# Dynamically Subarray-Connected Hybrid Precoding Scheme for Multiuser Millimeter-Wave Massive MIMO Systems

Authors: Guangyan Liao; Feng Zhao
Journal: Wireless Communications and Mobile Computing
 (2021)

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## Abstract

Hybrid precoding is widely used in millimeter wave (mmWave) massive multiple-input multiple-output (MIMO) systems. However, most prior work on hybrid precoding focused on the fully connected hybrid architectures and the subconnected but fixed architectures in which each radio frequency (RF) chain is connected to a specific subset of the antennas. The limited work shows that dynamic subarray architectures address the tradeoff between achievable spectral efficiency and energy efficiency of mmWave massive MIMO systems. Nevertheless, in the multiuser hybrid precoding systems, the existing dynamic subarray schemes ignore the fairness of users and the problem of user selection. In this paper, we propose a novel multiuser hybrid precoding scheme for dynamic subarray architectures. Firstly, we select a multiuser set among all users according to the analog effective channel information of the base station (BS) and then design the subset of the antennas to each RF by the fairness antenna-partitioning algorithm. Finally, the optimal analog precoding vector is designed according to each subarray, and the digital precoding is designed by the minimum mean-squared error (MMSE) criterion. The simulation results show that the performance advantages of the proposed multiuser hybrid precoding scheme for dynamic subarray architectures.


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## Body

## 1. Introduction

Millimeter-wave massive MIMO, as a key technology in the next-generation wireless systems, can achieve multigigabit data rates benefiting from its abundant frequency resource [1] and provide highly directional beamforming gains to compensate for the path loss of mmWave signals [2]. For mmWave massive MIMO systems, hybrid precoding not only solves the problem that the fully digital precoding unsuitable for this systems, but also can achieve performance close to a full digital precoding [3, 4].The studies of hybrid precoding focused on the fully connected architecture [5–8] and partially connected architecture [9–11], and there are two main ways to solve the hybrid precoding problem. One is the joint analog domain precoding and digital domain precoding simultaneously [5–7, 9]. And another is the analog domain precoding designed by the channel gain at first and then the digital domain precoding designed by the analog equivalent channel [12–15], and this way is usually applies to multiuser mmWave systems. However, most previous works did not explore the problem of the tradeoff between spectral efficiency and energy efficiency.Limited works have been carried out for dynamic subarray-connected structures were proposed in [16–20]. This new structure is different from the previous structure, which uses a dynamic subarray connection structure to connect the RF chain and antenna according to the channel state information of the users. In [16], adaptive hybrid precoding (AHP) was developed to achieve the highest system spectral efficiency, but it is only applicable for the number of employed RF chains which is equal to the number of users. In [17], based on knowing the long-term channel characteristics, a technique of dynamically hybrid subarrays was developed, but the user’s fairness was ignored. In [18], consider the number of RF chains which is more than or equal to that of scheduled users and utilized AHP based on the singular value decomposition (SVD). However, that scheme does not study the problem of user selection when the number of users increases. Moreover, in [19], utilize the subchannel difference of the users, and a dynamically connected hybrid precoding was developed. This scheme allocates a different number of antennas for each RF chain, but sometimes some RF chains cannot allocate an antenna, which was unfair to some users. User selection was considered in [20], and a multiuser hybrid precoding framework based on codebooks was proposed, which adopted the antenna-partitioning algorithm based on the maximum increment of the signal-to-interference-noise ratio (SINR), but it has the same problem as [19].In this paper, we proposed a new antenna-partitioning algorithm and hybrid analog/digital precoding schemes for the dynamic subarray characteristic. The main contributions of this paper can be summarized as follows.(1)
Proposing a new antenna-partitioning algorithm for multiuser mmWave system with a dynamic subarray characteristic. The algorithm guaranteed fairness because each user can obtain approximately equal channel gain. The new antenna-partitioning algorithm proposed in this paper avoids the problem that the number of antennas in some subarrays which is zero may occur in [20], and it can be used to the number of transmitting antennas and the number of subarrays which is not multiple. However, the AHP scheme in [16] is only applicable when the number of transmitting antennas is a multiple of the number of subarrays(2)
The dynamic hybrid structure model is obtained by the new antenna partition algorithm, which is based on the maximum channel gain of each user rather than SINR. Therefore, the spectrum efficiency of the system can be improved by eliminating the interuser interference through digital baseband precoding. Firstly, we design the analog precoding based on phase-only precoding. Then, in order to eliminate the interuser interference and ensure the high performance of the system under low SNR, we design the digital baseband precoding based on the minimum mean square error criterion. The simulation results show the advancement of our schemeThe remainder of the paper is as follows: Section2 describes the multiuser massive MIMO system model and millimeter-wave channel model. The problem description is proposed in Section 3. Section 4 provides a detailed description of the proposed solution. Simulation results are presented in Section 5. Finally, conclusions are drawn in Section 6.Notation. In this paper, we use the notation as the following: X is a matrix, x is a vector, and x is a scalar. XT, XH, and X† denote the transpose, transpose-conjugate operation, and pseudo-inverse of X, respectively. XF and X denote the Frobenius norm and determinant of X. TrX and EX denote the trace and expectation of X. CNm,R is a complex Gaussian random vector with mean m and covariance R.


## 2. System Model and Channel Model

### 2.1. System Model

In this study, we describe three different multiuser hybrid precoding system models, the fully connected structures, the subconnected structures, and the dynamic subarray structures. Firstly, a fully connected multiuser downlink massive MIMO hybrid precoding system models as shown in Figure1, where the BS is equipped with NRF RF chains and NT antennas. We assume that the BS simultaneously communicates with K mobile users. Each mobile user is equipped with one RF chain and NR antennas, where K≤NRF≤NT. The BS antennas are tagged as 1,2, …, NT, and let Ξ=1,2,⋯,NT. The baseband precoding matrix FBB=FBB1,FBB2,⋯,FBBK, where FBBk∈ℂNRF×NR is the baseband precoding matrix for k th mobile user. The analog precoding weight is a NT×NRF matrix FRF=fRF1,fRF2,⋯,fRFNRF, where fRFk∈ℂNT×1 is the analog precoding vector for the k th RF chain. FRF and FBB are linked by the total power constrains, i.e, FRFFBBF2=K. The sample transmitted signal can be represented as [21].
(1)x=FRFFBBs,where s=s1,s2,⋯,sKT is the K×1 vector of transmitted symbols, which satisfy EssH=P/KIK, and P is the average total transmitted power.Figure 1
Fully connected architecture of the multiuser hybrid precoding in mmWave massive MIMO system.Generally, the received downlink signal of the mobile userk can be represented as [21].
(2)yk=wkHkFRFfBBksk+∑i≠kwkHkFRFfBBisi+wknk,where Hk∈ℂNR×NT is the channel matrix from the BS to the k th mobile user. Let H=H1,H2,⋯,HNT. nk is an additive Gaussian white noise vector such that nk~CN0,δ2I. wk∈ℂ1×NR denotes the digital vector at the k th mobile user.Then, we describe the two structures shown in Figure2 and define the form of their analog precoding matrices. The conventional subarray architecture in which each RF chain is connected to a specific subset of the antennas as shown in Figure 2(a), and each RF chain connected to the antenna is fixed in the order. Therefore, we can define the analog precoding matrix FRF as [9].
(3)FRF=fRF10⋯00fRF20⋮⋱⋮00⋯fRFNRFNRFM×NRF,where fRFj=fjj−1M+1,fjj−1M+2,⋯,fjjMT denotes the analog weighting vector for the j th RF chain, M=NT/NRF, and j=1,2,⋯,NRF.Figure 2
Two subconnected architectures of the multiuser hybrid precoding in mmWave massive MIMO system: (a) conventional subarray architecture, where each RF chain is connected to a fixed subset of the antennas; (b) dynamic subarray architecture, where each RF chain is connected to interlaced antennas.
(a)(b)The dynamic subarray architectures are shown in Figure2(b), which each RF chain is connected to a dynamic subset of the antennas. The analog precoding matrix FRF=fRFS1,fRFS2,⋯,fRFSNRF is derived with
(4)fRFSjt,j=fjt,t∈Sj,0,t∉Sj,where fRFSj∈ℂNT×1 is the analog precoding vector for the j th RF chain, and fRFSjt,j denotes the element of t th antenna connected to the j th RF chain. Assume that fjt is a value of fRFSjt,j. The set Sj contains label of antenna connected to the j th RF chain, and let set S=S1,S2,⋯,Sj,⋯,SNRF, ∑j=1NRFSj=NT.


### 2.2. mmWave Channel Model

Millimeter wave channels with high free-space path loss and large dense antenna arrays are unlikely to follow the rich scattering model at low frequencies. According to the characteristics of millimeter wave propagation, the dense array makes the antenna highly correlated, and the high path loss in high free-space leads to the limited space selectivity. Thus, the millimeter-wave channel is considered to have finite scattering [1]. Traditional MIMO channel modeling methods are not accurate in millimeter wave channel. The geometric channel model can embody the low rank and spatial correlation characteristics of mmWave communications [9]. Thus, we consider a geometric channel model with Lk scatterers for the k th user and assume that each scatterer contributes a single propagation path between the BS and the user [18, 21]. Under this model, the k th user’s channel can be represented as
(5)Hk=NTNRLk∑l=1Lkβl,kaRθl,kRaTHθl,kT,where Lk is the number of scatterers and βl,k is the complex gain of the l th path between the BS and the k th user. θl,kR and θl,kT, respectively, represent the azimuth of arrival (AOA) and departure (AOD) of the l th path between the BS and the k th user, and the range of values for both θl,kR and θl,kT is 0,2π. aTθl,kT and aRθl,kR are the antenna array response vectors of the base station and k th user, separately. For a uniform linear array (ULA), the array response vector can be written as
(6)aTθl,kT=1NT1,ej2πd/λsinθl,kT,⋯,ejNT−12πd/λsinθl,kTT,where λ is the signal wavelength and d=λ/2 denotes the spacing distance between two adjacent antennas. The array response vector aRθl,kR takes a similar fashion.


## 2.1. System Model

In this study, we describe three different multiuser hybrid precoding system models, the fully connected structures, the subconnected structures, and the dynamic subarray structures. Firstly, a fully connected multiuser downlink massive MIMO hybrid precoding system models as shown in Figure1, where the BS is equipped with NRF RF chains and NT antennas. We assume that the BS simultaneously communicates with K mobile users. Each mobile user is equipped with one RF chain and NR antennas, where K≤NRF≤NT. The BS antennas are tagged as 1,2, …, NT, and let Ξ=1,2,⋯,NT. The baseband precoding matrix FBB=FBB1,FBB2,⋯,FBBK, where FBBk∈ℂNRF×NR is the baseband precoding matrix for k th mobile user. The analog precoding weight is a NT×NRF matrix FRF=fRF1,fRF2,⋯,fRFNRF, where fRFk∈ℂNT×1 is the analog precoding vector for the k th RF chain. FRF and FBB are linked by the total power constrains, i.e, FRFFBBF2=K. The sample transmitted signal can be represented as [21].
(1)x=FRFFBBs,where s=s1,s2,⋯,sKT is the K×1 vector of transmitted symbols, which satisfy EssH=P/KIK, and P is the average total transmitted power.Figure 1
Fully connected architecture of the multiuser hybrid precoding in mmWave massive MIMO system.Generally, the received downlink signal of the mobile userk can be represented as [21].
(2)yk=wkHkFRFfBBksk+∑i≠kwkHkFRFfBBisi+wknk,where Hk∈ℂNR×NT is the channel matrix from the BS to the k th mobile user. Let H=H1,H2,⋯,HNT. nk is an additive Gaussian white noise vector such that nk~CN0,δ2I. wk∈ℂ1×NR denotes the digital vector at the k th mobile user.Then, we describe the two structures shown in Figure2 and define the form of their analog precoding matrices. The conventional subarray architecture in which each RF chain is connected to a specific subset of the antennas as shown in Figure 2(a), and each RF chain connected to the antenna is fixed in the order. Therefore, we can define the analog precoding matrix FRF as [9].
(3)FRF=fRF10⋯00fRF20⋮⋱⋮00⋯fRFNRFNRFM×NRF,where fRFj=fjj−1M+1,fjj−1M+2,⋯,fjjMT denotes the analog weighting vector for the j th RF chain, M=NT/NRF, and j=1,2,⋯,NRF.Figure 2
Two subconnected architectures of the multiuser hybrid precoding in mmWave massive MIMO system: (a) conventional subarray architecture, where each RF chain is connected to a fixed subset of the antennas; (b) dynamic subarray architecture, where each RF chain is connected to interlaced antennas.
(a)(b)The dynamic subarray architectures are shown in Figure2(b), which each RF chain is connected to a dynamic subset of the antennas. The analog precoding matrix FRF=fRFS1,fRFS2,⋯,fRFSNRF is derived with
(4)fRFSjt,j=fjt,t∈Sj,0,t∉Sj,where fRFSj∈ℂNT×1 is the analog precoding vector for the j th RF chain, and fRFSjt,j denotes the element of t th antenna connected to the j th RF chain. Assume that fjt is a value of fRFSjt,j. The set Sj contains label of antenna connected to the j th RF chain, and let set S=S1,S2,⋯,Sj,⋯,SNRF, ∑j=1NRFSj=NT.


## 2.2. mmWave Channel Model

Millimeter wave channels with high free-space path loss and large dense antenna arrays are unlikely to follow the rich scattering model at low frequencies. According to the characteristics of millimeter wave propagation, the dense array makes the antenna highly correlated, and the high path loss in high free-space leads to the limited space selectivity. Thus, the millimeter-wave channel is considered to have finite scattering [1]. Traditional MIMO channel modeling methods are not accurate in millimeter wave channel. The geometric channel model can embody the low rank and spatial correlation characteristics of mmWave communications [9]. Thus, we consider a geometric channel model with Lk scatterers for the k th user and assume that each scatterer contributes a single propagation path between the BS and the user [18, 21]. Under this model, the k th user’s channel can be represented as
(5)Hk=NTNRLk∑l=1Lkβl,kaRθl,kRaTHθl,kT,where Lk is the number of scatterers and βl,k is the complex gain of the l th path between the BS and the k th user. θl,kR and θl,kT, respectively, represent the azimuth of arrival (AOA) and departure (AOD) of the l th path between the BS and the k th user, and the range of values for both θl,kR and θl,kT is 0,2π. aTθl,kT and aRθl,kR are the antenna array response vectors of the base station and k th user, separately. For a uniform linear array (ULA), the array response vector can be written as
(6)aTθl,kT=1NT1,ej2πd/λsinθl,kT,⋯,ejNT−12πd/λsinθl,kTT,where λ is the signal wavelength and d=λ/2 denotes the spacing distance between two adjacent antennas. The array response vector aRθl,kR takes a similar fashion.


## 3. Problem Formulation

In this section, we propose the problem of multiuser hybrid precoding scheme based on dynamic subarray architectures. As shown in Figure2(b), the antennas of each subarray are not adjacent to each other, and the optimal set S∗ needs to get at first. Then, design the hybrid precoding matrix FRF and FBB for the dynamic subarray architectures at the BS. In this paper, we assume that the channel state information (CSI) is perfect. The SINR of the k th user is written as
(7)SINRk=PH~kFRFfBBkF2Kσk2+P∑i≠kH~kFRFfBBiF2,where H~k=wkHk, and wk=UkH is obtained by SVD of Hk, which is Hk=UkΛkVkH.The achievable rate of thek th user can be written as
(8)Rk=log21+SINRk.We aim to jointly design the S∗, FRF∗, and FBB∗ to maximize the achievable sum-rate of dynamic subarray architectures, which can be formulated as
(9)S∗,FRF∗,FBB∗=argmaxFRF,FBB,S∑k=1Klog21+SINRk,s.t.fRFt,jF2=1,t=1,2,⋯,NT,FRFFBBF2=K.The optimization problem of (9) is a nonconvex problem, and it is almost impossible to solve the global optimal solution directly. We decompose problem (9) into three subproblems. Firstly, we design the initial precoding matrix for each user to get the optimal set S∗. Secondly, we design the optimal analog precoding matrix FRF∗ according to S∗, and finally, we design the optimal digital precoding matrix FBB∗ through the equivalent channel matrix H^k=H~kFRF. The detail of the scheme will be explained in the next section.


## 4. Hybrid Precoding Strategy

In this section, we introduce a dynamically subarray-connected hybrid precoding scheme for multiuser mmWave massive MIMO systems, including analog precoding initialization and user selection, dynamic antenna partitioning, and the hybrid precoding optimize for the dynamic subarray structures.


### 4.1. Initialization Analog Precoding and User Selection

In the multiuser mmWave massive MIMO system, the mmWave channel has a finite number of strong beam directions; thus, the best beams of the users lie on their own scattering paths. When the base station serves all users in the same time slot, the performance of the system will be degraded with the increase of interuser interference. Increasing the number of served users at the same time slot will result in performance degradation [22, 23]. In this paper, we select a group of users with minimum interuser interference and maximum objective channel gains. Generally, maximizing the SINR is the criterion for user selection [15].Firstly, we define a set of all candidate usersN=1,2,⋯,N and an empty set U. The initial analog precoding vector fn0 of n th user (n=1,2,⋯,N) can be obtained by solving
(10)fn0=argmaxHnfnF2,s.t.fnt,n=1,∀n,t.The SVD of channelHn can be expressed as Hn=UnΛnVnH. The unconstrained analog precoding vector for user n can be written as fnopt=vn1, where vn1 is the first columns of Vn.The optimization problem (10) can be rewritten as
(11)fn0=argminfnopt−fnF2,s.t.fRFt,n=1,∀n,t.The solutions of similar problems with (11) are given in [9]. Therefore, the solution of (11) can be obtained directly.
(12)fn0=ej∠fnopt=ej∠vn1.Algorithm 1: Multiuser selection based on SINR.
1:Input:H, N, U, K2:Initialize:H=H1,H2,⋯,HNT, N=1,2,⋯,N, U=∅3:forn =1:N do4: Calculate the SVD ofHn to obtain Wn and Vn, wn=WnH1,:, fn0=ej∠vn1, where vn1 is the first columns of Vn.5:∇n=wnHnF26:end for7:U1=argmaxN∇n8:U⟵U∪U1,N⟵N\U19:fork=2:K do10:Uk=argmaxN,UPwNnHNnfNn0F2/KσNn2+P∑i=1k−1wNnHNnfUi0F211:U⟵U∪Uk,N⟵N\Uk12:end for13:output:UThen, the process of multiuser selection is described as follows. Using the initial analog precoding vectors, the SINR of the usern written as
(13)SINRn=PwnHnfn0F2Kσn2+P∑i≠nwnHnfi0F2.The first userU1 selected by maximum channel gain wnHnF2 from set N, and then, we update both set N and U. The remaining users are selected from all candidate users set N according to the maximum SINR criterion, which can be described as follows:
(14)Uk=argmaxN,UPwNnHNnfNn0F2KσNn2+P∑i=1k−1wNnHNnfUi0F2,k≤K≤NRF,i=1,2,⋯,k−1,where HNn is the mmWave channel of the n th user in set N, fNn0 and fUi0 denote the initial analog precoding vector of the n th user in set N and the initial analog precoding vector of the i th user in set U, respectively. The user Uk with maximum SINR value is removed from N and added to U. The update strategy is as follows:
(15)U⟵U∪Uk,N⟵N\Uk.The process of multiuser selection is summarized in Algorithm1.


### 4.2. Dynamic Subarray for Multiuser mmWave System

The dynamic subarray hybrid architecture is a more efficient structure, in which the antenna elements are dynamically partitioned to each RF chain based on the long-term channel information [16, 18], and it addresses the tradeoff between the achievable spectral efficiency and energy efficiency. The existing antenna partition algorithm for dynamic subarray hybrid architecture is proposed in [20], but it has poor user fairness because of insufficient conditions. It uses the maximum increment of the SINR for each selected user, the conditions to ensure that each RF chain can be allocated one antenna at least have not been set, resulting in some RF chains not being partition of any antenna under special circumstances, such as some users can obtain the maximum SINR increment in all loop. In this subsection, based on the dynamic subarray hybrid structure proposed in Section 2, we design the new antenna partition algorithm to obtain higher spectral efficiency and guarantee user fairness.The objective of this subsection is to address the optimalSk∗ of the user k. We assume that the number of chain NRF is equal to the number of user K, and the BS transmits data to every user via only one stream. As shown in Figure 2(b), there are two constraints as
(16)∑j=1NRFfRFt,j=1,(17)∑t=1NTfRFt,j≥N,where N=NT−NTmodNRF/NRF, and the constrain of (16) ensures that each antenna is only connected to one RF chain, and the constrain of (17) ensures that each RF chain is connected to N antennas at least.Then, in order to ensure that each user gets maximize channel gain and ensure fairness among users, we neglect the resulting interference among users in this stage, and the user interference elimination problem is solved in the next subsection. Directly solving three unknown matrix variablesS∗, FRF∗, and FBB∗ will lead to high computational complexity. Therefore, we use the initial analog precoding vectors as significant information for partitioning antenna elements and let the dynamic subarray of each user start out as an empty set. The antenna-partitioning problem is described as
(18)t∗=argmaxt∈ΞwkHkfSk∪t0F2,where fSk0 is a NT×1 vector, which is obtained by fk0 according to (4), and Sk∪t represents the antenna t being added to the subarray Sk of k th user. Then, set Ξ and subarray Sk are updated as follows:
(19)Sk⟵Sk∪t∗,Ξ⟵Ξ\t∗,Algorithm 2: Multiuser dynamic subarray partitioning.
1:Input:H, U, K, S=S1,S2,⋯,Sk,⋯,SK, fn02:Initialize:H=H1,H2,⋯,HNT, U and fn0 are obtain by Algorithm 1, K=U,S1=S2=⋯=SK=∅, N=NT−NTmodNRF/NRF3:forloop=1:N do4:   fork=1:K do5:t∗=argmaxt∈ΞwkHkfSk∪t0F26:Sk⟵Sk∪t∗,Ξ⟵Ξ\t∗7:Δk=wkHkfSk0F28:   end for9:end for10:whileΞ≠∅ do11:k∗=argmink=1,⋯,KΔk12:t∗=argmaxt∈Ξwk∗Hk∗fSk∗∪t0F213:Sk∗⟵Sk∗∪t∗,Ξ⟵Ξ\t∗14:Δk∗=wk∗Hk∗fSk∗0F215:end while16:output:S1,S2,⋯,SKWithin each inner loop of Algorithm2, due to the constrain in (17), (18), and (19), will be carried out for all users once, and this inner loop will be carried out for N times. After the N loop, if Ξ≠∅, it means that there are still remaining antennas unallocated, and we allocate the remaining antennas to the user with the smallest channel gain. The details of the process are described in Algorithm 2.For the simplicity of the process analysis, we assume that the number of chainNRF is equal to the number of user K, and the BS transmits data to every user via only one stream. In fact, by changing the constraints (16) and (17), this new antenna partition algorithm can easily be extended to user multistream transmission.


### 4.3. Hybrid Precoding Design

In this subsection, we will design the hybrid precoding for dynamic subarray hybrid architecture that obtained in the previous subsection. For the dynamic subarray hybrid architecture, the beam shape and width are changed after the antenna partitioning of this architecture, and the initial analog precoding vector is invalid. Therefore, the analog precoding needs to be redesigned.Utilize the subarrayS1,S2,⋯,SK, the channel matrix for each user can be updated as
(20)HkSk:,t=Hk:,t,t∈Sk,0,t∉Sk,H¯k=HkSk,where H¯k is a NR×NT channel matrix for k th user, and define the effective channel ek=wkH¯k for k th user, where wk can obtain by SVD of H¯k.In order to obtain the large array gain, we consider the phase-only control at the RF domain [24] and perform the analog precoding for each user according to
(21)fkSk=ej−φk,where φk is the phase of the k th element in the vector ek and −φk is the conjugate transpose of φk.In the stage of digital baseband precoding, we focused on the elimination of interuser interference. Some classical digital precoding algorithms are used to eliminate the interuser interference, such as ZF and BD. However, in multiuser systems with a low signal-to-noise ratio (SNR), ZF or BD precoding performs poorly [12]. Thus, we use the minimum mean-squared error criteria to design digital baseband precoding. According to the received signal model in (2), the estimated symbol of k th user is given by
(22)s^k=wkH¯kFRFFBBksk+wkH¯k∑i≠kKFRFFBBisi+wknk.The estimated symbols for all users given (22) can be stacked into one vector.
(23)s^=WH¯FRFFBBs+Wn,where H¯=H¯1T,H¯2T,⋯,H¯KTT, W=blkdiagw1,w2,⋯,wK, FRF=f1S1,f2S1,⋯,fKSK, FBB=blkdiagFBB1,FBB2,⋯,FBBK, and n=n1T,n2T,⋯,nKTT. The aim is to get estimated symbols s^ close to s. The mean-squared error cost function can be expanded as
(24)Es−γs∧2=Es−γWH¯FRFFBBs−γWn2=IK−WH¯FRFFBBF2+Kγσ2,where γ is the scaling factor.Then, the problem of minimum mean-squared error can be expressed as(25)minimizeFRF,FBB,γIK−WH¯FRFFBBF2+Kγσ2s.t.TrFRFHFRFFBBFBBH≤γP.The analog precoding matrixFRF can be obtained by (21); thus, the problem (25) will be transformed into a convex optimization problem. The closed-form optimal solution of the similar problem is given in [25]. Its optimal solution is given by
(26)FBB=THT+Kδ2PIK†TH,where T=WH¯FRF, the optimal scaling factor γ=FRFFBBF2/P, normalizes F^BB=1/γFBB.In the process of MMSE-based digital precoding computation, the main computational complexity comes from formula (26). In (26), the complexity of matrix multiplication is OK2NRNTNRF, and the complexity of pseudo-inverse is OK3. Therefore, the total complexity is OK2NRNTNRF+K3. Since the baseband digital precoding design in hybrid precoding is carried out in low dimensions, the increase in the number of base station transmitting antennas will not bring huge computational complexity. In practical application, this method may bring some challenges when the number of users increases rapidly. In general, the MMSE-based baseband digital precoding is an effective method.


## 4.1. Initialization Analog Precoding and User Selection

In the multiuser mmWave massive MIMO system, the mmWave channel has a finite number of strong beam directions; thus, the best beams of the users lie on their own scattering paths. When the base station serves all users in the same time slot, the performance of the system will be degraded with the increase of interuser interference. Increasing the number of served users at the same time slot will result in performance degradation [22, 23]. In this paper, we select a group of users with minimum interuser interference and maximum objective channel gains. Generally, maximizing the SINR is the criterion for user selection [15].Firstly, we define a set of all candidate usersN=1,2,⋯,N and an empty set U. The initial analog precoding vector fn0 of n th user (n=1,2,⋯,N) can be obtained by solving
(10)fn0=argmaxHnfnF2,s.t.fnt,n=1,∀n,t.The SVD of channelHn can be expressed as Hn=UnΛnVnH. The unconstrained analog precoding vector for user n can be written as fnopt=vn1, where vn1 is the first columns of Vn.The optimization problem (10) can be rewritten as
(11)fn0=argminfnopt−fnF2,s.t.fRFt,n=1,∀n,t.The solutions of similar problems with (11) are given in [9]. Therefore, the solution of (11) can be obtained directly.
(12)fn0=ej∠fnopt=ej∠vn1.Algorithm 1: Multiuser selection based on SINR.
1:Input:H, N, U, K2:Initialize:H=H1,H2,⋯,HNT, N=1,2,⋯,N, U=∅3:forn =1:N do4: Calculate the SVD ofHn to obtain Wn and Vn, wn=WnH1,:, fn0=ej∠vn1, where vn1 is the first columns of Vn.5:∇n=wnHnF26:end for7:U1=argmaxN∇n8:U⟵U∪U1,N⟵N\U19:fork=2:K do10:Uk=argmaxN,UPwNnHNnfNn0F2/KσNn2+P∑i=1k−1wNnHNnfUi0F211:U⟵U∪Uk,N⟵N\Uk12:end for13:output:UThen, the process of multiuser selection is described as follows. Using the initial analog precoding vectors, the SINR of the usern written as
(13)SINRn=PwnHnfn0F2Kσn2+P∑i≠nwnHnfi0F2.The first userU1 selected by maximum channel gain wnHnF2 from set N, and then, we update both set N and U. The remaining users are selected from all candidate users set N according to the maximum SINR criterion, which can be described as follows:
(14)Uk=argmaxN,UPwNnHNnfNn0F2KσNn2+P∑i=1k−1wNnHNnfUi0F2,k≤K≤NRF,i=1,2,⋯,k−1,where HNn is the mmWave channel of the n th user in set N, fNn0 and fUi0 denote the initial analog precoding vector of the n th user in set N and the initial analog precoding vector of the i th user in set U, respectively. The user Uk with maximum SINR value is removed from N and added to U. The update strategy is as follows:
(15)U⟵U∪Uk,N⟵N\Uk.The process of multiuser selection is summarized in Algorithm1.


## 4.2. Dynamic Subarray for Multiuser mmWave System

The dynamic subarray hybrid architecture is a more efficient structure, in which the antenna elements are dynamically partitioned to each RF chain based on the long-term channel information [16, 18], and it addresses the tradeoff between the achievable spectral efficiency and energy efficiency. The existing antenna partition algorithm for dynamic subarray hybrid architecture is proposed in [20], but it has poor user fairness because of insufficient conditions. It uses the maximum increment of the SINR for each selected user, the conditions to ensure that each RF chain can be allocated one antenna at least have not been set, resulting in some RF chains not being partition of any antenna under special circumstances, such as some users can obtain the maximum SINR increment in all loop. In this subsection, based on the dynamic subarray hybrid structure proposed in Section 2, we design the new antenna partition algorithm to obtain higher spectral efficiency and guarantee user fairness.The objective of this subsection is to address the optimalSk∗ of the user k. We assume that the number of chain NRF is equal to the number of user K, and the BS transmits data to every user via only one stream. As shown in Figure 2(b), there are two constraints as
(16)∑j=1NRFfRFt,j=1,(17)∑t=1NTfRFt,j≥N,where N=NT−NTmodNRF/NRF, and the constrain of (16) ensures that each antenna is only connected to one RF chain, and the constrain of (17) ensures that each RF chain is connected to N antennas at least.Then, in order to ensure that each user gets maximize channel gain and ensure fairness among users, we neglect the resulting interference among users in this stage, and the user interference elimination problem is solved in the next subsection. Directly solving three unknown matrix variablesS∗, FRF∗, and FBB∗ will lead to high computational complexity. Therefore, we use the initial analog precoding vectors as significant information for partitioning antenna elements and let the dynamic subarray of each user start out as an empty set. The antenna-partitioning problem is described as
(18)t∗=argmaxt∈ΞwkHkfSk∪t0F2,where fSk0 is a NT×1 vector, which is obtained by fk0 according to (4), and Sk∪t represents the antenna t being added to the subarray Sk of k th user. Then, set Ξ and subarray Sk are updated as follows:
(19)Sk⟵Sk∪t∗,Ξ⟵Ξ\t∗,Algorithm 2: Multiuser dynamic subarray partitioning.
1:Input:H, U, K, S=S1,S2,⋯,Sk,⋯,SK, fn02:Initialize:H=H1,H2,⋯,HNT, U and fn0 are obtain by Algorithm 1, K=U,S1=S2=⋯=SK=∅, N=NT−NTmodNRF/NRF3:forloop=1:N do4:   fork=1:K do5:t∗=argmaxt∈ΞwkHkfSk∪t0F26:Sk⟵Sk∪t∗,Ξ⟵Ξ\t∗7:Δk=wkHkfSk0F28:   end for9:end for10:whileΞ≠∅ do11:k∗=argmink=1,⋯,KΔk12:t∗=argmaxt∈Ξwk∗Hk∗fSk∗∪t0F213:Sk∗⟵Sk∗∪t∗,Ξ⟵Ξ\t∗14:Δk∗=wk∗Hk∗fSk∗0F215:end while16:output:S1,S2,⋯,SKWithin each inner loop of Algorithm2, due to the constrain in (17), (18), and (19), will be carried out for all users once, and this inner loop will be carried out for N times. After the N loop, if Ξ≠∅, it means that there are still remaining antennas unallocated, and we allocate the remaining antennas to the user with the smallest channel gain. The details of the process are described in Algorithm 2.For the simplicity of the process analysis, we assume that the number of chainNRF is equal to the number of user K, and the BS transmits data to every user via only one stream. In fact, by changing the constraints (16) and (17), this new antenna partition algorithm can easily be extended to user multistream transmission.


## 4.3. Hybrid Precoding Design

In this subsection, we will design the hybrid precoding for dynamic subarray hybrid architecture that obtained in the previous subsection. For the dynamic subarray hybrid architecture, the beam shape and width are changed after the antenna partitioning of this architecture, and the initial analog precoding vector is invalid. Therefore, the analog precoding needs to be redesigned.Utilize the subarrayS1,S2,⋯,SK, the channel matrix for each user can be updated as
(20)HkSk:,t=Hk:,t,t∈Sk,0,t∉Sk,H¯k=HkSk,where H¯k is a NR×NT channel matrix for k th user, and define the effective channel ek=wkH¯k for k th user, where wk can obtain by SVD of H¯k.In order to obtain the large array gain, we consider the phase-only control at the RF domain [24] and perform the analog precoding for each user according to
(21)fkSk=ej−φk,where φk is the phase of the k th element in the vector ek and −φk is the conjugate transpose of φk.In the stage of digital baseband precoding, we focused on the elimination of interuser interference. Some classical digital precoding algorithms are used to eliminate the interuser interference, such as ZF and BD. However, in multiuser systems with a low signal-to-noise ratio (SNR), ZF or BD precoding performs poorly [12]. Thus, we use the minimum mean-squared error criteria to design digital baseband precoding. According to the received signal model in (2), the estimated symbol of k th user is given by
(22)s^k=wkH¯kFRFFBBksk+wkH¯k∑i≠kKFRFFBBisi+wknk.The estimated symbols for all users given (22) can be stacked into one vector.
(23)s^=WH¯FRFFBBs+Wn,where H¯=H¯1T,H¯2T,⋯,H¯KTT, W=blkdiagw1,w2,⋯,wK, FRF=f1S1,f2S1,⋯,fKSK, FBB=blkdiagFBB1,FBB2,⋯,FBBK, and n=n1T,n2T,⋯,nKTT. The aim is to get estimated symbols s^ close to s. The mean-squared error cost function can be expanded as
(24)Es−γs∧2=Es−γWH¯FRFFBBs−γWn2=IK−WH¯FRFFBBF2+Kγσ2,where γ is the scaling factor.Then, the problem of minimum mean-squared error can be expressed as(25)minimizeFRF,FBB,γIK−WH¯FRFFBBF2+Kγσ2s.t.TrFRFHFRFFBBFBBH≤γP.The analog precoding matrixFRF can be obtained by (21); thus, the problem (25) will be transformed into a convex optimization problem. The closed-form optimal solution of the similar problem is given in [25]. Its optimal solution is given by
(26)FBB=THT+Kδ2PIK†TH,where T=WH¯FRF, the optimal scaling factor γ=FRFFBBF2/P, normalizes F^BB=1/γFBB.In the process of MMSE-based digital precoding computation, the main computational complexity comes from formula (26). In (26), the complexity of matrix multiplication is OK2NRNTNRF, and the complexity of pseudo-inverse is OK3. Therefore, the total complexity is OK2NRNTNRF+K3. Since the baseband digital precoding design in hybrid precoding is carried out in low dimensions, the increase in the number of base station transmitting antennas will not bring huge computational complexity. In practical application, this method may bring some challenges when the number of users increases rapidly. In general, the MMSE-based baseband digital precoding is an effective method.


## 5. Simulation Results

In this section, simulation results are given to illustrate the performance advantages of the proposed new antenna-partitioning algorithm and hybrid analog/digital precoding scheme in spectrum efficiency, energy efficiency, and user fairness. We compare the performance of three hybrid precoding schemes with different structures, namely, conventional subconnected structure, existing dynamic subarray-connected structure [20], and fully connected structure. Firstly, we consider a mmWave channel and dynamic subarray architecture described in Section 2. The carrier frequency is set as 28 GHz [1]. The number of scatterers for each user is Lk=3. The transmit antenna arrays are ULAs with antenna spacing d=λ/2. We assume that both the AoAs and AoDs are following the uniform distribution within −π,π. Finally, SNR is defined P/σ2.


### 5.1. Performance Comparisons of Spectrum Efficiency

Figure3 shows the spectral efficiency comparison in different architectures, where the BS is equipped with 64 antennas and 2 RF chains serving two users, and each user is equipped with two antennas. We observe from Figure 3 that the performance of the proposed new antenna-partitioning-based hybrid precoding scheme outperforms the existing dynamic subarray precoding scheme [20] in the whole simulated SNR range. For instance, when SNR=0 dB, our proposed scheme can achieve 86.1% of the spectral efficiency obtained by fully connected architecture, and compared with the conventional subarray architecture, the spectral efficiency of the proposed scheme is improved by 11.1%. Moreover, compared with the existing dynamic subarray architecture scheme, the spectral efficiency of the proposed scheme is also improved; especially at low SNR, the spectral efficiency is improved by 4.7% when SNR=0 dB.Figure 3
Spectral efficiency comparison for the mmWave MIMO system withNT=64, NR=2, K=2, and NRF=2 RF chains.In Figure4, the BS is equipped with 128 antennas and 2 RF chains serving two users. Similar to the trend in Figure 3, the spectrum efficiency of all architectures is improved, and our proposed scheme is still superior to the existing dynamic subarray architecture scheme. As can be seen from Figures 3 and 4, our proposed scheme is closer to the performance achieved by fully connected architecture, and the advantages are obvious at low SNR.Figure 4
Spectral efficiency comparison for the mmWave MIMO system withNT=128, NR=2, K=2, and NRF=2 RF chains.Figure5 provides a spectrum efficiency comparison against the number of BS transmitting antennas. The BS is equipped with different antennas and two RF chains serving two users. As observed form Figure 5, the spectral efficiency increases with the increase of the number of antennas, but the rate of increase slows down when the SNR=−10 dB and the SNR=0 dB. More importantly, the existing dynamic subarray schemes [20] do not show significant advantages when the BS equipped small transmitting antennas, such as 16 and 32. Moreover, our proposed scheme in this paper has obvious advantages in the whole simulation range.Figure 5
Spectral efficiency comparison for the mmWave MIMO system with different numbers of transmitting antennas.


### 5.2. Performance Comparisons of Energy Efficiency

The energy consumption model was proposed in [26], and based on this model, the energy efficiency η can be defined as
(27)η=RsumPtotal=∑k=1Klog21+SINRkPt+NRFPRF+NPSPPSbps/Hz/W,where Ptotal is the total energy consumption, Pt is the transmission power, PRF is the energy consumed by RF chain, and PPS is the energy consumed by phase shifter. NRF and NPS are the number of required RF chains and phase shifters, respectively.In this paper, we usePRF=250mW [26] and PPS=1mW [27] in simulation, Pt is constrained to FRFFBBF2=K. The energy efficiency comparison against the number of RF chains NRF is shown in Figure 6, where NT=128, NR=2, and SNR=0dB. We can see that withFigure 6
Energy efficiency comparison against the numbers of RF chains, whereNT=128, SNR=0 dB, NR=2, and K=2.the increase of the number of RF chains, the energy efficiency gradually decreases, which is consistent with the theoretical analysis. Moreover, it is clear that the dynamic subarray-connected architectures have higher energy efficiency than the fully connected architectures. In addition, compared with the existing dynamic subarray schemes, the energy efficiency of the proposed scheme is improved by 3.5% on average.Figure7 shows the energy efficiency comparison against the number of transmitting antennas of BS, where NT=16, 32, 64, 128, and 256, SNR=0 dB, NRF=2, and K=2. As the number of base station antennas increases, the energy efficiency first increases and then decreases, especially for fully connected architectures, which are faced with rapid performance degradation. In addition, the proposed schemes achieve better energy efficiency than existing dynamic subarray schemes and can maintain a high and stable energy efficiency between 64 and 256 antennas.Figure 7
Energy efficiency comparison against the numbers of transmit antennas, whereSNR=0 dB, NR=2, and K=2.


### 5.3. User Fairness Analysis

As an indicator of the allocation of shared resources, fairness is widely used in various fields, for example, data collection in cognitive radio networks [28, 29] and data transfer in wireless communication systems. In a multiuser communication system, it is important to ensure that each user has access to roughly the same data transfer rate. In Section 4, we introduce user selection to provide better service for a small number of users, where it is important to ensure fairness among users. In order to investigate the fairness of each user, we introduce the Jain fairness index [30]. The Jain fairness index is a very effective measure of distributive fairness. Jain fairness index defines fairness among multiple users as
(28)JR=∑k=1KRk2K∑k=1KRk2,where JR is the coefficient of fairness between K users, JR∈1/K,1. According to the definition of the Jain fairness index, if the value of JR tends to 1/K, the fairness is worse; on the other hand, if the value of JR tends to 1, the fairness is better. In Figure 8, we compared the fairness of different schemes with different numbers of users K, where NT=128, SNR=−10 dB, and NR=2. We observe that our proposed scheme can make the fairness index close to 1 under the different number of users. However, the fairness index of the existing dynamic subarray schemes [20] decreases with the increase of the number of users.Figure 8
Fairness index comparison against the numbers of usersK, where NT=128, NR=2, and SNR=−10 dB.


## 5.1. Performance Comparisons of Spectrum Efficiency

Figure3 shows the spectral efficiency comparison in different architectures, where the BS is equipped with 64 antennas and 2 RF chains serving two users, and each user is equipped with two antennas. We observe from Figure 3 that the performance of the proposed new antenna-partitioning-based hybrid precoding scheme outperforms the existing dynamic subarray precoding scheme [20] in the whole simulated SNR range. For instance, when SNR=0 dB, our proposed scheme can achieve 86.1% of the spectral efficiency obtained by fully connected architecture, and compared with the conventional subarray architecture, the spectral efficiency of the proposed scheme is improved by 11.1%. Moreover, compared with the existing dynamic subarray architecture scheme, the spectral efficiency of the proposed scheme is also improved; especially at low SNR, the spectral efficiency is improved by 4.7% when SNR=0 dB.Figure 3
Spectral efficiency comparison for the mmWave MIMO system withNT=64, NR=2, K=2, and NRF=2 RF chains.In Figure4, the BS is equipped with 128 antennas and 2 RF chains serving two users. Similar to the trend in Figure 3, the spectrum efficiency of all architectures is improved, and our proposed scheme is still superior to the existing dynamic subarray architecture scheme. As can be seen from Figures 3 and 4, our proposed scheme is closer to the performance achieved by fully connected architecture, and the advantages are obvious at low SNR.Figure 4
Spectral efficiency comparison for the mmWave MIMO system withNT=128, NR=2, K=2, and NRF=2 RF chains.Figure5 provides a spectrum efficiency comparison against the number of BS transmitting antennas. The BS is equipped with different antennas and two RF chains serving two users. As observed form Figure 5, the spectral efficiency increases with the increase of the number of antennas, but the rate of increase slows down when the SNR=−10 dB and the SNR=0 dB. More importantly, the existing dynamic subarray schemes [20] do not show significant advantages when the BS equipped small transmitting antennas, such as 16 and 32. Moreover, our proposed scheme in this paper has obvious advantages in the whole simulation range.Figure 5
Spectral efficiency comparison for the mmWave MIMO system with different numbers of transmitting antennas.


## 5.2. Performance Comparisons of Energy Efficiency

The energy consumption model was proposed in [26], and based on this model, the energy efficiency η can be defined as
(27)η=RsumPtotal=∑k=1Klog21+SINRkPt+NRFPRF+NPSPPSbps/Hz/W,where Ptotal is the total energy consumption, Pt is the transmission power, PRF is the energy consumed by RF chain, and PPS is the energy consumed by phase shifter. NRF and NPS are the number of required RF chains and phase shifters, respectively.In this paper, we usePRF=250mW [26] and PPS=1mW [27] in simulation, Pt is constrained to FRFFBBF2=K. The energy efficiency comparison against the number of RF chains NRF is shown in Figure 6, where NT=128, NR=2, and SNR=0dB. We can see that withFigure 6
Energy efficiency comparison against the numbers of RF chains, whereNT=128, SNR=0 dB, NR=2, and K=2.the increase of the number of RF chains, the energy efficiency gradually decreases, which is consistent with the theoretical analysis. Moreover, it is clear that the dynamic subarray-connected architectures have higher energy efficiency than the fully connected architectures. In addition, compared with the existing dynamic subarray schemes, the energy efficiency of the proposed scheme is improved by 3.5% on average.Figure7 shows the energy efficiency comparison against the number of transmitting antennas of BS, where NT=16, 32, 64, 128, and 256, SNR=0 dB, NRF=2, and K=2. As the number of base station antennas increases, the energy efficiency first increases and then decreases, especially for fully connected architectures, which are faced with rapid performance degradation. In addition, the proposed schemes achieve better energy efficiency than existing dynamic subarray schemes and can maintain a high and stable energy efficiency between 64 and 256 antennas.Figure 7
Energy efficiency comparison against the numbers of transmit antennas, whereSNR=0 dB, NR=2, and K=2.


## 5.3. User Fairness Analysis

As an indicator of the allocation of shared resources, fairness is widely used in various fields, for example, data collection in cognitive radio networks [28, 29] and data transfer in wireless communication systems. In a multiuser communication system, it is important to ensure that each user has access to roughly the same data transfer rate. In Section 4, we introduce user selection to provide better service for a small number of users, where it is important to ensure fairness among users. In order to investigate the fairness of each user, we introduce the Jain fairness index [30]. The Jain fairness index is a very effective measure of distributive fairness. Jain fairness index defines fairness among multiple users as
(28)JR=∑k=1KRk2K∑k=1KRk2,where JR is the coefficient of fairness between K users, JR∈1/K,1. According to the definition of the Jain fairness index, if the value of JR tends to 1/K, the fairness is worse; on the other hand, if the value of JR tends to 1, the fairness is better. In Figure 8, we compared the fairness of different schemes with different numbers of users K, where NT=128, SNR=−10 dB, and NR=2. We observe that our proposed scheme can make the fairness index close to 1 under the different number of users. However, the fairness index of the existing dynamic subarray schemes [20] decreases with the increase of the number of users.Figure 8
Fairness index comparison against the numbers of usersK, where NT=128, NR=2, and SNR=−10 dB.


## 6. Conclusion

In this paper, we propose a new antenna-partitioning algorithm and a multiuser hybrid precoding scheme for the dynamic subarray of multiuser mmWave massive MIMO systems. The idea is to ensure that each user has the same opportunity to obtain the optimal antenna according to the maximum channel gain of the selected user. Then, we design the hybrid precoding with high performance for dynamic subarray architecture. Simulation results show that the proposed scheme in this paper can make the spectrum efficiency closer to the fully connected architectures and the energy efficiency closer to the traditional subconnected architectures. Moreover, the scheme proposed in this paper, especially at low SNR, is more advanced than the existing dynamic subarray schemes. In terms of user fairness, our scheme can almost make every user reach the same achievable rate. For future work, we are interested to evaluate performance in the dynamic subarray architecture with uniform planar arrays (UPAs) and under imperfect channel state information.



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*Source: 5528522-2021-04-28.xml*