Pattern Nulling of Linear Antenna Arrays Using Backtracking Search Optimization Algorithm

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In this paper, backtracking search optimization algorithm (BSA) [44] is presented for pattern nulling by controlling only the amplitude, phase, and position of the antenna array elements.The BSA is a metaheuristic search optimization algorithm based on an iterative process.It was introduced by Civicioglu in 2013 [44].Contrary to many other metaheuristic algorithms, BSA has a single control parameter.Many optimization methods are very difficult and complex to understand, but BSA has a very simple structure to adapt different optimization problems.In [44], BSA was statistically compared with artificial bee colony (ABC), PSO, covariance matrix adaptation evolution strategy (CMAES), CLPSO, adaptive DE (JDE), and self-adaptive DE (SADE) algorithms.Three real world benchmark and seventy-five boundary constrained problems were used for comparison.The simulations and comparisons in [44] indicate that BSA can provide optimal solutions for benchmark problems and that the results of BSA are more efficient than the compared algorithms.BSA was used in [45] and [46] for circular and concentric circular antenna array designs, respectively.BSA was also employed for solving different optimization problems [47][48][49][50][51].
The study will continue in the following order.The general problem formulations of linear antenna arrays synthesis are given in Section 2. A brief explanation of the BSA is presented in Section 3. In Section 4, numerical results are given.Finally, conclusions of the paper are given in Section 5.

Problem Formulation
We assume a linear antenna array with  elements located throughout the -axis.The far field array factor (AF) can be indicated as where  is the number of wave ( = 2/),  is the angle of scanning, and   ,   , and   are the location, phase, and amplitude of the th element, respectively.When the linear antenna array elements are located symmetrically and excited around the array center and the number of antenna array elements is even  = /2, it is possible to indicate array factor in the following way: The main aim of the study is to find a set of optimum element amplitude, phase, or position values in order to produce a radiation pattern with low maximum side-lobe level (MSL) and deep nulls placed in desired directions.For this reason, the following cost function will be minimized by BSA: where AF  () is the array factor value achieved by BSA and AF  () is the desired array factor value. () and ESL() are utilized for controlling the null depth level (NDL) and the MSL, respectively.

Backtracking Search Optimization Algorithm (BSA)
BSA is an innovative metaheuristic algorithm to solve different complex and nonlinear optimization problems [44].
BSA is based on an iterative process which finds global minimum in the solution space.BSA has five evolutionary steps including initialization, selection-I, mutation, crossover, and selection-II.The structure of BSA is quite simple, so it can be applied to different complex optimization problems with ease.The general structure of BSA is shown in Figure 1.
Equation ( 6) enables BSA to redefine the historical population at the beginning of each iteration: where  and  are the uniform real numbers in the range of [0, 1].The following equation is used to change the arrangement of the members randomly in historical population by using shuffling function: where the permuting() function represents a random shuffling function.

Mutation.
In the mutation process, mutants, initial form of the trial populations, are generated by where  is scale value to control the amplitude of the direction of search.BSA produces a trial population depending on old values.

Crossover.
In the crossover step of BSA, final form of the trial-population  is produced.The crossover process involves two steps.The first step uses mix-rate and the second step imposes that just one arbitrary individual mutates in each trial.

Selection-II.
All trial populations    with optimal values better than the relevant    are utilized to update the    by using the greedy selection strategy.If  best 's value is better than the global minimum value, the new global minimum value is changed with the best individual of .Further details of the BSA can be found in [44].

Numerical Results
To show the efficiency and performance of the BSA for steering single, multiple, and wide nulls with the prescribed directions by controlling amplitude only, phase only  The maximum iteration number of the BSA is selected as 10.000.The mix-rate and step size amplification parameter of BSA are set to 1 and  = 3⋅rndn, where rndn ∼ (0, 1), respectively.The population size is fixed to 30.The simulations are carried on a computer which has 2.8 GHz i7 processor and 4 GB RAM.The software used for the optimization applications of BSA is MATLAB.
In the first group of examples, BSA is used to achieve radiation patterns with imposed nulls at interference directions by controlling the amplitude only.In the second group of examples, desired nulls are placed on the array pattern by controlling phase only.In the third group of examples, only the element positions are controlled for pattern nulling.
4.1.Amplitude Only Control.In the first example, Chebyshev pattern is considered with a single null at   = 14 ∘ .The parameters values of the cost function are selected as The radiation pattern achieved by BSA is shown in Figure 3. Table 1 illustrates NDL, MSL, and DRR values of the radiation pattern obtained by using BSA.Table 1 also shows the results of MTACO [13], BA [16], PGSA [26], HSA [35], and BFA [19].It can be clearly observed from Table 1 that NDL, MSL, and  [13], BA [16], PGSA [26], HSA [35], and BFA [19] for one null imposed at 14 ∘ .
Figure 4 shows the convergence curve.The curve is the average values obtained by running the algorithm 30 times.It is apparent from this figure that 1000 iterations are needed to find the optimal solutions.
To illustrate the resilience of BSA, in the second example, () is redefined as follows: Figure 5 shows the radiation pattern produced by the BSA.The NDL and MSL values of the radiation pattern in Figure 5 are −182.3dB and −29.08 dB, respectively.In Figure 5, the NDL value of array pattern is better than the NDL value of the pattern in Figure 3.However, compared to the improvement in the NDL, the MSL value of the radiation pattern in Figure 5 is seen worse than the MSL of the pattern in Figure 3.There is a trade-off between the NDL and MSL; usually, performance cannot be improved considerably for one without sacrificing the other.
In third and fourth examples, to test the capability of BSA for the radiation pattern synthesis with multiple nulls at any direction, only the AF  () given by ( 9) is changed for synthesizing the array patterns with double null at 14 ∘ and 26 ∘ and triple null at 14 ∘ , 26 ∘ , and 33 ∘ , respectively.Figures 6 and 7 show the radiation patterns obtained by BSA for the multiple nulls, as it is seen from Figures 6 and 7 that each imposed null achieved by BSA is lower than −120 dB.
In the fifth example, the radiation pattern with a wide null placed at 30 ∘ with Δ = 5 ∘ is obtained and is illustrated in Figure 8.The NDL value of the radiation pattern achieved by BSA is deeper than −59 dB.
The normalized amplitude values of array elements for the radiation patterns showed in Figures 2, 3, and 5-8 are given in Table 2.It is clearly seen that, with respect to the main beam, the radiation patterns in Figures 3 and 5-8 are symmetric.This is a result of the even symmetry of the element amplitudes around the array center.For this reason, while a null occurs at the one side of the main beam, an image null imposes at the other side.In the sixth example, the null direction   is identified as −10 ∘ .The radiation pattern obtained by BSA is given in Figure 9.In Table 3, the NDL and MSL values of the radiation pattern obtained by BSA are compared with those of the CLONALG [21], BA [29], HSA [35], and BFA [24].As it can be seen from this table, the NDL and MSL values of the pattern obtained by BSA are better than the other compared algorithms.
Double null at  1 = −10 ∘ and  2 = 20 ∘ is achieved by controlling only the phase of the antenna array elements in the seventh example.The radiation pattern obtained by BSA is given in Figure 10.The MSL and NDL values of the radiation pattern obtained by BSA are compared with the results of SOA [32], CLONALG [21], and BFA [24] in Table 4.  [21], BA [29], HSA [35], and BFA [24] for one null imposed at −10 ∘ .BSA (Figure 9) CLONALG [   It is possible to see from Table 4 that the results of BSA are better than those of other optimization methods.
In the eighth example, the radiation pattern with triple null ( 1 = −10 ∘ ,  2 = 20 ∘ , and  3 = −33 ∘ ) is achieved by BSA and is illustrated in Figure 11.In Table 5, the MSL and NDL values of the radiation pattern achieved by BSA are compared with the MSL and NDL values of the radiation patterns achieved by BA [29], CLONALG [21], and SOA [32].It is clearly seen from Table 5 that MSL and NDL values of the pattern obtained by BSA are better than those of BA [29], CLONALG [21], and SOA [32] except that NDL value of CLONALG at −10 ∘ is deeper than NDL value of BSA.
The phase values of the array elements for the patterns obtained by BSA in Figures 9-11 are given in Table 2.

Position Only Control.
In the remaining three examples, nulls in the radiation pattern are obtained by controlling only the positions of uniform linear antenna array elements.The ninth example shows the synthesis of a 22-element antenna array for low MSL with desired null at 9 ∘ .In Figure 12, the radiation pattern obtained by BSA is illustrated.The NDL and MSL values of the radiation pattern obtained from BSA are given in Table 6.Table 6 also shows the NDL and MSL results of TSA [30], GA [30], MA [30], HSA [35], PSO [30], NSGA-2 [30], MODE [30], and MOEA/D-DE [30] for a comparison.According to Table 6, it is evidently seen that the NDL and MSL values of the radiation pattern obtained from BSA are better than those of other metaheuristic  [29], CLONALG [21], and SOA [32] for triple null imposed at −33 ∘ , −10 ∘ , and 20 ∘ .BSA (Figure 11 [30], GA [30], MA [30], HSA [35], PSO [30], NSGA-2 [30], MODE [30], and MOEA/D-DE [30] for 22-element linear antenna array with imposed one null at 9 ∘ .BSA (Figure 12) TSA [30] GA [30] MA [30] HSA [35]   In the tenth example, BSA is utilized to synthesize a 28element linear antenna array which has a pattern with low MSL and nulls at 30 ∘ , 32.5 ∘ , and 35 ∘ .The radiation pattern generated by BSA is illustrated in Figure 13.In Table 7, the MSL and NDL values of the radiation pattern of BSA are compared with the results of CLPSO [31] and MVMO [43].It can be seen from Table 7 that the results of BSA are better than those of CLPSO [31] and MVMO [43].It was emphasized in [31] that the performance of CLPSO is better than the common PSO algorithms and a real-coded GA.
In the last example, BSA is performed to synthesize 32element linear antenna array that has a null at the direction  of 9 ∘ and low MSL.The radiation pattern achieved by BSA is given in Figure 14.In Table 8, NDL and MSL values of the pattern obtained by BSA are compared with NDL and MSL values of the patterns obtained by PSO [15], HSA [35], and QPM [15].It is apparent from Table 8 that NDL and MSL values of the radiation pattern obtained by BSA are better than those of PSO [15], HSA [35], and QPM [15].
The position values of the antenna array elements calculated by BSA for the radiation patterns in Figures 12-14 are given in Table 9.  [15], HSA [35], and QPM [15] for 32-element linear antenna array with imposed one null at 9 ∘ .BSA (Figure 14

Conclusions
BSA is successfully used for synthesizing linear antenna arrays with single, multiple, and wide nulls imposed at the directions of interferences.The pattern nulling is achieved by controlling the amplitude only, the phase only, and the position only.The comparisons made between the results obtained by BSA and seventeen different algorithms show that BSA is very competitive algorithm for the pattern nulling.BSA exhibits good performance in terms of MSL and NDL as compared to other seventeen algorithms.Thus, BSA has a good potential for solving other electromagnetic problems.

Figure 1 :
Figure 1: General flow chart of BSA.
, and position only, eleven examples of a linear antenna array have been performed.In the first eight examples, a 30 dB Chebyshev pattern, having 20 equispaced elements with 0.5 interelement spacing, is utilized as the initial radiation pattern, as shown in Figure 2. In the last three examples, the initial pattern is chosen as a uniform array pattern for 22, 28, and 32 equispaced elements with 0.5 interelement spacing.The main aim is to obtain antenna array patterns having deep nulls at desired directions and low MSLs.The dynamic range ratio (DRR = | max / min |) is also taken into account for practical considerations.

Figure 3 :
Figure 3: Radiation pattern achieved by controlling the amplitude only with one null at 14 ∘ .

Figure 4 :
Figure 4: Convergence curve of the nulling pattern achieved by the BSA.

Figure 5 :Figure 6 :Figure 7 : 50 Figure 8 :
Figure 5: Radiation pattern achieved by controlling the amplitude only with a null depth level deeper than that of the first example having one null at 14 ∘ .

Figure 9 :
Figure 9: Radiation pattern found by phase only control with one null at −10 ∘ .0

Figure 10 :
Figure 10: Radiation pattern found by phase only control with double null at −10 ∘ and 20 ∘ .

Figure 12 :
Figure 12: 22-element array pattern found by position only control with one imposed null at 9 ∘ .

Figure 14 :
Figure 14: 32-element array pattern found by position only control with one imposed null at 9 ∘ .

Table 1 :
NDL, MSL, and DRR values obtained by BSA and other optimization techniques MTACO

Table 2 :
The element amplitudes (  ) for the array patterns given in Figures2, 3

Table 3 :
NDL and MSL values obtained by BSA and other optimization techniques CLONALG

Table 5 :
MSL and NDL values obtained by BSA and other optimization techniques BA

Table 6 :
NDL and MSL values obtained by BSA and other optimization techniques TSA

Table 8 :
NDL and MSL values obtained by BSA and other optimization techniques PSO