Performance Analysis of Uplink Massive Spatial Modulation MIMO Systems in Transmit-Correlated Rayleigh Channels

Qiyishu Li,Xiangbin Yu,Mingfeng Xie,Ning Li,Xiaoyu Dang

College of Electronic and Information Engineering,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China

Abstract:In this paper,the performance of uplink multiuser massive multiple-input multipleoutput (MIMO) system with spatial modulation over transmit-correlated Rayleigh fading channel is investigated,where a large number of antennas are deployed at the base station and linear zero-forcing(ZF)receiver is employed for detection.By taking the transmit correlation and the randomness of shadow fading in to account,the bit error rate(BER)performance of the system is analyzed.According to the performance analysis,an approximated expression of overall average BER of the system is attained.Besides,asymptotic performance is studied and the corresponding BER expression at high signal-to-noise ratio is derived.On this basis,the diversity gain of the system can be obtained for performance evaluation.Simulation results show that the derived theoretical expressions match the simulated values well,which verifies the correctness of our analysis.

Keywords:bit error rate;massive MIMO;spatial modulation;transmit correlation;shadow fading;ZF receiver

I.INTRODUCTION

Multiple-input multiple-output (MIMO) technology[1,2]has been studied extensively over the past decade,which employs the Bell Laboratories Layered Space-Time (BLAST) scheme to transmit multiple independent data streams.Thus,MIMO systems could achieve higher capacity than single-input singleoutput (SISO) systems.However,there are several problems in the development of multi-antenna transmission schemes,such as high complexity of signal processing at transmitter,high complexity detection algorithm at receiver,as well as inter-channel interference(ICI).To solve these problems,at the transmitter,spatial modulation(SM)is employed for dealing with these issues.SM technology was first proposed in[3],as one of the MIMO transmission techniques,SM uses a single antenna in a time slot for transmission to avoid ICI,since only one radio frequency (RF) chain is required,the complexity of the hardware implementation and signal processing design can be reduced [4–6].Moreover,the additional information is conveyed by the index of the active antenna besides constellation symbols in SM,which lead to an improved energy efficiency(EE)and spectral efficiency(SE).Besides,some recent research combined the SM with the concept of physical layered security (PHY),transmit antenna selection (TAS) schemes were also adopted for improving the security of the SM based systems[7–9].These studies have helped SM in becoming a secure,efficient communication technology.At the receiver,some low complexity detector such as minimum mean square error (MMSE) and zero-forcing(ZF) can be used for detection.In [10],the approximated bit error rate(BER)expressions of conventional MIMO systems with ZF receivers were derived.Further,the MIMO based multi-cell multiuser scenario was considered in [11],exact analytical expressions for the symbol error rate,uplink rate as well as the outage probability were derived,ZF receiver was utilized to retrieve the transmitted data.The analysis in the literatures demonstrated that ZF was more suitable in the multiuser scenario because of its ability of decoupling the multiuser transmission.More importantly,compared with MMSE,the ZF receiver provides significant performance degradation tolerance,additional interference statistics information is not acquired to achieve a good performance[12].

With the wide application of the smart terminals,the requirement of the data rate grows exponentially,a number of related studies emerged to satisfy much higher SE demand of fifth-generation (5G) network.Authors in[13,14]combined Internet of Things(IoT)with 5G to improve spectrum utilization by sharing the spectrum,which could tackle the spectrum shortage problem.Besides,massive MIMO is another technique in 5G to meet the need for high SE,EE and throughput [15–17].In massive MIMO systems,the base station (BS) is equipped with large antenna arrays and many users are served at the same time frequency resource simultaneously.Due to the deployment of large number antennas,the favorable propagation is achieved and the effect of the small-scale fading can be eliminated.Additionally,multi-antenna technique has the advantages of increased diversity gain,coverage and capacity.In summary,as an improvement of MIMO technology,massive MIMO can reap all the benefits from the conventional MIMO on a larger scale.The performance of massive MIMO systems has been investigated from many perspectives.In [15],the uplink SE of massive MIMO systems with low-resolution analog-to-digital converters(ADCs)was derived over Rician fading channels.The downlink achievable rate of massive MIMO was investigated in[16],the approximated expression of the achievable rate was obtained.In order to improve the reliability of the system,a data detection algorithm was proposed to reduce the complexity of the detection while achieving acceptable BER performance[17].The SE analysis of the massive MIMO systems in different scenarios have been studied widely in many literatures,but there are few literatures consider the BER performance,BER is an important performance criterion of the system reliability,it reveals the nature of the system performance but is difficult to compute.For this reason,the closed-form BER expressions of the massive MIMO systems have not been investigated.

Moreover,the literatures about massive MIMO mentioned above assume that users are equipped with single antenna,but in fact some larger sized user devices such as laptop and the tablet could be equipped with multiple antennas to achieve better performance.Combine the SM technique with massive MIMO,a novel concept of massive SM-MIMO is proposed.The abilities of reducing power consumption and cost of implementation at transmitter make SM technique suitable for uplink transmission.In [18],the applications of forward error correction coding and soft-information-aided multiuser detection were introduced to massive SM-MIMO systems,and BER performance was presented,however,closed-form expressions of BER were not derived.The achievable rate of multi-cell multiuser massive SM-MIMO systems was analyzed in[19],The analytical relationships between the achievable rate and the number of antennas,the SNR and the interference were also presented.The above-mentioned studies reveal that massive SMMIMO will combine the benefits of both the SM and massive MIMO to obtain superior performance.

Furthermore,the system design often assumes that the fading is independent,in realistic massive SMMIMO systems,there is enough space at the BS for the antennas to be spaced more than half a wavelength apart,spatial correlation can be avoided.However,employing multiple antennas at each user equipment(UE)leads to transmit correlation,this is due to space limitation of packing antennas in small UE and experiencing scattering environments.Some recent works have analyzed the influence of the channel correlation of the system.Considering the spatial correlation in practice and accounting for path loss,shadow fading,environmental parameters as well as antenna characteristics,a comprehensive channel model was proposed,the SE and EE of massive MIMO systems in both centralized and distributed configurations were analyzed in [20].Utilizing SM scheme,the achievable SE of massive SM-MIMO system was analyzed in [21],impact of the transmit correlation and imperfect channel state information(CSI)were investigated.Authors in[22]proposed a transmit antenna grouping scheme in massive generalized SM-MIMO systems to deal with the channel correlation,but multiuser scenarios were not considered.

Although the performance of massive SM-MIMO system with spatial correlation has been studied,these works mainly focus on the SE of the system,and the BER performance has not been addressed yet.More importantly,the shadow fading coefficient is always assumed to be constant over many coherent intervals or perfectly known to the BS in the literatures,the influence of shadow fading on the system is not analyzed.Motivated by the above analysis,we will analyze the BER performance of uplink multiuser massive SM-MIMO system with transmit correlation and the randomness of shadow fading is considered.Our contributions are summarized as follows:

1) An analytical framework is presented for evaluating the BER performance of the multiuser massive SM-MIMO system.Considering the transmit correlation as well as the randomness of shadow fading in practical scenarios,Kronecker model and gamma distribution are employed to represent the correlation and shadow fading coefficient distributions,respectively.Then,an easy-to-use and closed-form probability distribution function(PDF) of the correlated Rayleigh fading channel is deduced.

2) Utilizing ZF receivers for detection,the error probability of antenna index detection and the error probability of symbol detection are derived in sequence.Then,the approximated overall average BER expression of the multiuser massive SMMIMO system in the presence of transmit correlation is obtained.

3) For large SNR,the asymptotic BER expression is derived to assess the asymptotic performance of the multiuser massive SM-MIMO system with transmit correlation,and the corresponding diversity gain of the system is attained.Moreover,the theoretical expressions above are also applicable to evaluate the BER performance of single-cell massive SM-MIMO system without spatial correlation as a special case.

Notations:In this paper,(·)H,(·)−1and (·)†stand for the conjugate transpose,inverse operators and the pseudo-inverse operators,respectively.Cm×n(Rm×n)denotes am×ncomplex(real) matrix.Vectors and matrices are represented by boldface lower-case and upper-case symbols,respectively.Inis an identity matrix withnrows andncolumns,and[G]i,jdenotes the(i,j)-th entry of a matrix G.[G](n,:)and[G](:,n)represent then-th row and then-th column of the matrix G.Notation‖·‖means the Frobenius norm.Q(·) is theQ-function,Finally,CN(0,Θ),is used to denote the complex Gaussian distribution with zero mean and covariance matrix Θ.

II.MASSIVE MULTIUSER SM-MIMO SYSTEM WITH TRANSMIT CORRELATION

2.1 System Model

Figure1.Uplink multiuser massive SM-MIMO system.

Consider an uplink multiuser massive SM-MIMO system as illustrated in Figure1.This system consists ofKUEs,each UE hasNtantennas.A BS is placed at the center of the hexagonal cell withNrantennas servingKUEs simultaneously.Due to the limited size of the UE,the physical distance between the antennas in a UE becomes smaller.Therefore,the antennas in a UE become correlated.Besides,since the UEs in a cell is randomly distributed,the antennas in different UEs are regarded as independent.Let us denote the spatially-correlated channel matrix between the BS andKUEs as G∈CNr×KNt,where G=[G1,G2,...,GK],Gk=[gk,1,gk,2,...,gk,Nt]∈CNr×Ntdenotes the channel matrix between the BS andk-th UE of cell.Using the Kronecker model[23,24],the transmit correlated channel matrix can be expressed as G=HD1/2Rt1/2,in which H=[H1,H2,...,HK]∈CNr×KNtis the small scale fading matrix whose entries according toCN(0,1)and Hk=[hk,1,hk,2,...,hk,Nt]∈CNr×Nt.D∈RKNr×KNtis a diagonal matrix with D=diag{β1IK,β2IK,...,βKIK},whose elements describing the large-scale fading,βk=zk/(dk/d0)α,zkis shadow fading coefficient.A bit of labor shows thatzkcan be approximately modeled as gamma distribution [25–27]for analytical tractability,the PDF ofzkis expressed as

whereaandbdenote the shape and scale coefficients,respectively.Furthermore,dkdenotes the distance between thek-th UE and BS in cell,also,d0corresponds to the minimum reference distance between UE and BS,αdenotes the path-loss exponent.Rt=diag{Σ1,Σ2,...,ΣK}∈RKNt×KNtis a block diagonal matrix,the correlation between two antennas of UEkis modeled as exponentially decay with distance,i.e.,[Σk]i,j=ρt|i−j|,where Σkis the long term stable transmit correlation matrix for UEk,ρtrepresents transmit correlation coefficient[28].

2.2 Uplink Data Transmission

We assume that the system operating in time division duplexing(TDD)mode,BS has perfect knowledge of the CSI by means of uplink training,and ZF receiver is utilized to retrieve the transmitted data.At the transmitter,each UE uses SM for its signal transmission,and all UE send their data to the BS with equal powerpu,normalizing the noise variance to one,we define the average SNR aspu.Hence,the received signal at the BS is

where W=G（GHG）−1∈CNr×KNt,then,it can be further obtained thatis the [(k −1)Nt+nk]−thcolumn of W.Similarly,we define thatis the[(k −1)Nt+nk]−thcolumn of V.From (3) the received signal vector associated with UEkcan be expressed by

rkis then −thelement of ˜y.In order to detect the transmitted data correctly,based on the maximum likelihood(ML)criterion and utilize(4),the estimated active antenna indexand transmitted symbolcan be determined by

Intuitively,the average BER of the SM based system hinges on the detection errors of antenna indices and transmitted symbols,the transmitted bits can be recovered correctly only if both estimates are correct.

III.BER ANALYSIS

In this section,we will provide the BER analysis of uplink multiuser massive SM-MIMO systems over Rayleigh channel in the presence of transmit correlation,an approximated closed-form expression of overall average BER is deduced.In order to compute the overall average BER of the system (Pe),we need to calculatePa,kandPd,kfirstly.Pa,krepresents the average error probability of antenna index detection given that the symbol is accurately estimated of UEkandPd,kdenotes the average error probability of symbol detection given that the antenna index is correctly detected of the UEk,then the overall average BER of the systemPecan be attained by using the obtainedPa,kandPd,k.

3.1 Error Probability of Transmit Antenna Index Estimation

In this subsection,we give the analysis of the error probability of transmit antenna index estimation of UEk,since the transmit symbolx˙qkis known,the estimated active antenna index can be given by

whereD˙nk,nkindicates the decision metric of estimating the ˙nk −thtransmit antenna index asnk.Conditioned on G andx˙qk,the probability of the event that the antenna index ˙nkis determined as ˆnk,which is defined as conditional pairwise error probability(CPEP)can be given by

with the obtained APEP,the error probability of transmit antenna index estimation for UEkcan be approximately expressed as(11),

whereN(→) represents the number of bits in error between the activated antenna indexand estimated transmit antenna indexNote that(11)is an upper bound ofPa,k,and the tightness of the approximated expression will be degraded by higher order modulation scheme as well as more transmit antennas[29].

3.2 Error Probability of Transmitted Symbol Estimation

Assuming that transmit antenna index is known,the error probability of transmitted symbol estimation for UEkis analyzed in this subsection.Under this premise,the received SNR for UEkis obtained as

whereπ(M),µ(m),v(m)are uniquely determined by the modulation orderM[31].Then,we take the randomness ofzkinto account,the received SNR is represented asWith(1),(13),(14)and utilizing the similar approach as analyzingPa,k,the BER of the symbol estimation for UEkis given as(15).

3.3 Average Overall BER

In this subsection,average overall BER of the multiuser massive SM-MIMO system with transmit correlation is derived by using the results obtained in (11)and (15).The overall BER of UEkcan be approximated as

the similar methodology can be found in [3,32].With(16),the average overall BER in the cell can be given by averaging thePe,kofKUEs

IV.ASYMPTOTIC BER ANALYSIS AND DIVERSITY GAIN

In this section,the asymptotic expressions ofPa,kandPd,kare respectively derived in the sequel for large SNR,thereafter,the asymptotic expression ofPecan be attained,moreover,the diversity gain of the system is analyzed,which is useful in demonstrating the asymptotic performance.According to (7.1.23) in[33],the complementary error functionerfc(x) can be approximated for largex,that is

Consequently,for sufficiently high SNR,the asymptotic expression of APEP can be obtained by replacingQ(x)in(10)withand utilizing(18),the result is given as(19),

wherew=Nr −KNt+ 0.5,andLVU(Z) represents the Laguerre polynomial,asymptotic APEP is calculated with the help of (3.383.5) in [34].Subsequently,the asymptotic expression ofPa,kis obtained as

Besides,by substituting (18) into (15),the asymptotic expression ofPd,kat large SNR can be expressed as(21),

With(20)and(21),the asymptotic overall BER of the single cell massive SM-MIMO with transmit correlation for large SNR is obtained as

Based on(22)the diversity gain can be derived as

It is clear that the diversity gain of this system depends on the coefficient of shadow fading,the numbers of UE as well as the numbers of antennas.

V.SIMULATION RESULTS

In this section,we will evaluate the BER performance of the multiuser massive SM-MIMO system over transmit correlated Rayleigh fading channels by computer simulation to verify the effectiveness of the theoretical analysis.For the simulation setup,we consider a single hexagonal cell with radius ofR=500m,K=3 UEs are uniformly distributed,and the number of transmit antenna of each UE is set asNt=2,4 or 8.A BS is located in the center of the cell,and the receive number of the BS isNr=32,64 or 128.Besides,the reference distance is set asd0=100m,the path-loss coefficientαis 3.8,and the parameters of shadow fading area=1/b=3.3,5 or 10.The simulation results were obtained by using 108channel realizations.

Figure2.Error probability of transmit antenna index estimation of the system with different transmit correlation coefficients.

Figure3.Error probability of transmitted symbol estimation of the system with different transmit correlation coefficients.

Figure4.BER performance of the system with different transmit correlation coefficients.

Figure5.BER performances of the massive SM-MIMO and traditional massive MIMO.

In Figure2 and Figure3,we plot the error probability of transmit antenna index estimation(Pa)and the the error probability of transmitted symbol estimation (Pd) of the system,respectively,with different transmit correlation coefficientρtforNt=4,Nr=64,64QAM is adopted for modulation anda=1/b=3.3.The theoreticalPaandPdare obtained by averagingPa,k)andPa,k)in (11) and (15),respectively.As shown in these figures,theoreticalPdmatch the simulated results well while theoreticalPais an upper bound of the corresponding simulated ones.Specifically,thePa(d)withρt=0.5 is higher than that withρt=0.3,this is because the transmit correlation will lead to the deterioration of the error performance.Due to the same reason,thePa(d)withρt=0.7 is higher than that withρt=0.5.Asρt=0,no correlation happens,so the error performance of the system withρt=0 is superior to the other cases with transmit correlation.

Figure4 illustrates the overall average BER of the single cell massive SM-MIMO system with transmit correlation,simulation parameter settings are consistent with that in Figure2 and Figure3.Besides,the asymptotic BERs are plotted as well.ThePevalues are computed by using (16) and (17),the asymptotic BERs are obtained from (22).It can be seen in Figure4 that thePecurves is very close to the corresponding simulated values,and the performance degrades as the transmit correlation coefficients increases,as expected.The asymptotic BERs become tighter to the simulated values at large SNR due to better approximation.It also can be observed that all of the BER curves have the same slop at high SNR,which indicates that the transmit correlation has no impact on the system diversity gain.

In Figure5,we plot the BER performances of massive SM-MIMO and traditional massive MIMO over Rayleigh fading channel,wherea=1/b=3.3,Nt=4,Nr=32,ρt=0,andM=16,64.The simulated BERs are compared.For the same modulation orderM=16,it can be seen that the massive SM-MIMO has inferior BER performance than the corresponding traditional massive MIMO.The reason is that the massive SM-MIMO system needs to detect not only the transmitted symbol but also the antenna index,while the traditional massive MIMO system only performs the estimation of transmitted symbol.However,under the same transmission rate,the massive SM-MIMO system with 16QAM outperforms the traditional massive MIMO system with 64QAM because the former can use the lower-order modulation scheme.Figure6 illustrates the overall average BERPeand the asymptotic BER of the massive SM-MIMO system with transmit correlation for different modulation modes and different parameters of shadow fading,whereNt=2,Nr=64,ρt=0.3.Firstly,we can observe that the BERs withM=16 have smaller values than those withM=64,and have larger values than those withM=4,it is reasonable because high modulation order will lead to the reduction of the Euclidean distance,which can deteriorate the BER performance.Secondly,the BER curves under the same shadow fading parameter a have the same diversity order,which shows that the modulation order does not affect the diversity gain of the system.Moreover,in this case,the diversity gain of a is attained,which accords with the analysis in(23).

Figure6.BER performance of the system with different modulation modes and parameters of shadow fading.

Figure7.BER performance of the system with different numbers of antennas.

Figure8.BER performance of the system with different numbers of antennas and shadow fading parameters.

In Figure7,we plot the overall average BER and asymptotic BER with different numbers of transmit and receive antennas,where the parameters of shadow fading are set asa=1/b=5,M=64 andρt=0.3.From Figure7,it is found that the BER curves withNt=2 are lower than that withNt=4,and the BER performance gap between the theoreticalPeand simulated ones will be large whenNt=4.This is because that more transmit antennas will degrade the tightness of the bound at low SNR.Besides,with the increase ofNr,the system will achieve better performance.As a result,the system withNr=128 has the best performance.In order to clarify the factors affecting system diversity,Figure8 depicts the theoretical overall average BER and asymptotic BER under different numbers of antennas and parameters of shadow fading,whereρt=0.3 and 64QAM is adopted.To show the diversity gain of the system clearly,the range of the vertical axis is set from 10−10to 100in a log scale,and the simulated results are not presented.It is noted that the effect of system diversity gain is multifactorial.With the fixed number of UEs,the number of antennas and the shadow fading coefficient will affect the diversity of the system.The results in Figure.8 verify the validity of derived theoretical expression in(23),that is,asKNt −Nr+a ≤1,the diversity gain of the system isNr −KNt+1,and asKNt −Nr+a<1,the diversity gain ofais achieved.

VI.CONCLUSION

The BER performance of the multiuser massive SMMIMO system over transmit correlated Rayleigh fading channel is analyzed in this paper.The expressions of error probability of antenna index detection and the error probability of symbol detection are derived firstly.Based on this,the approximated close-form expression of the overall average BER of the system is obtained.To evaluate the asymptotic performance of the system at large SNR,an asymptotic BER expression is deduced by means of the asymptotical expansion of the complementary error function.Simulation results show that the theoretical expressions could match the simulated values well,which verify the effectiveness of the theoretical analysis.Besides,the impact of transmit correlation,modulation order,the number of antennas and shadow fading parameter on the system are analyzed.It is found that the BER performance of the system degrades in the presence of transmit correlation,and high modulation order,while large receive antenna arrays will bring about a superior BER performance.Moreover,with the asymptotic BER expression,the diversity gain of the system is derived,and it is determined by the minimum values ofNr −KNt+1 and shape parameters of shadow fadinga.The result shows that the number of antennas and shadow fading coefficient as well as the numbers of UE have strong effect on the asymptotic performance of the system.

ACKNOWLEDGEMENT

The authors would like to thank the anonymous reviewers for their valuable comments which improve the quality of this paper greatly.This work was supported by National Natural Science Foundation of China(61971220,62031017,61971221),Natural Science Foundation of Jiangsu in China (BK20181289),Open Research Fund of Nanjing University of Aeronautics and Astronautics (kfjj20200414),Open Research Fund Key Laboratory of Wireless Sensor Network and Communication of Chinese Academy of Sciences(2017006).

APPENDIX A

We extend the CPEP given in(7)as

Thereby,the result in(7)can be obtained,this completes the derivation.

APPENDIX B

According to(10)the APEP is expressed as

Substituting (27) into (26),the APEP is rewritten as(28),