Wireless Communications and Mobile Computing

Wireless Communications and Mobile Computing / 2021 / Article
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Smart Antennas and Intelligent Sensors Based Systems: Enabling Technologies and Applications 2021

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Research Article | Open Access

Volume 2021 |Article ID 9915420 | https://doi.org/10.1155/2021/9915420

Qiankun Liang, Bin Chen, Huaning Wu, Chaoyi Ma, Senyou Li, "A Novel Modified Sparrow Search Algorithm with Application in Side Lobe Level Reduction of Linear Antenna Array", Wireless Communications and Mobile Computing, vol. 2021, Article ID 9915420, 25 pages, 2021. https://doi.org/10.1155/2021/9915420

A Novel Modified Sparrow Search Algorithm with Application in Side Lobe Level Reduction of Linear Antenna Array

Academic Editor: Fawad Zaman
Received18 Mar 2021
Revised17 May 2021
Accepted04 Jun 2021
Published28 Jun 2021


Antenna arrays play an increasingly important role in modern wireless communication systems. However, how to effectively suppress and optimize the side lobe level (SLL) of antenna arrays is critical for communication performance and communication capabilities. To solve the antenna array optimization problem, a new intelligent optimization algorithm called sparrow search algorithm (SSA) and its modification are applied to the electromagnetics and antenna community for the first time in this paper. Firstly, aimed at the shortcomings of SSA, such as being easy to fall into local optimum and limited convergence speed, a novel modified algorithm combining a homogeneous chaotic system, adaptive inertia weight, and improved boundary constraint is proposed. Secondly, three types of benchmark test functions are calculated to verify the effectiveness of the modified algorithm. Then, the element positions and excitation amplitudes of three different design examples of the linear antenna array (LAA) are optimized. The numerical results indicate that, compared with the other six algorithms, the modified algorithm has more advantages in terms of convergence accuracy, convergence speed, and stability, whether it is calculating the benchmark test functions or reducing the maximum SLL of the LAA. Finally, the electromagnetic (EM) simulation results obtained by FEKO also show that it can achieve a satisfactory beam pattern performance in practical arrays.

1. Introduction

The latest development trend of wireless communication systems is to realize the antenna array system with strong directivity and maneuverability, so that it can radiate and receive energy to the maximum extent in a specific direction and reduce the waste of energy by suppressing the SLL in noninteresting directions [1]. According to the pattern multiplication theorem, the pattern of an antenna array can be obtained by multiplying the element pattern by the array factor. By adjusting the spacing, excitation amplitude, and phase of array factor, the antenna array can have the characteristics of high gain, narrow beam, low SLL, and easy electrical scanning [2]. In the field of scientific research and engineering practice, most of the problems encountered can be attributed to solving optimization problems, and the design and optimization of the antenna array are no exception [3]. With the development of computer technologies and computational electromagnetics, some intelligent optimization algorithms that simulate the behavior mechanism of biological groups or the laws of natural phenomena have begun to appear in the vision of many scholars. With its unique advantages in solving large-scale, nonlinear, and other complex optimization problems, the design and optimization technology of antenna arrays based on an intelligent optimization algorithm has always been a research hotspot in the field of EM optimization [4].

In the past few decades, various intelligent optimization algorithms have been implemented to optimize and design antenna arrays, such as genetic algorithm (GA) [5], particle swarm optimization (PSO) [6], bees algorithm (BA) [7], biogeography-based optimization (BBO) [8], firefly algorithm (FA) [9], cat swarm optimization (CSO) [10], cuckoo optimization algorithm (COA) [11], backtracking search algorithm (BSA) [12], symbiotic organisms search (SOS) [13], grey wolf optimization (GWO) [14], extended GWO (GWO-E) [15], spider monkey optimization (SMO) [16], gravitational search algorithm (GSA) [17], invasive weed optimization (IWO) [18], elephant swarm water search algorithm (ESWSA) [19], grasshopper optimization algorithm (GOA) [20], and equilibrium optimization algorithm (EOA) [21]. These algorithms have been successfully applied in this field. However, according to no free lunch (NFL) theorem, no algorithm can perform best on all kinds of optimization problems [22]. Therefore, finding and researching more efficient algorithms are still a problem worthy of attention in the field of EM optimization.

SSA is a new intelligent optimization algorithm proposed by Xue and Shen in 2020. It is mainly inspired by the foraging and antipredation behaviors of sparrows. It has the characteristics of simple implementation, few adjustment parameters, and high expansibility and has been successfully applied in many scientific research and engineering practice fields [23]. For example, [24] proposed an improved SSA to solve the path planning problem of UAV under constraint and proved that the route generated by this method is better than that by the other four algorithms in the same environment. Wang and Xianyu used SSA to solve the optimal configuration model of distributed power supply for the first time and verified the effectiveness and superiority of this method through the IEEE 33 distribution system [25]. Reference [26] established the SSA model based on empirical mode decomposition to optimize the parameters of the kernel extreme learning machine (KELM) and achieved higher prediction accuracy in blood glucose prediction. Kumaravel and Ponnusamy improved the control parameters of the power controller based on SSA to optimize the power flow management of the smart grid system, which realized the real-time energy management in the microgrid [27]. In [28], the improved model based on SSA can track the distributed maximum power point more accurately and quickly and has good robustness, thus effectively solving the problem of power mismatch loss in a photovoltaic microgrid system. Zhu and Yousefi introduced an adaptive strategy on the basis of SSA and applied it to the optimization of proton exchange membrane fuel cell (PEMFC) stack model parameters. The validity of the proposed method in determining the maximum output power and optimal operating state of the stack is verified through four cases [29]. With the help of the adaptive SSA, Liu and Rodriguez took a residential building as an example and took the energy load demand and investment cost as the optimization objectives to get the best combination scheme of the integrated sustainable energy systems, which achieved the purpose of reducing energy consumption and saving economic expenses [30].

LAA is known as the basic and one of the most practical geometric configurations of antenna array in which all elements are arranged in a straight line. Aimed at the design and optimization of LAA, a new modified SSA is proposed in this paper. The main contributions are briefly highlighted as follows: (i)Proposition of the MSSA to further enhance the performance of SSA(ii)Application of the algorithm to the electromagnetics and antenna community for the first time(iii)Design of a few different scenarios of LAA for the maximum SLL reduction by optimizing the element spacing and excitation amplitude(iv)Consideration of an additional constraint on the total length of the antenna array to preserve the features of the beam pattern(v)Comparison of the results with several classical and well-known algorithms in function optimization and antenna array design problem(vi)EM simulation with Altair FEKO 2019 to test the validity of the experiment results in practical conditions

The rest of this paper is structured as follows. Section 2 describes the basic principle of SSA, introduces the improvement strategies, and gives the pseudocode of the modified algorithm. Section 3 verifies its effectiveness and convergence performance. In Section 4, the mathematical description of the LAA optimization problem is given, and the simulation results are discussed and analyzed, and then, the EM verification experiment is carried out. Finally, Section 5 draws the conclusion of this paper.

2. The Sparrow Search Algorithm and Its Modification

2.1. Standard Sparrow Search Algorithm

A sparrow is an intelligent social animal, which keeps alert and stays a safe distance at all times. They also show great unity when encountering the enemy. According to their different behavior rules, sparrows can be divided into three roles: producers, scroungers, and scouters. Assuming that there are sparrows in a -dimensional search space, the position of the th sparrow can be expressed as . The following is an introduction of the three updating methods.

2.1.1. Producers

Producers are sparrows with better fitness values in the population. They have a wide search range and are responsible for searching and providing foraging directions for the whole population. The mathematical expression of producers is described as follows: where represents the current iteration number and represents the maximum number of iterations. denotes the position of the th sparrow in the th dimension. is a uniform random number in the range , and and represent the alarm value and the safety threshold respectively, where and . is a random number with normal distribution. is a one-dimensional matrix with all elements of 1. When , it means that the surrounding environment is safe and they can search for food extensively. When , it means that there is danger at this time, and all sparrows have to fly to other safe areas quickly.

2.1.2. Scroungers

Scroungers are sparrows except all the producers and keep an eye on the producers. If they find that the producers have found better food, they will immediately leave their present position to fight for food and make themselves the producers. The position of the scroungers is updated as follows: where represents the global worst position of the current population and represents the best position occupied by the producers in the current iteration process. is a one-dimensional matrix with elements of 1 or -1, and . If , it indicates that the th scrounger with low fitness value is in an unfavorable position and needs to expand his flight range to obtain food; if , the th scrounger will find a random place near the optimal location and perform local search.

2.1.3. Scouters

The scouters are randomly generated between the producers and the scroungers and can perceive whether there is danger in the foraging area. The model of scouters can be formulated as follows: where represents the global optimal position of the current population, and it is also the safest location. As a step size control parameter, is a random number subject to standard normal distribution. indicates the direction of movement of the sparrows. , , and represent the fitness value of the th sparrow, the global optimal, and the worst fitness value of the current population, respectively. is a minimal constant that avoids zero division error. indicates that the sparrow is at the edge of the population, vulnerable to predators, and needs to move to a safe area. indicates that the sparrow is in the middle of the population, but it is aware of the danger and needs to be close to other sparrows to reduce the risk of predation.

2.2. Modified Sparrow Search Algorithm

Compared with other representative intelligent optimization algorithms in recent years, although SSA has strong competitiveness in convergence speed, accuracy, and stability [31], it is still inevitable to fall into local optimum at the later stage of iterations, resulting in insufficient convergence accuracy [32]. In order to further improve the performance of the algorithm, this paper proposes a new modified algorithm based on chaotic adaptive inertia weight and improved boundary constraint.

2.2.1. Homogeneous Chaotic System

The quality of initial solution directly affects whether the algorithm can find the optimal solution. Chaos is a kind of random phenomenon with ergodicity, inherent regularity, and long-term unpredictability [33]. Within the search range of feasible solutions, chaotic sequences are widely used in population initialization of optimization algorithms; this is because they can traverse all states without repetition [34]. The research in Reference [35] shows that the homogeneous chaotic system has better random effect in variable initialization. Its function is expressed as where is the initialization sequence and is completely chaotic in . The formula for transforming chaotic sequence into the solution space is as follows: where and , respectively, represent the upper and lower boundary values of the optimized variables.

The statistical histogram obtained from the numerical statistics of ordinary random sequences and sequences generated by the homogeneous chaotic system is shown in Figure 1. It can be seen from the figure that the chaotic system has better homogenization, that is, better randomness. Hence, when the diversity of the sparrows increases, the quality of initial solution can be improved.

2.2.2. Adaptive Inertia Weight

All intelligent optimization algorithms include two processes of global exploration and local exploitation. An efficient algorithm needs to balance the global exploration capability and the local exploitation capability [36]. From the basic principle of SSA in Section 2.1, it is not difficult to find that the producer’s search ability plays a vital role in whether the algorithm can find the optimal solution. Therefore, inertia weight is introduced to adjust it adaptively. The mathematical expression is shown in

Take and . The schematic diagram is illustrated in Figure 2.

As can be noticed, as the number of iterations increases, the inertia weight decreases adaptively. The update formula of the producers’ position is modified as follows:

By introducing the adaptive inertia weight, sparrow individuals can search favorable regions in the global range with a larger step size in the early stage of search and strengthen the ability of global exploration; in the later stage of the search, a smaller weight can ensure sparrows to do fine search near the extreme points and strengthen the ability of local exploitation, so that the algorithm has a greater probability of converging to the global optimal value.

2.2.3. Improved Boundary Constraint

In the standard SSA, the processing strategy for sparrows overstepping the boundary is generally given by

In this method, the individual that oversteps the boundary is simply assigned to the boundary value of the search range, which is equivalent to giving up the individual’s search information. In the iterative process, if more individuals overstep the boundary, the positions of sparrows will accumulate more boundary values, resulting in the decrease in population diversity, which directly affects the convergence accuracy of the algorithm [37]. The strategy of improved boundary constraint handling is shown in Algorithm 1. Sparrows that overstep the boundary will randomly determine a position near the optimal position of the population, which enhances the diversity of the population and improves the global optimization ability of the algorithm to a certain extent.

Input: (Position that overstep the boundary), Ub (Upper boundary), Lb (Lower boundary),
    Xbest (Global optimal position), Xgood (Current optimal position)
Output: (The new position)
1: if < Lb || X i,j > Ubthen
2:   temp = Xbest + | Xbest - Xgoodrand(1);
3:   ifLbtempUbthen
4:     = temp;
5:   else
6:     = Xgood;
7:   end if
8: end if
9: return

The SSA modified by the above strategies is named MSSA, and the pseudocode outlining the steps of its implementation is shown in Algorithm 2.

Input:N (Population size), D (Dimension size), PNum (Producers size), SNum (Scouters size), ST
   (Safety threshold), t (Initial iteration), tmax (Maximum iterations)
Output:Xbest (Global optimal position), Fbest (Fitness of global optimal position)
1: /Initializing/
2: Randomly generate the positions of N sparrows Xi,j by homogeneous chaotic system
  (i=1,2,…,N, j=1,2,…,D);
3: Calculate the fitness of each sparrow Fi;
4: Find Xbest and Fbest;
5: /Iterating/
6: whilet < tmaxdo
7:   Sort the Fi and find global worst position Xworst;
8:   R2 = rand(1);
9:   fori = 1 : PNum
10:    Evaluate and calculate the adaptive weight w;
11:    Update the position Xi,j by (7);
12:    Check and adjust position that overstep the boundary by Algorithm 1;
13:   end for
14:   Sort the Fi and find the best position of producers Xp;
15:   fori = (PNum + 1) : N
16:    Update the position Xi,j by (2);
17:    Check and adjust position that overstep the boundary by Algorithm 1;
18:   end for
19:   fori = 1 : SNum
20:    Update the position Xi,j by (3);
21:    Check and adjust position that overstep the boundary by Algorithm 1;
22:   end for
23:    Evaluate and update Xbest and Fbest;
24:    t = t + 1;
25: end while
26: returnXbest and Fbest;

3. Performance Analysis

In this section, MATLAB is used to verify and analyze the computational performance of MSSA on benchmark test functions. In order to verify the effectiveness and superiority of the modified algorithm, classical PSO and representative algorithms in recent years, including PSOGSA, WOA, GOA, and MTDE, are adopted for comparative experiments. All the parameters of these algorithms are in accordance with the original papers, as shown in Table 1.


PSO [38, 39]Individual learning factor 2
Social learning factor 2
Inertia weight 1.05

PSOGSA [40]Individual learning factor 0.5
Social learning factor 1.5
Weighting function
Gravitational constant 1
Alpha 20

WOA [41]Variable
Constant defining logarithmic spiral shape 1

GOA [42]Minimum reduction factor 0.00004
Maximum reduction factor 1
Attraction intensity 0.5
Attractive length scale 1.5

MTDE [43]Number of portions divided by iterations 20
best-history size 5
Nonlinear decreased coefficient
Dimension-dependent value
Mean value of improved scale factors 0.5
Variance of improved scale factors 0.2

SSA [23]Number of producers 20%
Number of scouters 10%
Safety threshold 0.8

The population size of all algorithms is set to 30, and the number of iterations is 500, for fairness. All numerical experiments are implemented on Intel(R) Core(TM) i5-9400U CPU with 2.90 GHz and 8 GB RAM.

3.1. Benchmark Test Functions

In order to comprehensively evaluate the global and local optimization ability of the algorithm, the experiment uses three types of benchmark test functions, in which the unimodal function has only a global optimal value but no local optimal value, so it can better test the local exploitation ability of the algorithm; the multimodal function has many local optimal values, so it can test the ability of the algorithm to jump out of the local extreme value and the global exploration ability. Table 2 shows the benchmark test functions for the experiment.

Function’s typeFunction’s nameFunction’s equationSearch space

Schwefel 2.2230
Schwefel 1.230


Fixed-dimension multimodalGoldstein–Price2
Shekel’s foxholes4

3.2. Experimental Results and Analysis

Considering the randomness of algorithm operation, 50 tests were run independently in order to make the results more convincing and universal. The best value and the worst value can show the exploration ability of the algorithm, and the mean value and standard deviation can show the accuracy and stability of the algorithm. Therefore, the best value, the worst value, and the mean value of 50 experimental results were statistically analyzed, and the standard deviation was calculated. The numerical results are shown in Table 3, where the optimal value is expressed in bold.