AF(θ,ϕ) is the array factor. ELP(θ, ϕ) is element pattern of each x-directed horizontal thin dipole antenna. The element pattern of the horizontal dipole antenna is given below considering ϕ=90 o for vertical plane:  kl sin( ) cos( )   kl  cos n   cos n  2    2  ELP(  , )  2 2 ( 1  sin( ) cos( ) )

……. (2)

II. BASIC THEORY AND ANALYSIS A linear array of 2N very thin wire dipole antennas is considered along y-axis. All the dipoles are placed parallel to x-axis and are assumed non-identical. The far-field pattern in the vertical (y-z) plane relies not only on the geometry of the array but also on the excitation currents applied at the center of the dipoles. The geometry represents the lengths of the antenna elements and the distances from origin to the center of dipoles, while the excitations represents the current amplitudes applied to the array elements by proper feeding network. Elements of the array are located symmetrically on each side of the origin. The geometry of the array is given by the lengths l n (n = 1… N) of the dipoles and the distance from origin to center of dipole d n (n = 1,. . . ,N). The free-space far-field pattern [1] FP(θ,ϕ) in the vertical plane (y-z) with symmetric amplitude distributions is given by equation (1).

Fig.1. Geometry of linear array of x-directed parallel unequal length dipole antennas along y-axis.

The voltage distribution matrix of size (1N) on the antenna is obtained by [1]: V=IZ where I  I n e

 jkd sin( o )

…….. (3)

, I is the current matrix

of size (1 N) applied to the dipole antennas and

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Z is the mutual coupling impedance matrix of size (N×N). Self-impedances and mutual impedances of Z are calculated by induced electro-motive force (EMF) method [1], which assumes the current distribution on the dipoles to be sinusoidal.

The value of mutual coupling matrix Z depends on the geometry of the dipoles as well as the distance between them. The integration is solved using 16-point Gauss-Legendre quadrature integration formula. The voltage across the nth dipole [1] is obtained by:

Vn  Z nn I n   Z nm I m

…..(4)

m n

where Znn is the self-impedance of dipole n and Znm is the mutual impedance between dipoles n and m. The input impedance [1] of dipole n, ZnA is given by:

Z nA  V n / I n  Z nn 

Z

nm ( I m

/ I n ) ….(5)

 IZA0  IZA d , if  IZA0  IZA d  F2   0 d  0 , if  IZA  IZA 

......(8)

 SL  SL d , if  SL o  SL d F3   0  0 , if  SL o  SL d

......(9)

 NFo  NFd , if  NFo  NFd F4   0, if  NFo  NFd

…..(10)

The coefficients wet1, wet2, wet3 and wet4 are the relative weights applied to each term in (6), where RZA o and RZA d are the obtained and desired values of real part of the input impedance, IZA o and IZA d are the obtained and desired values of imaginary part of the input impedance, SL0 and SLd are the obtained and desired values of side lobe level, NFo and NFd are the obtained and desired values of maximum null filling respectively. mse refers to the mean squared error.

mn

III. QUANTUM PARTICLE SWARM OPTIMIZATION

In the end, the real and imaginary parts of the input impedance is calculated for all the elements. The aim is now to obtain the set of current excitations, spacing between the elements and the length of the antenna elements using QPSO that helps in minimizing the following cost function to generate the free space radiation pattern with low side lobe level as well as main lobe tilting including uniform null filling for impedance matching.

QPSO detailed in [12-13] is proposed by Sun et al in 2004 and is based on the elemental theory of particle swarm and rules of quantum mechanics. Here all the particles have the features of quantum deportment. Moreover, researches in the past shows that global optimization performance of QPSO is superior to that of the standard PSO algorithm.

 wet1  (mse( F1 )) 0.5  wet 2  (mse( F2 )) 0.5 Cost   2 2  wet 3  F3  wet 4  F4

   (6)

where  RZA 0  RZA d , if  RZA 0  RZA d  ….(7) F1   0 d  0 , if  RZA  RZA 

The protocol dealing with the movement of the particles in QPSO is quite different from the particles in standard PSO. In accordance with the unpredictability theory of quantum world, particle becomes visible at any position of search space with a certain probability resulting in the fact that the position and velocity of that particle cannot be found at the same instant. Different steps associated with QPSO are detailed below: 1: Generate initial population of particles randomly between the minimum and the

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maximum operating limits in the D-dimensional space. 2: Estimate the fitness value of every particle. 3: Present fitness value of the particle is compared with the personal best (pbest) of every particle. If the present fitness value of the particle is superior, then allocate the present fitness value to pbest and allocate the present coordinates to pbest coordinates. 4: Compute the mean best position ( meanbest ) of all W particles using the equation written below: meanbest 

1 W  pbesti W i 1

particles from detonation if they try to come out of the required domain of interest.

….(11)

5: In the overall population, determine the present best fitness and its coordinates. If the present best fitness value is superior to global best (gbest), then allocate the present best fitness value to gbest and allocate the current coordinates to gbest coordinates. 6: Compute the vector local focus of the particle:

 





it  rnd 1it  pbest  1  rnd 1it  gbest  VLfid id id id

….(12) 7: Update position (Posid) of the dth dimension of the ith particle utilizing the following equations: it it  VLf it  (  1 ) ceil ( 0 . 5  rnd 2id )   Pos id id it  1  log ( 1 / rnd 3 it )  meanbest  Pos id e id …..(13) it  Pos d If Pos id min

Then



it  Pos d  0.25  rnd 4 it  Pos d d Pos id max  Pos min min id



(14) it d If Pos id  Pos max

Then it  Pos d  0.25  rnd 5it  ( Pos d  Pos d ) Posid max max min id

(15) where rnd1, rnd2, rnd3, rnd4 and rnd5 are uniform random numbers between 0 and 1. The term it refers to the current iteration. Equations (14) and (15) are applied along in each dimension d , Pos d within Pos max min to clamp the position. These techniques are necessary to stop the





Fig.2. Flow chart of QPSO algorithm.

8: Repeat steps from 2 to 7 till the maximum number of iterations being completed. The parameter VLf is the local attractor of each particle. To avoid untimely convergence meanbest is considered as the barycenter of all particles. It contains only one control parameter called contraction and expansion coefficient (α), which can be tuned easily by trial and error method to control the convergence speed of the algorithm. Setting the value of α in the interval (0.5, 0.8), can generates good results, see literature report [16]. In this case, the value of α is given by:

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  (  max   min ) 

(max gen  it ) (max gen  1 )

  min

where αmax =0.76 and α min =0.73 are the maximum and minimum values of α, maxgen is the maximum number of iteration (generation) and it is the current generation. Flow chart of the algorithm is detailed in Fig. 2 IV. RESULTS A. Simulation A linear array of 14 (Example 1) and 20 (Example 2) dipole antennas of radius 0.003λ are considered along y-axis. For both the examples: To generate a far-field pattern in the vertical plane (y-z plane), excitation current amplitude, spacing between the elements and the length of the each antenna element are varied in the range of 0 to 1, 0.4λ to 1.1λ and 0.4λ to 0.6λ respectively. The far-field pattern is generated by considering the excitations and geometry symmetric from the center of the array.

Example 1: For the generation of far-field pattern in vertical plane with main lobe tilting at 3°, QPSO and PSO are run for five times each with 500 iterations with a particle size of 40. Due to symmetry, only seven current amplitudes, seven spacing between the elements and seven antenna heights are to be optimized. Algorithms have been utilized to generate a vector of 21 values, first 7 values to obtain the spacing between the elements; next 7 values for antenna length and the last 7 values for current amplitudes. Algorithms generate one best scoring individual in each run. A global best individual is regarded as best among such five best scoring individuals (best fitness values). The other design settings for parameter values are obtained from Table 1.

Example 2: Similar to the above, to obtain the design variables and generation of far-field pattern with main lobe tilting at 5°, QPSO and

PSO are run for five times each with 500 iterations with a population size of 40. As done in example 1, a vector of 30 real values is obtained using QPSO and PSO.

The other design setting parameter values are obtained from Table 1. It is assumed that wet1=12, wet2=3 (to provide more importance to real and imaginary impedances than SLL and NF), wet3 =1 and wet4 =1 for example 1 and wet1=11, wet2=3 (to provide more importance to real and imaginary impedance than SLL and NF), wet3 =1 and wet4 =1 for example 2. Table 1. Settings for the QPSO and PSO algorithms

Design Parameters Population size Generation Total number of Runs α decreasing linearly Acceleration coefficients (C1 & C2) C1 decreasing linearly with iterations C2 increasing linearly with iterations Inertia weight (W) Changing randomly between

B.

Example 1 (14 Elements) QPSO PSO 40 40

Example 2 (20 Elements) QPSO PSO 40 40

500 5

500 5

500 5

500 5

from 0.76 to 0.73 —

—

from 0.76 to 0.73 —

—

from 2.5 to 1

from 1 to 2.5

—

0.4 to 0.9 with iterations

from 2.5 to 1

from 1 to 2.5

—

0.4 to 0.9 with iteration s

FEKO Assessment

1. Build the antenna array geometry in CADFEKO and build geometry to describe surrounding geometry in CADFEKO. 2. Meshing of designed antenna array and the surrounding geometries.

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3. Request for types of solution and setting solution parameters, run the FEKO solver, read in and illustrate results using PostFEKO. All the steps are detailed in FEKO tutorial [17]. Voltage excitation on the antenna elements is calculated from simulation by equation (3). Here we consider voltage excitations, length of antenna elements and distance from the origin to the center of antenna elements obtained from simulation using Matlab as excitation voltage, length and distance of antenna elements from origin. Then the far-field pattern is generated in PostFEKO.

normalized power pattern in dB obtained from QPSO for example1, example 2; pattern obtained from FEKO analysis using QPSO for example 1 and example 2. Fig.10, Fig.11, Fig. 12 and Fig. 13 shows the normalized power pattern in dB obtained from PSO for example1, example 2; pattern obtained from FEKO analysis using PSO for example 1 and 2. Obtained results using QPSO is better than PSO in terms of the antenna parameters like side lobe level, null filling level and impedance matching as well as computational time. Table 2 gives the complete comparative details about the results obtained using simulations.

Fig.3. Constructing geometry of linear array of xdirected non-uniformly spaced unequal length centerfed dipole antennas along y-axis on CADFEKO.

Fig.4. Best fitness value of five runs versus iteration number obtained from QPSO. 800

Best Fitness value

The program is written in Matlab. Computational time is measured here using a PC with Intel(R) Core (TM) i5-4690 processor of clock frequency 3.50 GHz and 4 GB of RAM. Table 2 shows the obtained and desired results from simulation using QPSO, PSO and FEKO analysis. Tables 3 and 4 shows the excitation current amplitudes, length of the each antenna element and the distance from the origin to the center of the antenna elements obtained from QPSO and PSO for example 1 and 2. Tables 5 and 6 shows the real and imaginary values of input impedance of all the elements obtained from simulations and FEKO analysis for example 1 and 2. Fig. 4 and Fig.5 shows the best fitness value of five runs versus iteration number using QPSO and PSO. Fig.6, Fig.7, Fig.8 and Fig. 9 shows the

Example1 Example2

700 600 500 400 300 200 100

0

100

200

300

400

500

Number of Iterations

Fig.5. Best fitness value of five runs versus iteration number obtained from PSO.

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Normalized Power in dB

0 Example1

-10 -20 -30 -40 -50 -60 -70 -90 -70 -50 -30 -10

10

30

50

70

90

 in degrees

Fig.6. Normalized power pattern in dB at 3 ° tilt angle obtained from QPSO.

Fig.9. Normalized power pattern in dB at 5 ° tilt angle obtained from FEKO using QPSO.

Normalized Power in dB

0 Example1 -10 -20 -30 -40 -50 -60 -70 -90 -70 -50 -30 -10

10

30

50

70

90

 in degrees

Fig.7. Normalized power pattern in dB at 5° tilt angle obtained from QPSO.

Fig.10. Normalized power pattern in dB at 3 ° tilt angle obtained from PSO.

Normalized Power in dB

0 Example2 -10 -20 -30 -40 -50 -60 -70 -90 -70 -50 -30 -10

10

30

50

70

90

 in degrees

Fig.8. Normalized power pattern in dB at 3° tilt angle obtained from FEKO using QPSO.

Fig.11. Normalized power pattern in dB at 5 ° tilt angle obtained from PSO.

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Fig.12. Normalized power pattern in dB at 3° tilt angle obtained from FEKO using PSO.

Fig.13. Normalized power pattern in dB at 5 ° tilt angle obtained from FEKO using PSO.

Table 2: Desired and obtained results from simulation and FEKO

Design Parameters

Peak Side Lobe Level (dB) Maximum Null Filling (dB) Computation Time (Seconds)

Desired Value

Example 1 (14 Element) QPSO

Example 2 (20 Element)

PSO

QPSO

PSO

Simulation

FEKO

Simulation

FEKO

Simulation

FEKO

Simulation

FEKO

-20

-14.16

-14.12

-12.63

-12.61

-18.93

-18.66

-17.50

-17.19

-30

-27.51 (11 o to 18 )

-27.50

-19.72 (11 o to 18 )

-20.10

-29.87 (12 o to 18 )

-30.20

-28.20 (12 o to 18 )

—

23396.65

—

23503.75

—

45423.16

o

o

o

44594.91

o

-28.60

Table 3: Excitation current amplitude, antenna height and spacing for the array

Element No ±1 ±2 ±3 ±4 ±5 ±6 ±7

Obtained From QPSO (Example1) Current Length of Spacing Excitation Antenna 0.4839λ 0.4080λ 0.8010 0.4900λ 0.9466λ 0.7850 0.4844λ 1.6628λ 0.7578 0.4939λ 2.2125λ 0.7298 0.4739λ 2.8577λ 0.4096 0.4818λ 3.9172λ 0.4649 0.4875λ 4.4693λ 0.5207

Obtained From PSO (Example1) Current Length of Spacing Excitation Antenna 0.4838λ 0.4182λ 0.5182 0.4958λ 0.9609λ 0.6536 0.4915λ 1.5220λ 0.5512 0.4948λ 2.0651λ 0.2689 0.4961λ 2.9752λ 0.3595 0.4784λ 3.5179λ 0.4583 0.4628λ 4.4333λ 0.3685

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Table 4: Excitation current amplitude, antenna height and spacing for the array Element No ±1

Obtained From QPSO (Example2) Current Excitation Length of Antenna Spacing 0.4814λ 0.4001λ 0.8590

±2 ±3 ±4 ±5 ±6 ±7 ±8 ±9 ±10

0.8638 0.7355 0.6425 0.5177 0.4987 0.3786 0.3339 0.3841 0.3188

0.4867λ 0.4828λ 0.4981λ 0.4914λ 0.4978λ 0.4774λ 0.4775λ 0.5007λ 0.4820λ

Obtained From PSO (Example2) Current Excitation Length of Antenna Spacing 0.4762λ 0.4161λ 0.7474

0.9791λ 1.6880λ 2.2332λ 2.7712λ 3.2996λ 3.8562λ 4.7732λ 5.3115λ 5.8496λ

0.7804 0.7629 0.5849 0.4284 0.5584 0.1732 0.3669 0.3609 0.3302

0.4998λ 0.4817λ 0.4869λ 0.5066λ 0.4787λ 0.4867λ 0.4939λ 0.4880λ 0.4576λ

0.9871λ 1.6278λ 2.2902λ 2.7674λ 3.3376λ 3.9548λ 4.7206λ 5.2467λ 5.9617λ

Table 5: Real and imaginary value of the input impedance (in Ω) for example1 Element No 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Obtained Value From Simulation (QPSO) Real Imaginary 44.3403 3.3847 55.2914 3.8312 41.9157 1.9278 42.1034 3.4993 49.6407 0.5271 37.3957 4.3531 56.6654 3.2510 44.6013 1.1875 48.9150 0.6369 47.7617 1.0582 42.9806 2.1412 47.8198 2.3504 43.5490 4.0310 54.0769 3.9517

Obtained Value From FEKO (QPSO) Real Imaginary 49.790 2.6863 53.849 1.5724 53.054 2.2629 47.259 3.5051 54.666 4.9273 49.139 6.0320 59.284 -3.5317 62.689 -3.1456 41.523 6.2573 55.293 0.2077 46.210 5.1994 47.825 4.9880 61.458 -3.7906 49.473 5.0745

Obtained Value From Simulation (PSO) Real Imaginary 58.3735 4.2595 43.8102 2.3021 55.7317 2.6675 44.9997 9.6956 48.6187 2.7640 46.7986 0.2667 49.7821 4.1207 43.5439 0.2913 49.8036 1.9466 47.6142 5.2280 51.6146 8.2399 43.2784 4.5505 52.5381 1.1333 52.6265 7.6010

Obtained Value From FEKO (PSO) Real Imaginary 48.637 1.5767 54.075 2.4766 52.443 5.8887 55.506 -7.9878 47.418 6.4657 59.950 2.1714 61.141 -5.5507 56.081 5.1219 50.828 1.0010 53.369 3.3455 49.642 11.130 59.729 -2.2246 49.750 4.4517 68.001 -2.3148

Table 6: Real and imaginary value of the input impedance (in Ω) for example 2 Element No

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Obtained From Simulation (QPSO) Real Value 55.5464 44.3135 54.9001 39.4494 50.9120 47.6308 48.4710 44.1837 35.1473 49.1504 34.1551 49.2059 41.3016 51.9341 49.1110 46.6607 51.6631 39.0166 55.6573 47.4817

Imaginary Value 0.1873 0.1797 0.1217 0.7024 0.3732 0.5950 0.2236 1.0158 1.0896 1.4113 2.9824 0.5714 1.1083 0.1673 0.4683 0.1623 0.8723 0.1942 0.0251 0.3473

Obtained From FEKO (QPSO) Real Value 38.435 54.393 46.031 55.828 53.326 50.663 58.477 44.047 59.262 53.336 55.020 39.251 49.686 52.232 51.980 54.779 44.482 62.260 47.732 61.742

Imaginary Value 5.5977 0.8841 2.7771 0.0268 -0.5578 0.5688 1.7097 1.5535 0.3328 0.0529 0.0001 3.3614 2.5917 0.4385 0.6953 0.8168 1.3327 0.7120 0.6721 0.0458

Obtained From Simulation (PSO) Real Value 42.1927 39.6252 62.4973 44.7493 55.0627 50.9900 49.2945 40.3748 35.3297 53.0019 40.6987 44.7704 36.9030 47.6183 53.1491 42.2801 41.1005 52.9309 46.9816 46.7908

Imaginary Value 10.5004 4.4954 3.2767 14.2274 2.0905 11.6186 9.4587 0.3673 7.5459 1.7754 4.8861 6.1065 14.6382 10.7080 13.1668 8.0857 1.3841 6.6683 1.2249 16.4047

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Obtained From FEKO (PSO) Real Value 45.971 48.511 42.498 52.579 56.893 47.501 44.156 57.687 52.748 54.206 60.823 38.634 45.349 55.694 52.975 61.759 55.752 67.542 43.701 49.090

Imaginary Value -3.2291 7.8656 18.948 -10.653 13.649 -6.8737 0.6049 7.2128 2.7494 -14.026 3.1044 9.9106 1.6643 11.107 -11.320 3.9220 5.9032 -4.9201 6.9587 -7.4917

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V. CONCLUSIONS This paper presented a technique using QPSO to generate a far-field pattern with the above requirements for broadcasting technology. Results obtained from the above Tables proved that the algorithm was well suitable for synthesis of power pattern with main lobe tilting in presence of mutual coupling. For validating the obtained results from simulation of the algorithm FEKO was successfully utilized here to generate the free space far-field pattern. Impedance matching was well obtained for all the antenna elements of an array by matching the real and imaginary part of the input impedance to the specified value. It was shown by the obtained values from simulation and FEKO analysis in Table 5 and 6. Results obtained from simulation and FEKO analysis nearly matched to each other. This method can be extended to other antenna array configurations also.

[8]

[9]

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