Hang Yuan,Xiang Wang,Kai Yang,*,Jianping An

1 School of Information and Electronics,Beijing Institute of Technology,Beijing 100081,China

2 National Innovation Institute of Defense Technology,Chinese Academy of Military Science,Beijing 100000,China

Abstract:Terahertz(THz)wireless communication has the capability to connect massive devices using its ultra-large spectrum resource.We propose a hybrid precoding scheme for the cluster-based multi-carrier beam division multiple access(MC-BDMA)to enable THz massive connections.Both the inter-beam interference and inter-band power leakage in this system are considered.A mathematical model is established to analyze and reduce their effects on the THz signal transmission.By considering the peculiarities of THz channels and characteristics of THz hardware components,we further propose a three-step hybrid precoding algorithm with low complexity,where the received signal power enhancement,the inter-beam interference elimination,and the inter-band power leakage suppression are conducted in succession.Simulation results are presented to demonstrate the high spectrum efficiency and high energy efficiency of our proposed algorithm,especially in the massive-connection scenarios.

Keywords:terahertz wireless communications;multicarrier hybrid precoding;power leakage suppression

I.INTRODUCTION

Terahertz(THz)wireless communication is envisioned as a key enabler in the future wireless networks,as its usable bandwidth is ultra large which easily exceeds 100 GHz[1,2].Due to the severe propagation loss,wireless communications in the THz band require massive multiple-input multiple-output(MMIMO)[3]to provide high beamforming gain and extend the transmission distance[4].However,the high cost and hardware complexity of THz devices in radio frequency(RF)chains[5,6]make the conventional fully digital precoding nearly impractical[7].Against this background,hybrid precoding where the signal processing is divided into digital domain at baseband and analog domain in RF chains was investigated[8-10].

Hybrid precoding architecture could provide high data rate while maintaining ultra-low hardware complexity,as the number of RF chains can be drastically reduced.In[8],the spatial sparsity of channels as well as large tightly-packed antenna arrays were leveraged to design the hybrid precoding.The results in[8]indicate that the degree of freedom(DoF)in the hybrid precoding architecture is jointly restricted by the rank of channel matrix and the number of RF chains,instead of the number of antennas.In order to increase the DoF,dynamic-subarray hybrid precoding architecture,where subsets of antennas are adaptively connected to RF chains,was developed[11].An enlightening criterion to construct dynamic subarrays was obtained which implies that each subarray should be formed with highly correlated antenna elements.Both the studies in[8,11]were conducted in a single-user system where the channel responses are highly correlated.

Actually,investigating the multiuser hybrid precoding problem is of special interest,as the diverse channels among users provide more DoF to exploit[12,13].However,it is challenging to design one single analog precoder serving all users.In[14],a transmission scheme where users are scheduled at the entire band for each time slot was provided,which indicates that the channels of different subcarriers can be highly correlated due to the sparsity of channels.In[15],a low-complexity hybrid precoding algorithm was developed for multiuser communications over frequency selective fading,where an extra digital precoding matrix was proposed to compensate for the DoF loss in the analog precoder.It is worthwhile to emphasize that,in these studies,the number of served users cannot exceed the number of RF chains.This limitation becomes an obstacle to implementing the hybrid precoding in massive-user systems.

Apart from the ultra-high data rate in the order of Tera-bit-per-second(Tbps),THz wireless communications also has the capability to provide a massive number of device connections.To break the limitation mentioned above,we proposed a cluster-based multicarrier beam division multiple access(MC-BDMA)transmission scheme in[16],where a user grouping strategy with rough beam pre-scanning and an iterative power allocation strategy were developed.Despite that[16]provides an insight into the inter-beam interference elimination in massive-user scenarios,the power leakage problem which may degrade the system performance severely was not considered.

To address this problem,we develop a new hybrid precoding algorithm for power leakage suppression in THz cluster-based MC-BDMA systems.The proposed algorithm enables massive connections in THz communications using the ultra-large THz spectrum resource.The main contributions of this paper are summarized as follows:

·We exploit the multi-carrier hybrid precoding architecture to develop a cluster-based MC-BDMA transmission scheme.In this scheme,massive users are grouped into several clusters.The users in the same cluster are covered by one single beam,corresponding to one single RF chain,and divided by the distance-aware multi-carrier(DAMC)modulation.This scheme could serve massive users using the hybrid precoding with a limited number of RF chains.

·We consider both the inter-beam interference and the inter-band interference which is caused by the power leakage from the side-lobes of other bands.A mathematical model of the side-lobe power leakage is established.We also provide an orthogonal precoding method for the inter-band power leakage suppression.

·We propose a three-step hybrid precoding algorithm to maximize the achievable sum-rate of the system with low complexity.In the design,we take the peculiarities of THz channels and characteristics of THz transceivers into account.The analog precoding is optimized to enhance the received signal power in the first step.The digital precoding is designed to eliminate the inter-beam interference and suppress the inter-band power leakage in the second and third steps,respectively.

The performance of our proposed algorithm and the cluster-based MC-BDMA transmission scheme is evaluated by simulations and compared with both the existed linear hybrid precoding and fully digital precoding schemes.Aided by simulation results,we clearly demonstrate the benefits of our proposed hybrid precoding algorithm in terms of high spectral efficiency,high energy efficiency,and low complexity,especially in the massive-user scenarios.

We use the following notation throughout this paper.Given a matrix A,‖A‖denotes the Frobenius norm of A,A-1,AT,and AHdenotes the inverse,transpose,and Hermitian transpose of A,respectively.The vector andenotes thenth column of A and[A]r,cdenotes the entry in therth row andcth column of A.

II.SYSTEM MODEL

We consider a single-cell THz wireless network,as depicted in Figure 1,wherein a multi-antenna BS communicates withMmulti-antenna users in the THz band.The BS is located at the origin and exploits hybrid precoding architecture withNBSantennas andNRFRF chains to reduce the hardware complexity.Each user is equipped withNUEantennas and one single RF chain.We assume that the number of users is much larger than the number of RF chains at the BS,i.e.,M?NRF.The BS employs the cluster-based MC-BDMA transmission scheme to support massive connections.Specifically,each RF chain,corresponding to one single beam,serves many users in the same cluster who are divided by the multi-carrier modulation.Different clusters are served by different RF chains using beam-width control techniques[17].In this section,we describe the multi-carrier hybrid precoding architecture and the adopted wideband THz channel model.

Figure 1.Illustration of the THz cluster-based MC-BDMA network.

Figure 2.A block diagram of the multi-carrier hybrid precoding architecture at the BS.

2.1 Multi-Carrier Hybrid Precoding

As illustrated in Figure 2,the source signal is precoded at the BS by the hybrid precoding architecture before transmission.The symbol vector on thekth carrier,denoted by x[k],is expressed as

where FRFis anNBS×NRFanalog precoding matrix in the RF domain,FBB[k]is anNRF×Nsdigital precoding matrix at baseband,and s[k]is a symbol vector before the precoding.Here,we note that FBB[k]s[k]is the transformed signal in the time domain afterNRFparallelK-point inverse fast Fourier transform(IFFT).A guard interval(GI)is added on each RF chain to eliminate the inter-symbol interference(ISI)before going through the digital-to-analog converter(DAC).The analog precoding is implemented by THz phase shifters[18].We assume that those THz phase shifters have infinite resolution with no return loss,insertion loss,or phase error.

Without loss of generality,we consider a typical user who is scheduled into thenth cluster and thekth carrier.The received signal vector,denoted by yn[k],is presented as

wheresn[k]is the symbol transmitted to the typical user,Hn[k]is the THz channel response matrix,?n[k]denotes the inter-band interference from other carriers,which we will analyze in the next section,and nn[k]is the additive white Gaussian noise(AWGN)corrupting the received signal with the power ofσ2n.

2.2 Wideband THz Channel Model

We consider many non-overlapping carriers in a transmission window[19].Here,we use the Saleh-Valenzuela(S-V)method[20]to model the channel on each carrier.Considering the high reflection and scattering loss in the THz band,arrival paths generally consist of one dominant line-of-sight(LoS)path and a few non-line-of-sight(NLoS)paths.The channel response matrix corresponding to the typical user who is scheduled into thenth cluster and thekth carrier is presented as[21,22]

whereLNLis the number of NLoS paths,GtandGrare antenna gains at the transmitter and receivers,respectively,αis the complex gain,θis the angle of departure(AoDs),φis the angle of arrival(AoAs)at the receiver,and at(·,·)and ar(·,·)are antenna array response vectors at the BS and receiver,respectively.We now provide a brief but crucial introduction to some unique characteristics of THz channels.

2.2.1 Path Loss

At THz frequencies,the molecular absorption effect which is highly dependent of frequencies drastically affects the signal propagation.Thus,complex path loss of THz channels consists of the spreading loss and the molecular absorption loss.The amplitude of the LoS path loss is presented as

wherecis the speed of light andkabs(fk)is the frequency-dependent medium absorption coefficient which is determined by the composition of the transmission medium at the molecular level[22].To incorporate the losses of NLoS paths,the Fresnel reflection coefficient needs to be considered[23].We note that only up to the second order reflections need to be considered due to the high reflection loss in THz channels[24].

2.2.2 Antenna Array Response Vector

The antenna array response vector is only relevant to the antenna array architecture.The BS is equipped with a basic ULA.Accordingly,the antenna array response vector at the transmitter,at(θ),is expressed by

whereadenotes the space between two adjacent antennas andλcdenotes the wavelength.

2.2.3 High Spatial Sparsity

In the THz band,the reflection and scattering loss through the signal transmission is ultra large due to the small wavelength.As a result,THz channel shows unique high sparsity in the spatial domain,i.e.,THz channel has only a few NLoS paths and the gap between the LoS and NLoS path loss is significant,which is up to 15 dB[9].Hence,the LoS path shows absolute dominance in the THz signal propagation.

We note that our proposed scheme and corresponding analysis are not restricted to the channel model in(3),but general for THz communications.We introduce the THz channel model as some characteristics are important for understanding the design philosophy of our scheme proposed in Section IV.

III.INTER-BAND POWER LEAKAGE ANALYSIS

In this work,we consider many non-overlapping carriers in the ultra-large THz band.Thus,we assume perfect frequency synchronization and neglect the intercarrier interference caused by the carrier frequency offset[25].Instead,we focus on the inter-band interference caused by the power leakage from the sidelobes of other bands.In this section,we establish a mathematic model of side-lobe power leakage and analyze its effect on the signal transmission.After that,we provide an orthogonal precoding method for the inter-band power leakage suppression,which is a basis of the hybrid precoding design in Section IV.

Here,we consider a basic DFT-based hybrid precoding system with no special pulse shaping function.Hence,the DFT modulation on each RF chain can be treated as a rectangular window in the time domain.We note that this is a basic case and the corresponding analysis can be extended to many systems using special windowing functions.We denote the symbol duration asTand guard period asTg.By Fourier transforming the rectangular window,we obtain the frequency-domain representation of thekth carrier on frequencyf,which is presented as[26]

Actually,wk(f)is the inter-band interference coefficient from thekth carrier on the frequencyf.Hence,the inter-band interference on thekth carrier,denoted by?n[k],is expressed by

We note that?n[k]only includes the inter-band interference on thenth RF chain.The leakage power on other RF chains can be treated as the inter-beam interference in(2)and will be not discussed in this section.

We consider the spectral power leakage from all carriers and define the combined leakage in the RF domain as

wherePs=is the average transmit power.The second equation in(9)implies that the average leakage power is determined byThe optimal combined digital precoding matrixfor leakage power maximization should be designed such that the Frobenius norm ofis minimized.The optimization problem ofis expressed by

To solve this problem,we first assume thatis an orthogonal matrix.The columns ofcan be treated asNRForthogonal bases in CK.The restK-NRFbases,which are orthogonal to,organise aK×(K-NRF)matrix,named.Here,we construct three matrices in CK×K,i.e.,andWe find thatThus,the optimization problem in(10)can be transformed to min■which can be treated as approximatingwithWe introduce the Eckart-Young Low Rank Approximation Theorem to solve this problem,which is described as follows[26].If F is of rankaand=F+E is of rankb,b＞a,then the sum of the lastb-asingular values ofis smaller than or equal to the norm of E,i.e.,denotes theith singular value of?A.Hence,we have

By decomposing W using singular value decomposition(SVD)into W=UWΣW,we have the optimal digital precoding matrix to minimize the norm as

where VW,1∈CK×NRFconsists of the lastNRFcolumns in VWcorresponding to the smallestNRFsingular values and Q is anNRF×NRFunitary matrix.

The power leakage after the precoding can be calculated as

As such,the leakage power is not eliminated but suppressed intoPL,min.We note from(13)that the minimum leakage power only contains the smallestNRFsingular values of W andK-NRFbiggest ones are removed by the digital precoding.This also implies that the suppressed leakage power increases as the number of RF chains increases,but is irrelevant to the number of carriers.

IV.HYBRID PRECODING DESIGN

In this section,we investigate the hybrid precoding design problem in the THz cluster-based MC-BDMA system,where both the inter-beam interference and the inter-band interference are considered.Our objective is to design the analog and digital precoding matrices to maximize the achievable sum-rate of the system considering the characteristics of THz channels and the hardware constraints of THz devices.

According to(2)and(7),the signal-to-interferenceplus-noise ratio(SINR)is presented as

wheredenotes the power of inter-beam interference anddenotes the power of inter-band interference.

Hence,the achievable sum-rate of the system is given by

We aim to maximize the achievable sum-rate such that hardware constraints of the hybrid precoding architecture are satisfied.As such,the hybrid precoding design problem is formulated as

whereC1is the constant-modulus constraint for each entry of the analog precoder andC2is the normalized power constraint on the hybrid precoder.

We note that the problemPis in general intractable due to the non-convex constraint inC1and the coupling effect between analog and digital precoding matrices inC2.In this work,we exploit a greedy method to decouple the problemPinto three sub-problems and obtain a near-optimal solution with low complexity.To this end,we propose a three-step algorithm to design the hybrid precoding where the main idea is to optimize the analog precoder and digital precoders,respectively.In the first step,the analog precoder FRFis designed to maximize the average received power for all users,while ignoring the inter-beam interference and the inter-band interference.In the second step,the digital precoder FBB[k]is designed to eliminate the inter-beam interference.In the third stage,the digital precoder is then designed to minimize the inter-band interference using the rest DoF,according to the power leakage suppression method in(10).

4.1 Analog Precoding for the Desired Signal Power Maximization

It is challenging to design one common analog precoder to serve massive users in the hybrid precoding system since different carriers have diverse channel matrices.In this work,we explore the cluster-based MC-BDMA transmission scheme proposed in our previous work[16],where multiple users located in the same cluster are served by one single beam,which corresponds to one single RF chain,with the DAMC modulation[27].Due to the significant gap between the LoS and NLoS path loss,the impact of NLoS paths is ignored in this scheme.Hence,one RF chain can support many users who have approximately the same LoS AoDs.In this case,each RF chain can be seen as being independent of other RF chains.Hence,we propose to design the analog precoding matrix column by column,i.e.,design each RF chain individually.

Before transmission,the BS activates a subset of antennas to perform a rough but fast beam pre-scanning with a given angular separation.The angular separation is determined by the number of RF chains.With the pre-scanning,we assume that the location information of all users is available at the BS.These users falling into the same sector are assigned to the same user cluster.After the user clustering,the BS carries out analog precoding design for each cluster.We note that THz channel shows unique sparsity,i.e.,the transmit power is mainly focused on the LoS path.Thus,the statistical LoS AoDs of all users in the cluster are critical to design the analog precoding.We assume that the beam-steering vector,denoted by at(θ0),with an ideal LoS AoD,θ0,is adopted as the analog precoding vector.The effective channel at baseband,from one RF chain to the user who is scheduled into thenth cluster and thekth carrier,is expressed as

where

The optimal beam-steering vector on thenth RF chain can be designed such that the sum of the effective channel modulus for all users in the cluster is maximized,which is formulated as

Ignoring the extra hardware constraints,the optimal solution is obtained by the statistical eigen beamforming,i.e.,the dominant eigenvector ofConsidering the constantmodulus constraints,the analog precoding problem requires the optimization of beam-steering direction.In this case,we propose to select the beam-steering direction for each RF chain which is the same as that of the optimal statistical eigen beamforming vector.This idea is of a special interest as it provides a useful insight to the hybrid precoding design that the key role of analog precoding is to enhance the received power for each user by precise beamforming.Hence,the optimal beam-steering vector on thenth RF chain is presented as

4.2 Digital Precoding for the Inter-Beam Interference Elimination

With the analog precoder,we need to design the digital precoding in the second step to eliminate the interbeam interference.In conventional linear precoding systems,where each user is served by one beam,i.e.,NRF≥M,the low-complexity ZF precoding can be used to eliminate the interference.However,in our considered cluster-based MC-BDMA hybrid precoding system,M?NRF,which indicates that the pseudo-inverse of theNRF×Mcombined channel matrix does not exist.Thus,the conventional ZF method does not work in the massive-user scenario.

We organize the effective channel on each carrier to reduce the matrix dimension such that the ZF digital precoding does exist.With the analog precoder FRFat the BS and a basic maximal ratio combiner cn,kat the receiver,we organize the equivalent channel vector on thekth carrier as?hn[k]=cn,kHn[k]FRF,which can be seen as a MISO channel.The combinedNs×NRFequivalent channel matrix on thekth carrier can be organized asThen,the ZF digital precoding matrix can be generated on each carrier as

Each column of the digital precoding matrix needs to be normalized by fBB,n[k]=In our cluster-based MC-BDMA hybrid precoding scheme,users are grouped intoNsclusters.On each carrier,onlyNsusers need to be divided.Hence,the inverse operation in(21)is conducted on anNs×Nsmatrix,instead of anM×Mmatrix,which significantly reduces the computational complexity.

4.3 Digital Precoding for the Inter-Band Power Leakage Suppression

In the above subsection,we have obtained the optimal digital precoders,FBB[k],to eliminate the interbeam interference.Now,we need to re-design the digital precoding,in the third step,to suppress the inter-band power leakage.To this end,we first define a combined matrix across all carriers as a goal matrix,which is presented as GBB,n=In section III,we have concluded that the optimal digital precoder which minimizes the inter-band power leakage has the expression in(12).We also note that minimizing the inter-band leakage power costsK-NRFDoF and there areNRFDoF left,i.e.,the DoF in Q∈CNRF×NRF.Hence,the extraNRFDoF provide a opportunity to approximate VW,1Q to GBB,n.

If VW,1Q=GBB,the precoded signal would have both the ZF waveforms to eliminate the inter-beam interference and the orthogonal waveforms to suppress the inter-band power leakage.Hence,we now approximate the precoding matrix,VW,1Q,to the goal matrix,GBB,n,by using the leftNRFDoF in Q,which is expressed as

The above problem is an Orthogonal Procrustes Problem,whose optimal solution is given by[26]

Algorithm 1.Three-step hybrid precoding algorithm for the cluster-based MC-BDMA scheme.1:Input:Hn[k],cn,k,?k,n,and W.2:for n=1:NRF do 3:Rn=1 K images/BZ_98_1572_551_1620_597.pngK k=1 HHn[k]Hn[k].4:fRF,n= 1■NBS ej■(eig1(Rn)).5:end for 6:FRF=[fRF,1,fRF,2,...,fRF,NRF].7:for k=1:K do 8:for n=1:Ns do 9: ?hn[k]=cn,kHn[k]FRF.10:end for 11: ?H[k]=[?h1[k],?h2[k],...,?hNs[k]].12:FBB[k]=?HH[k]images/BZ_98_1722_1076_1749_1122.png?H[k]?HH[k]images/BZ_98_1965_1076_1992_1122.png-1.13:for n=1:Ns do 14: fBB,n[k]= fBB,n[k]‖F(xiàn)RFfBB,n[k]‖.15:end for 16:end for 17:W=UWΣWVHW.18:VW,1=[VW]:,K-NRF+1:K.19:for n=1:NRF do 20:GBB,n=images/BZ_98_1578_1562_1597_1607.pngfTBB,n[1],fTBB,n[2],...,fTBB,n[K]images/BZ_98_2155_1562_2174_1607.png.21:VHW,1GBB,n=?U?Σ?VH.22:FBB,n=VW,1?U?VH.23:end for 24:FBB[k]=images/BZ_98_1548_1808_1567_1853.pngfTBB,1[k];fTBB,2[k];...;fTBB,NRF[k]images/BZ_98_2142_1808_2161_1853.png.25:Output:FRF and FBB[k],?k.

V.SIMULATIONS AND DISCUSSIONS

In this section,we evaluate the performance of our proposed hybrid precoding scheme.Throughout this section,users are randomly distributed in anR-radius semicircle area following the Poisson Cluster Process(PCP)model.Specifically,cluster centers are randomly distributed with the Poisson Point Process(PPP)model and users in each cluster are independent and identically distributed(i.i.d.)in anr-radius area around the cluster center.The user clustering is assumed to be perfectly conducted using the location information.In all schemes,the number of active RF chains at the BS is assumed to be equal to the number of clusters.Also,the number of carriers is equal to the number of users in each cluster,which requires tens of non-overlapping carriers in the THz band to realize massive connections.Simulation parameters are listed in Table 1.We note that the molecular absorption coefficient is calculated using the HITRAN database[28].The power amplifier on each antenna is used to compensate for the power loss caused by the power divider.

Table 1.Simulation Parameters.

Figure 3 plots the spectral efficiency of our proposed cluster-based MC-BDMA hybrid precoding scheme against the average signal-to-noise ratio(SNR).To demonstrate the benefits of our proposed scheme,we compare our proposed scheme with the existing multicarrier hybrid precoding scheme in[15],i.e.,the linear ZF hybrid precoding scheme proposed for the multiuser case when the number of RF chains is larger than or equal to the number of users.We also consider two benchmarks:i)The unconstrained clusterbased MC-BDMA hybrid precoding scheme assuming no interference and ii)The hybrid precoding scheme with no inter-band power leakage suppression,i.e.,the inter-band power leakage is treated as noise.

We first observe from Figure 3 that our proposed cluster-based MC-BDMA hybrid precoding scheme significantly outperforms the existing scheme in[15],which clearly demonstrate the benefits of our proposed scheme in the massive-user scenario.This is because the linear ZF method does not work when the number of users is much larger than the number of RF chains.The inter-user interference(IUI)decreases the SINR at receivers severely.On the other hand,we propose to divide users in the same cluster using the multi-carrier modulation.In this case,,the inter-beam interference among clusters can be eliminated by digital precoding on each carriers.We also observe that the spectral efficiency of our proposed scheme is much higher than that of the scheme with no inter-band power leakage suppression and approaches that of the scheme assuming no interference.This indicates that our proposed scheme suppresses the inter-band power leakage,even though cannot eliminate it.

Figure 3.Spectral efficiency of hybrid precoding schemes versus the average SNR,where NBS=256,K=30,NRF=8,and M=240.

Figure 4 shows the data rate comparison of several hybrid precoding schemes mentioned above in terms of the number of served users.We note that the number of RF chains is always smaller than the number of users.We first observe from Figure 4(a)that the achievable sum-rate of our proposed scheme increases with the number of users almost linearly.This increase mainly comes from the multiplexing gain due to the use of multiple carriers.We then observe from Figure 4(b)that the average data rate per user of our proposed scheme increases slightly as the number of user increases,which is mainly resulting from the spatial multiplexing gain.However,in massive-user cases,the average data rate remains unchanged,even decreases as more users are included.This is because the limitation of degrees of freedom(DoF)in the beamspace causes the serve IUI,especially reflected in the benchmark scheme which ignores the interference.

We then plot the energy efficiency of all schemes against the transmit power at the BS,denoted byPs,in Figure 5.The energy efficiency here is defined asρ=Rsum/(K(Ps+NRFPRF)),wherePRFdenotes the power consumption of THz RF chains.We adopt a typical valuePRF=136 mW[7].The AWGN power is set as=-80 dBm.We observe that the energy efficiency of our proposed scheme is much higher than that of the existing hybrid precoding scheme and the fully digital scheme.We also notice that the energy efficiency of our proposed scheme reaches the maximum value whenPs=19 dBm.This observation suggests that we can improve the energy efficiency of the system by regulating the transmit power.

Figure 4.Achievable data rate of hybrid precoding schemes versus the number of users,where NBS=256,NRF=8,and SNR=0 dBm.

VI.CONCLUSIONS

Figure 5.Energy efficiencies of precoding schemes versus the transmit power at the BS,where NBS=256,K=50,NRF=6,and M=300.

We proposed a three-step hybrid precoding algorithm for the cluster-based MC-BDMA in THz wireless communications leveraging the high spatial sparsity of THz channels and capabilities of THz transceivers.The inter-beam interference among clusters and the inter-band power leakage among carriers are considered.Specifically,we established a mathematical model to analyze the side-lobe power leak-age from other bands,based on which an orthogonal precoding method to minimize the average leakage power was provided.We further formulated a hybrid precoding design problem for maximizing the achievable sum-rate.To address this problem,we proposed a low-complexity three-step hybrid precoding algorithm which is mainly accomplished by the EVD and SVD.Simulation results were provided to demonstrate the benefits of our proposed algorithm where the high spectrum efficiency and energy efficiency were highlighted.

ACKNOWLEDGEMENT

This work was supported by the National Natural Science Foundation of China under Grant No.61771054.