Formulation of Optimization Problems with Radiation Field Phases as Design Variables for Pattern Synthesis of Linear Antenna Arrays

Fei Zhao ,Jun He ,Sanyou Zeng ,Changhe Li ,Qinghui Xu ,Zhigao Zeng

1 School of Mechanical Engineering and Electronic Information,China University of Geosciences,Wuhan 430074,China

2 Science and Technology on Blind Signal Processing Laboratory,Southwest Electronics and Telecommunication Technology Research Institute,Chengdu 610041,China

3 School of Automation,China University of Geosciences,Wuhan 430074,China

4 Hubei Key Laboratory of Advanced Control and Intelligent Automation for Complex Systems,Wuhan 430074,China

5 College of Computer,Hunan University of Technology,Zhuzhou City,Hunan Province,412007,China

Abstract: Pattern synthesise of antenna arrays is usually complicated optimization problems,while evolutionary algorithms (EAs) are promising in solving these problems.This paper does not propose a new EA,but does construct a new form of optimization problems.The new optimization formulation has two differences from the common ones.One is the objective function is the field error between the desired and the designed,not the usual amplitude error between the desired and the designed.This difference is beneficial to decrease complexity in some sense.The second difference is that the design variables are changed as phases of desired radiation field within shaped-region,instead of excitation parameters.This difference leads to the reduction of the number of design variables.A series of synthesis experiments including equally and unequally spaced linear arrays with different pattern shape requirements are applied,and the effectiveness and advantages of the proposed new optimization problems are validated.The results show that the proposing a new optimization formulation with less complexity is as significant as proposing a new algorithm.

Keywords: pattern synthesis,evolutionary algorithms,linear antenna arrays.

I.INTRODUCTION

Antenna arrays synthesizing with a specific radiation field is usually a nonlinear optimization problem.The prominent problem of antenna arrays’pattern synthesizing is to determine a set of element excitations(i.e.amplitudes and phases),for generating an amplitude shape of a radiation field(literally the radiation pattern at below)as closely as possible to the desired one.

A number of advanced shaped pattern synthesis techniques have been proposed to deal with the problems.Various classical synthesis methods,have been put forward for the design of antenna arrays,and they have been proven successful,such as the Woodward—Lawson method [1],iterative sampling method[2],convex optimization method[3,4],and matrix pencil method[5].The authors in[6]introduced a new synthesis technique based on superposition principle and Taylor method to deal with the shaped beam radiation patterns.In [7],the approach based on semidefinite programming was presented to solve two array synthesis problems.The combination of convex optimization and Bayesian compressive sensing [8] was proposed for the shaped beam pattern synthesis,and the numerical assessment pointed out its advantages.However,traditional optimization methods are not well suited for optimizing a large number of parameters or discrete parameters[9],and what is worse is that the optimization of unequally spaced antenna arrays degrade the performance of them.In addition,traditional optimization methods are also easy to fall into local optimal in dealing with the highly constrained optimization problem for the design of antenna arrays.

EAs such as genetic algorithm (GA) [10,11],differential evolution (DE) [12,13],and particle swarm optimisation(PSO)[14,15],have been applied to deal with the design of antenna arrays.And EAs were regarded as powerful alternative tools which are capable of obtaining global optima for solving array designs[16].Some literatures show that EAs are not only capable of dealing with pattern synthesis with equally spaced antenna arrays [17,18],but also work well for synthesizing the pattern of unequally spaced arrays[19,20].In[18],a dynamic DE was applied to synthesis flat-top and cosecant-squared patterns by employing element rotation and phase optimization for the design of a linear dipole array.The number of design variables in problems of above literatures was not too large.With a large number of variables,there would be a serious performance degradation of EAs to solve the optimization problem.

Recent year,another kind of artificial intelligence,machine learning technology,is also used for antenna array pattern synthesises.Artificial Neural Network,as an instance,has been successfully used to predict the radiation pattern of antenna array[21,22].In[23],by establishing a pre-trained decoder,the encoderdecoder-based ANN framework synthesizes the beampattern fastly.For solving the phase-only antenna pattern synthesize problem,Radial Basis Function Neural Network(RBFNN)is proposed to predict the phase of excitation of antenna[24].While,though these machines learning technologies show virtue of an excellent learning-by-examples ability,but are limited by the lack of database.

Almost all papers on synthesizing pattern were to design algorithms to solve the problems,few to study how to construct problems.In this paper,a new optimization formulation for the antenna array synthesis is proposed.In the new formulation,the objective that is to approach given radiation field with information of both magnitudes and phases,not the given radiation pattern with information of only magnitudes.This is beneficial to decrease non-convexity in some sense.The design variables are the phases of radiation field within shaped-region,not the excitations.Fortunately,only the phases in the shaped-regions are needed to design.This leads to reduce the number of design variables.

The rest of this paper is organized as follows: Section II describes related formulations of pattern synthesises.Section III shows a new optimization form of pattern synthesises with reduced design variables by using LSM.Section IV introduces the dynamic constrained multi-objective EA for solving the problems in this paper.Section V presents some examples of shaped pattern synthesises for the equally and unequally spaced linear arrays and corresponding experimental results.Finally,Section VI draws the conclusions and future works.

II.RELATED FORMULATIONS OF PATTERN SYNTHESIS

2.1 Patterns of Antenna Arrays

The pattern synthesises of linear antenna arrays are considered in this paper.Suppose that a linear antenna array is composed ofNelements distributed along thexaxis as illustrated in Figure.1.

Figure 1.Geometric of a linear array along the x-axis.

Figure 2.The patterns of two linear antenna arrays.(a)Flat-topped sector antenna.(b)Cosecant squared antenna.

For the sake of discussion,each element can be treated as ideal particle source of radiation.

Definition 1.Theradiation fieldE(θ)of antenna ar-rays is written as

with

where θ is the direction angle,θ ∈[?90?,90?];λ denotes the wavelength;=(d1,d2,...,dN?1)T,in which di is the distance between element i and i+1;Di is the vector from the origin point to the location of ith elements,because the all elements lie in a line,these vectors can be treated as one-dimensional vectors,so Di here is the distance from the origin point to the location of ith elements,and Di=1,2,...,N ?1);is the N-dimensional steering vector;is the complex excitation vector.

The radiation pattern(pattern,for short)is the normalized amplitude of field.In some reality engineering which we call“pattern synthesises”,the users only concern on patterns in some situations,in which the phases are thought as undetermined variable.

In numerical computation,the region[?90?,90?]is divided intoM ?1 parts withMdirectionsθi,i=1,2,...,M.According to literature[25],when solving pattern synthesises problem of equally spaced antenna arrays,Mis chosen as about two or three times of the array sizeN.

Here,we introduce the notation:

And,

In pattern synthesises,the desired pattern regions are usually classified as two: the shaped regions and the sidelobe regions,and always the shaped regions are more significant.See the two regions in Figure.2,where the patterns of two commonly desired fields are shown: flat-topped sector pattern and cosecantsquared pattern.

Ideally,the designed pattern equals the desired pattern,but in almost all situations it is impossible to design a pattern as the desired one.Instead the goal of pattern synthesizing is to reduce the difference between the designed pattern and the desired pattern as possible.

2.2 Common Optimization Form of Pattern Synthesises

The pattern synthesises are usually regarded as constrained optimization problems (COPs).In the case of pattern synthesises of linear arrays,a minimization problem is constructed,the excitationsstand for the design variables.

Here we suppose the distribution of all elements is given.

A candidate objective for the COPs is to reduce the difference between the designed pattern and desired pattern,and a normalized“differnence”is constructed with the form of 2-norm:

The constraint on peak sidelobe level(PSLL)of the designed field is a candidate constraint for the COPs.The PSLL is expressed as

whereθSLis the angle within sidelobe peaks in the sidelobe regions,θis the angle within the whole region[?90?,90?].

III.NEW OPTIMIZATION FORM OF PATTERN SYNTHESISES

The motivation of this section is to construct a new approach.Different from finding a set of excitations to fit the desired radiation pattern directly,the new approaching try to fit the desired radiation field with considering both the magnitudes and the phases.In common approaching,optimizing excitations is a nonconvex optimal problem solved by EAs,while in new approaching both the excitations and the phases are considered as designed variables,as detail,optimizing phases are solved by EAs and optimizing excitations with precondition of known phases are solved by LSM(least square method).For reducing the computation complexity,phases of sidelobe region are not considered and only the phases of shaped regions are still the designed variables.

3.1 The Costruction of the Optimization Problem

For convenience of discussion in this paper,the fieldE(θ)matrix in Eq.(1)is rewritten in a real vector form as

Always,pattern synthesizing is to approach the desired pattern (shape of normalized amplitude) while the information of ideal phases is not considered.The opinion of the this paper suggests the phases of the ideal phases should be considered.By introducing phases of the desired radiation field,this paper tries to approach the ideal radiation field,specifically,approach both the desired magnitudes and the desired phases,not as the common to approach the radiation pattern with only approaching the desired magnitudes.

The new idea is shown in the following formulation of optimization problem Eq.(8).

The common form of optimization problem of pattern synthesise is listed as comparison:

It is easy to reveal that the objective of both problems in Eq.(8)and Eq.(9)are nonconvex.

As mentioned in Eq.(8),the design variables are consisting of excitations and phases.When phases are set as some hypothetical constants,the process to find optimal excitations can be seen as a process to find an optimal solution in the subspace limited byThe process finding the“best”phase is the process to find the subspace including the optimal solution of the whole solution space.

Then the optimization objective in Eq.(8)is reconstructed as a bi-level one as in Eq.(10):

3.2 The Reduction of Variables

As discussed in the last subsection,the variales of the nonconvex optimization problem is translated fromthis subsection will try to reduce the dimension of variable vector.

The detailed process to choosing the bestby EA is discussed in Section IV.

The number of sample directions in the shaped regions is the dimension of the phasesof the pattern in the shaped regions.Denote the width of shaped regions byWshaped,that of sidelobe regions byWsidelobe,the rate of width of shaped regions with that of sidelobe regions by

SupposeM=k×N,then,

It holds that

That is,the reduction of the number of design variables of the new COPs on that of the original COPs holds if and only ifr <2/(k ?2).It means that the method of formulation of new COPs for the pattern synthesises proposed in this paper works if and only if that the rate of shaped regions with sidelobe ones is less than 2/(k ?2).Furthermore,the less the rate,the larger the reduction of the number of design variables.

In real engineering,a sharp beam region is sometimes an important requirement.Always,a growing number of antenna elements are set,with the rate of width of shaped regions with that of sidelobe regions going to zero.At the same time,the number of design variables in the new optimization formulation has a lower-order increase than that in the common formulation.This means the reduction can be great significant.Otherwise,if ther >2/(k ?2),the variables of new formulation of optimization problem will be more than that of common one.

In the next section,a candidate algorithm will be introduced for solving both the original COPs in Eq.(9)and the new COPs in Eq.(14).They are two cases of the general COP form.

IV.A CANDIDATE ALGORITHM

The original COPs in Eq.(9) and the new COPs in Eq.(14)are nonlinear and belong to the general COPs.Traditional mathematical optimizers are easily trapped in local optima in solving the COPs.Evolutionary algorithms are promising in finding global optima for the COPs.An effective evolutionary algorithm,named a dynamic constrained multi-objective optimization algorithm (DCMOEA) [26],was proposed by our research group,to solve the COPs about the pattern synthesises of antenna arrays in this paper.

The DCMOEA is provided to deal with the general form of COPs.The details of DCMOEA are omitted for limited space.It can be obtained in[26].

V.NUMERICAL EXPERIMENTS

In last section,we construct a new form of optimization problem.With the help of LSM,we translate the original optimization problem with variables as excitationsto the new problem with variables as phasesTo illustrate the benefits of the LSM assisted new form of COPs with reduced design parameters,a set of widely used synthesises with flattopped and cosecant-squared patterns in equally and unequally spaced linear arrays are employed as the test problems.Both the common form in Eq.(9) and the LSM assisted one in Eq.(14) for the test problems are constructed and solved by a same EA,the DCMOEA in Section IV with same function evaluations,for comparisons.And for testing the performance of algorithms in OPs with more variables,cases in which the distances between each adjacent elements are also considered as variables with phases.

5.1 Formulations of the Test Problems

5.1.1 Formulations of Pattern Synthesis of Equally Spaced Arrays

In the pattern synthesis of equally spaced arrays,the spaces between adjacent elementsare constants,not the design variables.

5.1.2 Formulation of Pattern Synthesis of Unequally Spaced Arrays

5.2 Numerical Synthesises of the Test Problems

DCMOEA [26] is employed to solve the constructed COPs for the test problems.

5.2.1 Parameters Setting

DCMOEA parameters setting as in[26]: DE operator(DE/rand/1/bin) in [27] is used,the crossover factorCR=0.9,the scaling factorF=rndReal(0,1),which is a random value,the population sizepop=100,the mutation ratePm=0.01.According to the suggestions of [28,29],the number of function evaluationsFEs=n ?20,000 (nchooses the number of design variables of the LSM assisted new COPs in Eq.(14),not Eq.(9),in the purpose of comparisons).

Problem parameters setting: according to literature[25],for patterns synthesizing of equally spaced array,the sample pointsMin the radiation region for LSM should be chosen as about three times of the array sizeN.M=3Nin this paper for approaching the pattern of equally spaced arrays.Otherwise,for improving the precision of the approaching,M=6Nin this paper for patterns synthesizing of unequal spaced arrays.

5.2.2 Pattern Synthesis of Equally Spaced Arrays

In the first and second examples,the element excitation amplitudes and phases were optimized to obtain the desired patterns in equally linear arrays with 0.5λinter-element spacing.A top-flatted pattern was synthesized in[?20?,20?],and a cosecant-squared one in[?10?,20?]was generated by the linear array with 20-element.Corresponding to[6],the desired PSLL is set to?24.0dBfor the top-flatted pattern,and?23.0dBfor the cosecant-squared one in this paper.

In the case of top-flatted pattern,the number of design variablesn=2?20=40 of the common COP,n=of the LSM assisted one.The DCMOEA consumesFEs=14?20,000=280,000 for solving both the common COP and the LSM assisted COP.

In the case of cosecant-squared pattern,n=2?20=40 of the common COP,=10 of the LSM assisted one.The DCMOEA consumesFEs=10?20,000=200,000 for solving both the COPs.

The synthesized pattern in Figure.3 has a top-flatted pattern withinθ ∈[?20?,20?].Basing on the same setting,the DCMOEA finds feasible solutions for both the common COP and the LSM assisted COP in 25 independent runs.The statistics of the average value,standard deviation of best function values obtained by DCMOEA are listed in Table 2.The designed pattern of first run is shown in Figure.3,and the solution is shown in Figure.5(a).The corresponding normalized errors between the designed and the desired pattern in Eq.(4)found areeP=0.05006 for the common COP,andeP=0.02187(translated from the solution solved byeEfor the convenience of comparing)for the LSM assisted one.

Figure 3.Synthesized top-flatted patterns of 20-elements equally spaced arrays by solving the common COP and the LSM assisted one.

Figure 4.Synthesized cosecant-squared patterns of 20-elements equally spaced arrays by solving the common COP and the LSM assisted one.

Figure.4 shows a cosecant-squared pattern in the region ofθ ∈[?10?,20?].The DCMOEA finds feasible solutions for both the COPs in 25 independent runs as well.In first run,the errors areeP=0.14693 for the common COP,andeP=0.15542 (translated fromeE) for the LSM assisted one.In Figure.4,there are some fluctuates in shaped region,but the fluctuates are not very strong and less than-23dB.The optimal solution is shown in Figure.5(b).

Figure 5.Excitations of shaped patterns of 20-elements equally spaced arrays obtained from solving the LSM assisted COP.(a)Excitations of top-flatted pattern.(b)Excitations of cosecant-squared pattern.

5.2.3 Pattern Synthesis of Unequally Spaced Arrays

A top-flatted pattern which was presented in [20],was produced by optimizing the positions,rotation angles and excitation phases for a 26-element unequally spaced linear array.The shaped region is set as[?20?,20?].Corresponding to [20],the desired PSLL is set to?24.0dBin this paper.

The synthesized pattern in Figure.6 has a top-flatted pattern withinθ ∈[?20?,20?].The DCMOEA finds feasible solutions for both the common COP and the LSM assisted COP.The errors of first run areeP=0.05966 for the common COP,andeP=0.01567(translated fromeE) for the LSM assisted one.The values of optimization variables and the distance between two adjacent elements are illustrated in Figure.7.

Figure 6.Synthesized top-flatted patterns of 26-elements unequally spaced arrays by solving the common COP and the LSM assisted one.

Figure 7.Excitations and spaces of 26-elements unequally spaced arrays with top-flatted pattern obtained from solving the LSM assisted COP.(a)Excitations.(b)Spaces between two adjacent elements.

Figure 8.Synthesized cosecant-squared patterns of 19-elements unequally spaced arrays by solving the common COP and the LSM assisted one.

In the last example,we try to synthesize a cosecantsquared pattern which was found in [30] by finding a set of excitation amplitudes,phases and positions of a 19-element nonuniform spaced linear array.The shaped region was set as [?10?,20?].The desired PSLL was under?23.0 dB.n=2?19+18=56 of the common COP,n==37 of the LSM assisted COP.The DCMOEA consumesFEs=37?20,000=740,000 for solving both the COPs.

Figure.8 illustrates a cosecant-squared pattern in the region ofθ ∈[?10?,20?].The DCMOEA finds feasible solutions for both the common COP and the LSM assisted COP.The errors of first run areeP=0.1031 for the common COP,andeP=0.08044 (translated fromeE) for the LSM assisted one.The values of optimization variables and the distance between two adjacent elements are illustrated in Figure.9.

Figure 9.Excitations and spaces of 19-elements unequally spaced arrays with cosecant-squared pattern obtained from solving the LSM assisted COP.(a) Excitations.(b) Spaces between two adjacent elements.

5.3 Discussions

5.3.1 Comparison of Numbers of Design Variables

Table 1 lists the numbers of design variables of common COPs and the LSM assisted COPs for the four test problems.The rates of shaped regions with sidelobe ones in Eq.(15)r==0.29,0.20,0.29,0.20 respectively.

Table 1.The comparison about the number of design variables of the common COP and the LSM assisted new COP.

In Table 1,it is shown that the numbers of design variables of the LSM assisted COPs are greatly smaller than those of the common ones for all the four test problems.

5.3.2 Comparison of Precision

The objective is a normalized error between the designed and the desired pattern in Eq.(4).Table 2 presents average (shown as Avg+Std,which means average objective value+standard deviation) objective values obtained by DCMOEA on 25 independent runs for each of these four problems.To be fair,both settings for the common COPs and the LSM assisted COPs are the same.

As experimental results,the objective values of the LSM assisted COPs were smaller than or almost equal to those for the common ones for all the four test problems,while the standard deviation of the LSM assisted COPs were smaller than those for the common ones as well.The experimental results in Table 2 reveal that DCMOEA has higher theoretical accuracy and stability in solving LSM assisted COPs than in solving common COPs.

Table 2.Average best(shown as Avg+Std)fitness obtained by LSM-assisted COPs and common COPs in four problems.

5.3.3 Comparison of Convergence

Convergence of the DCMOEA in solving two formulation in all four problems on both population distance and objective are shown in Figure.10.In Figure.10,(a) -(d) present the change of population distance with evolutionary generation and in Figure.10,(e) -(h)present the chang of mean objective value.

Objectivevalue here is the errorePin Eq.(9),and mean objective value is calculated as the average of objective value in 25 runs.Population distance,calculated as the averge euclidean distance from each design variable vector in the population to the centroid of the whole population,as shown asPdin Eq.(20),can indicate the population diversity.Always,the population diversity contribute a lot for finding the optima.

Population distances in Figure.10 are the average of 25 independent runs,so as mean objective value.As in Figure.10,(a) -(d),the population distances in LSM assisted COPs are better (larger) than common COPs in most situation,which contribute the ability of finding better solution.

Figure 10.Population distances changing curves and Convergence objective function curves.(a)(b)Population distances changing curves of Equal-spaced Top-flatted synthesizing and Equal-spaced CSC-squared synthesizing.(c)(d)Population distances changing curves of Unequal-spaced Top-flatted synthesizing and Unequal-spaced CSC-squared synthesizing.(e)(f) Convergence objective function curves of Equal-spaced Top-flatted synthesizing and Equal-spaced CSC-squared synthesizing.(g) (h) Convergence objective function curves of Unequal-spaced Top-flatted synthesizing and Unequal-spaced CSC-squared synthesizing.

The mean objective values are shown in Figure.10,(e) -(h).In Figure.10(e),Figure10(g) and Figure.10(h),the mean objective value of LSM assisted COPs decline with slight vibration and converge to a better solution than that of common COPs.Significantly,as an EA for solving constrained optimization problem,the DCMOEA will make a tradeoff among the objective value,the constraint violation and the population diversity.In Figure.10(f),at first the solution of LSM-assisted COPs with better mean objective value but worse constraint violation were founds by DCMOEA,while with the iteration going on,these solutions were replaced.After all,as mentioned above,DCMOEA in solving LSM assisted COPs has better property of convergence than in solving common COPs.

5.3.4 Comparison of Time Cost

In EA research areas,the number of objectiveconstraints function evaluations is employed to measure EA complexity.Thus,the comparison of time costs in solving pattern synthesises can be revealed in average cost FEs for reaching the same objective function.In all four problems,DCMOEA costs less FEs for reaching the same objective value in solving LSM assisted COPs than in solving common COPs,in most situations.

The other way for discussing the time cost is the normal function evaluations numbers.As suggested in [28],the number of function evaluationsFEs=n ?20,000 for solving COPs.The numbers of design variables of the LSM assisted COPs in Eq.(14)aren=14,10,60,and 37 for the four test problems.The EAs solve the four LSM assisted COPs withFEs=280,000,200,000,1,200,000,and 740,000,respectively.Suppose using the common method of formulation of COPs in Eq.(9) for the four test problems,the numbers of design variable would ben=40,40,77,and 56,and function evaluations areFEs=800000,800000,1540000,and 1120000,respectively.As can be seen from Table 3,comparedwith the common formulation method,the proposed method greatly reduces the computational burden for the DCMOEA in solving the pattern synthesises.

Table 3.The reduction of fuction evaluations(FEs)of EAs.

For evolution algorithm,always it is hard to discuss the time complexity of the algorithm.

VI.CONCLUSION

In this paper,a new formulation of optimization problem is proposed.In common optimization problem of antenna array synthesizing,there is an approaching to the desired pattern by adjusting the excitations of each element,while the phases of the field are not considered.The new formulation try to find an optimal phases as supplementary information so that the approaching to desired pattern directly is replaced as an approaching to an ideal field.So,the design variables of nonconvex problem becomes the phases of ideal radiation field,while the objective becomes as approaching the ideal field.

In desired pattern,amplitudes of sidelobe regions are always considered as zeros,as the result phases of ideal field in such regions can be out of consideration.Then,the dimention of designed variable reduced,and especially in some some situations the shaped regions is narrow enough the reducing is more obvious.

And there are two points in the new optimization problem should be noticed:

1.Because the phases of sidelobe region are void in the formulation,the number of phases will reduce.What is more,in situations (the rate of shaped regions with sidelobe regions is less than one),the number of phases is less than the number of excitation.And the less the rate,the larger the reduction.

2.The objective of common optimal problem is a nonconvex one.In new formulation of optimal problem,when the phases of field are given,the process for optimizing excitation is a convex problem.So the non-convexity of the new optimal problem is about the phases.When the number of phases is less than the number of excitations,the nonconvexity of new optimal problem is less than the common one.

Several experiments have been conducted to validate the effectiveness of the proposed method for synthesizing different pattern shape requirements.Experimental results show that the proposed method can obtain satisfactory shaped pattern performances in equally and unequally spaced linear arrays.

A limitation of the proposed technique is that the desired pattern given in this paper could not actually be implemented.It causes the fail in well controlling the designed pattern.In a future work,the authors will replace the desired pattern with an amplitude mask with upper-lower boundaries,and then construct an improved new formulation of optimization problem to overcome the limitation.An other future research topic is to verify the proposition of this paper on synthesising patterns of planar arrays

ACKNOWLEDGEMENT

This work was supported in part by Major Project for New Generation of AI under Grant 2018AAA0100400,in part by Scientific Research Fund of Hunan Provincial Education Department of China under Grant 21A0350,21C0439,in part by the National Natural Science Foundation of China under Grant 61673355,in part by the Fundamental Research Funds for the Central Universities,China University of Geosciences(Wuhan) under Grant CUGGC02,in part by the Hubei Provincial Natural Science Foundation of China under Grant 2015CFA010,in part by the high-performance computing platform of the China University of Geosciences,in part by the 111 project under Grant B17040.