Mengting Lou，Jing Jin，Hanning Wang，Dan Wu，Liang Xia，Qixing Wang，Yifei Yuan，Jiangzhou Wang

1 Future Research Lab，China Mobile Research Institute，Beijing 100053，China

2 School of Engineering and Digital Arts，University of Kent，Canterbury，U.K.

*The corresponding author，email: loumengting@chinamobile.com

Abstract: Massive multiple-input multiple-output(MIMO)technology enables higher data rate transmission in the future mobile communications.However，exploiting a large number of antenna elements at base station (BS) makes effective implementation of massive MIMO challenging，due to the size and weight limits of the masssive MIMO that are located on each BS.Therefore，in order to miniaturize the massive MIMO，it is crucial to reduce the number of antenna elements via effective methods such as sparse array synthesis.In this paper，a multiple-pattern synthesis is considered towards convex optimization (CO).The joint convex optimization (JCO) based synthesis is proposed to construct a codebook for beamforming.Then，a criterion containing multiple constraints is developed，in which the sparse array is required to fullfill all constraints.Finally，extensive evaluations are performed under realistic simulation settings.The results show that with the same number of antenna elements，sparse array using the proposed JCO-based synthesis outperforms not only the uniform array，but also the sparse array with the existing CO-based synthesis method.Furthermore，with a half of the number of antenna elements that on the uniform array，the performance of the JCO-based sparse array approaches to that of the uniform array.

Keywords:B5G;6G;sparse array;joint convex optimization;massive MIMO;system-level simulation

I.INTRODUCTION

Massive multiple-input multiple-output (MIMO) is considered as a key technology for increasing the capacity in the beyond-fifth-generation(B5G)and sixthgeneration(6G)mobile communications[1-3].Using a large number of antenna elements at a base station(BS)to serve many user equipments(UEs)simultaneously，massive MIMO can effectively improve spectral efficiency (SE)，reliability and energy efficiency，which has attracted significant attention in academia and industry.In [4-8]，massive MIMO were innovatively integrated with non-orthogonal multiple access(NOMA)，millimeter wave(mmWave)and relays，which give a brand new method for achieving capacity improvements.However，a large number of antenna elements that are located in the panel with uniform array spacings may lead to a substantial increase in the weight，cost and complexity，which may hinder the practical deployment of massive MIMO[9].With the consideration of practical deployment，the number of antenna elements cannot be too large.Hence，it is crucial to reduce the number of antenna elements in massive MIMO while maintain good performance.

Sparse array synthesis[10，11]is a promising technique since it uses nonuniform array spacings to reconstruct the desired radiation pattern with a smaller number of antenna elements，thereby relieve the pressure of large-scale antenna elements integration in a BS.Sparse array has been widely used in radar and satellite communication systems，sensing and imaging systems and wireless communication systems.However，in the legacy mobile communication system，the number of antenna elements was usually small and within the applicable ability of the BS，there was no strong need to reduce the number of antenna elements.Therefore，sparse array has not been used in modern mobile communication system.There is an increasing demand for more antenna elements equipped on the BS when uniform array is considered.In order to make a feasible deployment of the massive MIMO on the BS，sparse array becomes important in terms of reducing of antenna elements.In this paper，we consider a sparse array based massive MIMO system.Owing to the ability of reducing antenna elements and radio frequency(RF)chains，sparse array can potentially offer much lighter weight and significant reduction in complexity of the baseband processor.In addition，since sparse array can relocate the antenna elements，it may provide better system performance than the uniform array with the same number of antenna elements.

1.1 Synthesis Issues

Extensive work has been devoted to developing synthesis algorithms over the past decades.The sparse array synthesis is complicated because of its nonlinear characteristics.Both single-pattern synthesis and multiple-pattern synthesis were well investigated.The former only needs one desired pattern while the latter usually needs at least two desired patterns.As a result，the sparse array synthesized from the former method may be well matched at the desired direction but badly matched when scanning angle changes.Evolutionary algorithm is a more general synthesis method for its advantages in multivariable optimization，including genetic algorithm(GA)[12-14]，modified genetic algorithm(MGA)[15，16]，particle swarm optimization(PSO)[17-19]，invasive weed optimization(IWO)[20，21]，differential evolution (DE) [22，23]，etc.However，performing the aforementioned stochastic methods are time-consuming，especially for a large-scale array.Besides，these methods did not consider the minimum number of antenna elements since the desired number of elements needs to be given as a priori.Fortunately，fast Fourier Transform(FFT)[24]，matrix pencil method(MPM)[25，26]and forward-backward MPM(FBMPM)[27]have been proposed to minimize the number of elements and enhance computation efficiency.However，the mininum array spacing was not considered in these methods，which may make these methods not feasible in practical systems.

Convex optimization(CO)is also a kind of effective approach since it has more freedom with nonuniform array spacings and optimize the number and the amplitude simultaneously.The work in[28]takes use of CO for sparse array synthesis where up to a few hundred antenna elements can be synthesized.In [29，30]，an iterative optimization is introduced to generate more stable results.The work in [31，32] conbines CO and compressive sensing(CS)for planar array synthesis.The above work belongs to single-pattern synthesis which is effective only in a specific scanning angle.Although single-pattern CO method is extended to multiple-pattern methhod[33，34]with reweightedl1minimization[35]，the scanning angle is fixed to the direction of the array boresight and the method is verified only for linear array.In [36]，for the first time，we adopted the single-pattern CO-based synthesis and evaluated the system performance of the sparse array in mobile communication.It is shown that the grating lobes are formed when scanning angle is far from the direction of the array boresight，and the system performance loss is mainly due to the interference caused by the grating lobes.Therefore，it is important to suppress grating lobes in mobile communications，thereby reducing the interference from neighboring cells.

In principle，sparse array synthesis gives rise to the possibility of the massive MIMO with a smaller number of antenna elements in the B5G and 6G mobile communications.Existing work usually focuses on the typical optimization characteristics，such as the matching accuracy with the desired radiation pattern，the side lobe level，the minimum number of antenna elements，the efficiency and the complexity of algorithm.However，the impacts on system in practice have not been well studied for system performance evaluation，especially in mobile communication system.In practical scenarios，the system performance is affected by multiple factors such as channel quality，beamforming and mobility，which may restrict the application of sparse array.Inspired by the above discussion，we propose to apply sparse array in massive MIMO and investigate the performance under realistic simulation environment.

1.2 Constributions

Since many mobile UEs need to be served in mobile communication system，beamforming is required to provide satisfied continuous services.The singlepattern synthesis usually only considers UEs that are located at the array boresight，that is，the array synthesized using this method may not be able to serve well to the UEs that are not located at the array boresight，as verified in [36].Therefore，this paper proposes an innovative joint multiple-pattern synthesis method and the main contributions are summarized as follows:

Firstly，a joint convex optimization (JCO) based multiple-pattern synthesis method is developed by combining the convex optimization algorithm and beamforming.Different from the single-pattern synthesis methods，the characteristics of the mobile communication system are well considered and multiple radiation patterns are applied for joint optimization.The weighting factor of each pattern can be flexibly configured as needed.In order to obtain multiple radiation patterns，a discrete Fourier Transform (DFT)based codebook is introduced for beamforming in the process of synthesis.

Secondly，a criterion for synthesis which contains multiple constraints is proposed for multiple desired radiation patterns.Generally，the main constraints of the traditional synthesis schemes include the radiation pattern at the array boresight，which ignores the loss of the scanning ability.In the proposed scheme，the constraints of multiple radiation patterns are considered.Using this criterion，the reconstructed sparse array is required to fulfill the corresponding constraint conditions.Thus，the proposed sparse array and the original uniform array have a good degree of matching in multiple radiation directions，which contributes to the effectiveness of suppressing the grating lobes during scanning.

Finally，the system performances of the uniform arrays and the sparse arrays are evaluated numerically.The results show that the proposed method outperforms the uniform array method and the CO-based synthesis method under the condition of the same number of antenna elements.Besides，when the number of antenna elements is reduced by half compared with the uniform array，no significant performance loss is observed by comparison.

The remainder of this paper is organized as follows.In Section II the array model and the CO-based synthesis method are presented.In Section III，the proposed JCO-based snthesis method is developed in details.The link-level and system-level simulation results and analysis are presented in Section IV and Section V，respectively.Section VI concludes the paper.

Notation:Matrices and vectors are denoted by bold upper case and lower case letters，respectively.(·)Tdenotes matrix transposition of(·)，while‖·‖and‖·‖Frepresent the Euclidean norm of a vector and the Frobenius norm of a matrix.⊗and◦denote the inner product and Hadamard product，respectively.

II.MATHMATICAL FORMULATION

2.1 Uniform Array Model

First，consider a uniform planar array (UPA)，a special case of which is uniform linear array (ULA).As shown in figure 1a，the UPA(M×N)contains a number of active antenna elements that are located ony-zplane.Here，M ×Nmeans there areMantenna elements in each row in the horizontal direction andNin each column in the vertical direction.We defineGas the number of antenna elements on the UPA，andG=MN.The radiation pattern of the UPA can be expressed as，

whereθis the elevation angle defined between 0°and 180° (90° represents perpendicular to the array)，andφis the azimuth angle defined between -90° and 90°(0°represents perpendicular to the array).wm，nis the excitation amplitude of the element at position (md，nd)，dis the array spacing，andλis wavelength.The steering vector e(θ，φ)and the excitation vector w are defined as(2)and(3)，respectively，

In(2)，u=sinθsinφ，v=cosθ.

2.2 Sparse Array Model

Next，consider a sparse planar array(SPA)that targets for reconstructing the desired radiation pattern in(1).As shown in figure 1b，we assume that the desired SPA withGactactive antenna elements is obtained through a delicate synthesis method，then the radiation pattern of the desired SPA can be expressed as，

wherews，lis thel-th entry of the excitation vector ws，and it is the excitation amplitude of the element at position(ys，l，zs，l).Thel-th entry of the steering vector es(θ，φ)can be denoted as

It can be seen that the radiation patterns in (1) and(4) are calculated based on the excitation amplitudes and the phases that are related to the position of the antenna element.Compared with the UPA，the number of antenna elements on the desired SPA is smaller，and the distribution of the antenna elements on the desired SPA is irregular，instead of being limited to the regular grid(md，nd).This is the most significant characteristic of SPA.By delicately utilizing such characteristic，the performance of the UPA is probably approached with a smaller number of antenna elements.

2.3 Convex Optimaization-based Synthesis

The CO-based synthesis process can be described as follows:

Step 1: Construct an initial SPA.For fairness，the sizes of the targetM×NUPA’s panel and the desired SPA’s panel should keep the same.The SPA is initially constructed as a virtual UPA withMs×Nsinactive antenna elements.We assumued that the horizontal and vertical array spacings are Δdyand Δdz(Δdy，Δdz ≪d)，respectively.Under the same size panel，MsandNscan be calculated as，

whereis the rounding down operation.We defineGsas the initial number of antenna elements on the initial SPA，andGs=MsNs(Gs ≫G).

Step 2: Sample reference radiation pattern.The reference radiation pattern of the UPA on the desired radiation pattern is required.Based on the reference radiation patternF(θ，φ)，we sampleKpoints (Kpairs ofθandφ) and form the sampled vector Frefwith thek-th entryF(θk，φk).

Step 3: Develop the synthesis problem.The basic idea of the CO-based synthesis is to replace thel0norm with thel1norm，and an error term will be introduced in the constraint.In the synthesis，the radiation pattern of the reconstructed SPA should be as close as possible to the pattern of the UPA，so we can use the difference between the two patterns as the error term in the constraint.Then，we need to construct the steering matrix of the initial SPA.Different from 1× Gvector e(θ，φ)，withKpairs ofθandφ，the steering matrix Esof the initial SPA is aK × Gsmatrix as shown in (7)，

whereuk=sinθksinφk，vk=cosθk(k=1，··· ，K)，and (yg，zg) is the position of theg-th element on the SPA(g=1，··· ，Gs).Meanwhile，in order to improve the sparsity of the solution，the synthesis problem by using iterative weightedl1norm[35]can be constructed as follows，

where E(p)s ⊗w(p)srepresents the radiation patterns of the SPA of thep-th iteration，w(p)sis the SPA’s excitation vector of thep-th iteration and w(p)s=Σ(p-1)sw(p-1)s.Σsis a diagonal matrix and itsi-th diagonal elementsiof thep-th iteration is updated by

hereinis thei-th element in theandξis a known constant that is used to ensure the denominator great than zero.is a configuable matching error，which usually can be chosen within 10-1～10-5[29，36-38].

When the process of(8)converges after several consecutive iterations，the process is terminated and the final result w(p*)scan be extracted.Since some elements in w(p*)sare negligible，a thresholdσis introduced to sieve out the elements larger thanσ.Therefore，the desired SPA withGactactive antenna elements is obtained and the sparsity factorηcan be computed as，

III.JOINT CONVEX OPTIMAIZATIONBASED SYNTHESIS AND BEAMFORMING DESIGN

The CO-based synthesis method suffers from the effects of the higher side lobes or grating lobes when the main beam scans.Since the UEs can move freely in the realistic mobile communication environment，some UEs in neighboring cells that are located near side lobes or grating lobes may experience strong interference from the current cell.In order to mitigate such effects on system performance，one key aspect regarding synthesis is to suppress the interference when the main beam scans.The following subsection formulates the general problem of jointly performing the sparse array synthesis and beamforming.

3.1 Codebook-based Beamforming

Beamforming technique can be used in sparse array synthesis to combat the degradation of scanning abilities.Analog precoding and digital precoding are two common methods for supporting beamforming mechanism.In this paper，we propose to synthesize sparse array that considers a set of desired radiation patterns by the digital precoding codebook.For the UPA，the codebook that containsQcodewords is denoted as C and theq-th codeword in C is defined as，

Similar to the CO-based synthesis method，the preliminary SPA withGsantenna elements is also needed to be constructed for the JCO-based synthesis method.For the preliminary SPA，the digital precoding codebook can be denoted as Csand theq-th codeword in Csis defined as，

3.2 Reference Patterns Sampling

The next step is to sample multiple radiation patterns instead of being limited to the radiation pattern at the array boresight.Based onQreference radiation patterns in(12)，we can sampleKpoints for each radiation pattern and theq-th sampled setcan also be generated as

3.3 Joint Optimization Problem Formulation

Sparse array synthesis and beamforming should be performed jointly in order to achieve the best possible performance.Due to the different probabilities of occurrence of the radiation directions，the weighting factorαqcan be introduced for theq-th desired radiation pattern and the sum of all the weighting factors is 1.WithQselected codewords corresponding to the desired radiation patterns，a criterion that contains multiple constraints should be developed.That is，for each desired radiation direction，the matching error between the reconstructed pattern formed by the sparse array and the desired radiation pattern yields a configuable value.Hence，(8)can be rewritten as，

For each set of initial excitation vector，，it is necessary to single out the elements larger thanσand mark down the corresponding indexes.It should be noted that the positions of antenna elements in different excitation vector should be the same while the excitations of elements can be different.With regards to this，we need to carry out intersection operation for theQsets of indexes.Based on the final indexes that allQsets contain，we figure outGactactive antenna elements and obtain theQsets of final excitation vectors as follows，

The JCO algorithm is summarized as Algorithm 1.

Algorithm 1. JCO algorithm for obtaining the solution to problem(16).1: Initialize M，N，d，Δdy，Δdz，K，ξ，σ，Q，αq，¯εq;2: Set p:=1，Σs :=I;3: for q =1 to Q do 4:Compute c(q) in(11)and c(q)s in(13);5:Compute F(q)(θ，φ)in(12)to get ¯Fref in(15);6: end for 7: repeat 8:Solve problem (16) to get an optimal solution{w(p，1)s ，··· ，w(p，Q)s};9:Set p:=p+1;10:Update w(p，q)sin the objective function by w(p，q)s:=Σ(p-1)sw(p-1，q)s;11:Update Σ(p)s through(9);12: until the predefined stopping criterion is met.13: Obtain an optimal {w(p*，1)s，··· ，w(p*，Q)s}.Seize out the elements larger than σ and mark down the indexes.14: Carry out intersection operation for the Q sets of indexes and obtain the final solution{^w(1)s ，··· ， ^w(Q)s }.

3.4 Reconstructed Pattern and Error

The reconstructed radiation patterns with beamforming of the SPA can be expressed as，

whereys，landzs，lindicate the position ofl-th antenna element on the reconstructed SPA.Thel-th entry of the steering vector es(θ，φ) for the reconstructed SPA，can be denoted asThe precoding codewordthat corresponding to the desired radiation direction，is a subset of c(q)sin(13)，and thel-th entry ofcan be denoted as

In order to evaluate the effectiveness of the proposed method，the reconstruction error is introduced that describes the approximation degree between the SPA and the UPA，and can be calculated as shown in (20).

Here，θmin，θmax，φmin，φmaxdonate the lower bound or upper bound inθandφdirection.

IV.LINK-LEVEL SIMULATIONS

This section evaluates the performance of SPA in conjunction with the beamforming design，and analyses the complexity of CO-based synthesis scheme and JCO-based synthesis scheme.In modern mobile communications，the number of antenna elements is usually a power of two，such as 32，64，etc.Here，the SPAs with 32 antenna elements，are synthesized from the UPA with 64(16×4)antenna elements(64-UPA)，followed by CO-based 32-SPA and JCObased 32-SPA.The UPA with 32 (8×4) antenna elements(32-UPA) is also adopted as a comparison，since it has almost the same number of antenna elements with the SPAs both in horizontal and vertical directions.

Given a cell in mobile communication，it is assumed that the azimuth scanning angle of each cell varies from-60°to 60°，and the elevation scanning angle is fixed to 90°.Due to the symmetry，we may only consider the optimizations with the azimuth scanning angle varying from 0° to 60°.Three sets of desired scanning anglesare considered，followed by(90°，0°)，(90°，20°)and(90°，40°).The weighting factors for each set are 0.6，0.2 and 0.2，respectively.Besides，Δdyis 0.1λand Δdzis 0.5λfor the initial SPA construction.¯εis set as 10%，same as that in[36].

4.1 Elements Distribution

Figure 2 shows the distribution of antenna elements on different arrays.It can be seen that both the CO-based 32-SPA and the JCO-based 32-SPA adopt nonuniform spacings and some array spacings exceed half-wavelength，which means that under the premise of the same number of antenna elements，the two SPAs may have higher resolution than that of the 32-UPA.

Figure 1. Structure diagram of UPA and SPA.(a)UPA.(b)SPA.

Figure 2. Distribution of antenna elements on different arrays.(a)64-UPA.(b)CO-based 32-SPA.(c)JCO-based 32-SPA.

Figure 3. Normalized radiation patterns of the arrays with various scanning angles.(a) ^φ1=0°.(b) ^φ2=20°.(c) ^φ3=40°.

Besides，the elements on the JCO-based 32-SPA show a centrally symmetric distribution while that of the CO-based 32-SPA does not.This is because for a symmetrical reference pattern，it can be formed by symmetrical excitation vector with symmetrical element distribution，or asymmetrical excitation vector with asymmetrical element distribution.However，it is difficult for the latter to meet multiple constraints in(16).Hence，for the sparse array synthesis，it may be more efficient to synthesize based on the symmetric part of the reference array，which is beneficial to the simplification of the synthesis process.

4.2 Radiation Characteristics

Figure 3 shows the normalized radiation pattern of the arrays at various scanning angles.Some useful results like 3dB beamwidth and max side lobe level (SLL) can be observed from figure 3 as Table 1 summarizes.

Figure 4. UE distribution in a cell.(a)Type A:line-based distribution.(b)Type B:circle-based distribution.(c)Type C:area-based distribution.

Figure 5. CDF of per-user net throughput for different systems，with UE distribution of type A.

Figure 6. CDF of per-user net throughput for different systems，with UE distribution of type B and R=ISD/6.

Figure 7. CDF of per-user net throughput for different systems，with UE distribution of type B and R=ISD/2.

Figure 8. CDF of per-user net throughput for different systems，with UE distribution of type C.

Table 1. 3dB Beamwidth，Max SLL and Reconstruction Error of the UPAs and the SPAs.

When the number of antenna elements is reduced by half(the arrays involved here are the 64-UPA and the JCObased 32-SPA)，the main lobes of JCO-based 32-SPA and 64-UPA are nearly overapped.Specifically，the values of 3dB beamwidth of JCO-based 32-SPA and 64-UPA are almost same.In the case of the same number of antenna elements(here refers to the 32-UPA and the two 32-SPAs)，the main lobe of JCO-based 32-SPA is a bit narrower than that of CO-based 32-SPA，and much narrower than that of the 32-UPA.

In figure 3(b)-(c)，for the CO-based 32-SPA，there appears some grating lobes that are higher than those of the others，especially when ^φqscans to 40°，the max SLL has risen to -2.21 dB，which is close to the main lobe level.Meanwhile，the JCO-based 32-SPA keeps relatively low SLL under the above scanning angles，and no more serious grating lobes appear.This implies that the JCO-based synthesis method can effectively mitigate the grating lobes introduced by scanning.

4.3 Reconstruction Error

Furthermore，we compare reconstruction errors of different SPAs synthesized from the 64-UPA.Here，the reconstruction errors are calculated based on the 64-UPA，and the values ofθmin，θmax，φmin，φmaxare 0°，180°，-60°，60°，respectively.

As Table 1 shows，all the reconstruction errors of JCObased 32-SPA at different scanning angles are smaller than that of the CO-based 32-SPA.For the CO-based 32-SPA，the reconstruction error increases significantly when the scanning angle increases to 40°，and the value of the error can reach 52.97%.However，for the JCO-based 32-SPA，due to the more balanced consideration of joint optimization of multiple scanning angles formed by beamforming，the max reconstruction error here is 14.02%，about a quarter of that of the CO-based 32-SPA.

It is worth mentioning that the error fluctuation between different scanning angles of JCO-based 32-SPA is smaller than that of the CO-based 32-SPA，which means JCO-based synthesis method has relatively more stable radiation capability and less potential interference to neighboring UEs.These advantages are very favorable for the deployment of ultra-large-scale MIMO equipped with JCO-based SPAs.

4.4 Complexity Analysis

Generally，for convex optimization problems，we can use the number of iterations or computing time to evaluate the complexity.For intuitive comparison，we have carried out 20 independent simulations for the CO-based synthesis scheme and JCO-based synthesis scheme，respectively.The average number of the iterations and the average computing time are recorded.In each independent simulation，Kis 160，Gsis 304 (calculated according to the setups aforementioned)andQis 3.The central processing unit we adopted is Intel(R)Core(TM)i5-6200U.The simulation results show that the average number of iterations of the CObased synthesis scheme is 4.92 while that of the JCO-based synthesis scheme is 5.42.The average computing time of CO-based synthesis scheme is 41.16 seconds while that of the JCO-based synthesis scheme is 96.03 seconds.It can be seen that the number of iterations of the two schemes are similar.However，the calculation efficiency of JCO-based synthesis scheme is somewhat lower than that of CO-based synthesis scheme due to the increase of constraints and multiplication operations.

V.SYSTEM-LEVEL SIMULATIONS

The massive MIMO system in this section includes four types that are applied with the arrays aforementioned，followed by 64-UPA system，32-UPA system，CO-based 32-SPA system and JCO-based 32-SPA system.The performances of these systems are evaluated via system-level simulations.

5.1 Simulation Setups

System-level simulations are conducted in the Urban Macro cell (UMa) at 2.4 GHz and the setups are defined by the 5G channel model in 3GPP TR 38.901[40].A wraparound model is assumed for network topology with realistic traffic and radio propagation models.19 BSs with 57 hexagonal cells are generated and the inter-site distance(ISD)is 350 meters.The BS can be equipped with the UPAs and the SPAs aforementioned.In each cell，UEs are randomly distributed outdoors by means of different types for certain purposes as figure 4 dipicted.In all simulations，time division duplex (TDD) system with 100 MHz bandwidth is considered.Other simulation parameters such as transmitter power are listed in Table 2.

Table 2. Simulation Assumptions.

Type A:UEs are randomly distributed at the array boresight.The radiation directions of beams keep the same inφdirection，which is used to intuitively compare the performance differences of the CO-based and JCO-based synthesis methods without scanning.

Type B:UEs are randomly distributed along the circle with the configuable radiusR.The radiation directions of beams vary inφdirection.The purpose of this case is to verify the effectiveness of the JCO-based synthesis method in suppressing the grating lobes that result in neighboring interference.

Type C:UEs are randomly distributed in each cell.The beams have freedom scaning angle in bothφandθdirections.The system performance in this case can be comprehensively evaluated，which will provide reliable guidance for practical deployment.

5.2 Results and Analysis

The cumulative distribution function(CDF)of the per-user net throughput is generated over 1000 time slots.The cell-edge user throughput is then obtained by accumulating the top 5% per-user net throughput.Moreover，the whole throughput of the entire system is accumulated and the average cell throughput is then calculated through dividing the whole throughput by the number of cells.

We first investigate the performances of different systems on the perspective of UE.figure 5-8 dipict the CDFs of the per-user net throughput of different systems with different types of UE distribution.Table 3 presents the cell-edge throughput of the above systems based on the curves in figure 5-8.In figure 5，the UE distribution of type A is adopted.When comparing under the condition of the same number of antenna elements，the cell-edge user throughput of the JCObased 32-SPA system is 13.36%higher than that of the 32-UPA system and 9.36% higher than that of the CO-based 32-SPA system.This is due to the fact that the main lobe radiation ability of SPAs here depends on the scanning ability inθdirection，which is related to the positions of antenna elements in vertical direction.The positions of antenna elements on the SPAs in each column are not in parallel toz-axis，which provide higher degree of spatial freedom and contribute to better cell-edge performance than that of the 32-UPA system.Furthermore，when comparing under the premise of the number of antenna elements which is reduced by half，the cell-edge user throughput of the JCObased 32-SPA system is close to that of the 64-UPA system.

Table 3. Cell-edge User Throughput of 64-UPA，32-UPA，CO-based 32-SPA and JCO-based 32-SPA Systems.

Table 4. Average Cell Throughput of 64-UPA，32-UPA，CO-based 32-SPA and JCO-based 32-SPA Systems.

In figure 6-7，type B of UE distribution is assumed and theRis equal to ISD/6 and ISD/2，respectively.In case of the same number of antenna elements，asRincreases，the gaps between the JCO-based 32-SPA system and the CO-based 32-SPA system or 32-UPA system are further increased.This is because a smallRmeans that most of the UEs are located near the BS，hence better services can be achieved in the above systems.But for a largeR，the possibility of potential interference from the other cells increases.In particular，whenRreaches to ISD/2，the celledge user throughput of the JCO-based 32-SPA system is 28.17%higher than that of the 32-UPA system and 17.86%higher than that of the CO-based 32-SPA system.As expected，JCO-based 32-SPA system benefits from the consideration of beamforming and joint multiple-pattern synthesis，and therefore，it suffers from less interference than CO-based 32-SPA system does，especially when UEs are located at the edge of the cell.Besides，in the case that the number of antenna elements is reduced by half，the per-user net throughput of JCO-based 32-SPA system is close to that of the 64-UPA system.

Figure 8 assumes type C of UE distribution.The performance of JCO-based 32-SPA system approaches to the 64-UPA system.The cell-edge user throughput of the JCObased 32-SPA system reaches to 59.29 Mbps，which is 11.74%higher than that of the 32-UPA system and 21.32%higher than that of the CO-based 32-SPA system.

Finally，we investigate the performances of different systems on the perspective of the cell.The average cell throughputs of the UPAs and the SPAs applied systems as Table 4 summarizes.Regardless of UE distribution，the JCO-based 32-SPA system outperforms CO-based 32-SPA system and 32-UPA system，and is the closest one to the 64-UPA system.Taking type C for illustrative instance，the average cell throughput of the JCO-based 32-SPA system is improved by about 5.12%compared with the 32-UPA system and about 8.68%compared with the CO-based 32-SPA system，and it is close to that of the 64-UPA system.

VI.CONCLUSION

In this paper，multiple patterns have been considered by proposing a new synthesis method，named joint convex optimization(JCO).By introducing beamforming，the proposed JCO-based synthesis method can reconstruct a suitable SPA with lower side lobes and grating lobes during scanning，which makes the reconstructed patterns more accurate.The effectiveness of the proposed method is verified by numerical link-level and system-level simulations.The comparison among the JCO-based 32-SPA system，the CO-based 32-SPA system and the 64/32-UPA systems were also carried out under different types of UE distribution.

The results have demonstrated that under the condition of the same number of antenna elements，the JCO-based 32-SPA system outperforms the 32-UPA system and the CObased 32-SPA system.In particular，the JCO-based 32-SPA system is much more robust against the interference from the neighboring cell.In addition，when the number of antenna elements is reduced by almost 50%，the performance of the JCO-based 32-SPA system is close to that of the 64-UPA system，which is attractive for practical deployment in the future B5G and 6G mobile communications.

In order to promote the practical deployment of sparse array，there are still some issues that need to be further studied in the future.In particular，the extreme large number of antenna elements on the future BSs may result in the challenge of algorithm complexity，which can be caused by the application of high frequency bands (such as millimeter waves，the number of antenna elements may exceed 1000).Therefore，more efficient sparse synthesis algorithms are required to be studied and the performance of sparse array under the high frequency bands may also need to be further evaluated.