# Iterative Multiuser Equalization for Subconnected Hybrid mmWave Massive MIMO Architecture

Authors: R. Magueta; V. Mendes; D. Castanheira; A. Silva; R. Dinis; A. Gameiro
Journal: Wireless Communications and Mobile Computing
 (2017)

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## Abstract

Millimeter waves and massive MIMO are a promising combination to achieve the multi-Gb/s required by future 5G wireless systems. However, fully digital architectures are not feasible due to hardware limitations, which means that there is a need to design signal processing techniques for hybrid analog-digital architectures. In this manuscript, we propose a hybrid iterative block multiuser equalizer for subconnected millimeter wave massive MIMO systems. The low complexity user-terminals employ pure-analog random precoders, each with a single RF chain. For the base station, a subconnected hybrid analog-digital equalizer is designed to remove multiuser interference. The hybrid equalizer is optimized using the average bit-error-rate as a metric. Due to the coupling between the RF chains in the optimization problem, the computation of the optimal solutions is too complex. To address this problem, we compute the analog part of the equalizer sequentially over the RF chains using a dictionary built from the array response vectors. The proposed subconnected hybrid iterative multiuser equalizer is compared with a recently proposed fully connected approach. The results show that the performance of the proposed scheme is close to the fully connected hybrid approach counterpart after just a few iterations.


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## Body

## 1. Introduction

A new generation of cellular network (fifth generation, 5G) is coming, and some innovative technologies are needed to ensure better performance and quality of service (QoS). Two enabling technologies have been considered to meet the QoS requirements for future wireless communication, massive MIMO (mMIMO), and millimeter wave (mmWave) communications [1]. By using mmWave bands, several tens of GHz of bandwidth become available for wireless systems [2], while the mMIMO allows the continued increasing demand of higher data rates for future wireless networks [3]. Comparing the mMIMO with conventional MIMO approaches, the mMIMO can scale up the conventional MIMO by orders of magnitude [4]. A survey on mMIMO, also identified as large-scale MIMO, with channel modelling, applications scenarios, and physical/networking techniques can be seen in [5]. The use of mmWave with mMIMO is very promising, because smaller wavelength compared to conventional communication systems allows the same volume to pack more antennas [6], which means that the terminals can support a large number of antennas.The mmWave mMIMO combination can be used to exploit new efficient spatial processing techniques, such as beamforming/precoding and spatial multiplexing, at both transmitter and/or receiver terminals [7]. These techniques are different than those used for sub-6 GHz bands due to limitations in hardware [8]. In these systems, it is not practical to have one fully dedicated radio frequency (RF) chain by antenna [9] as in sub-6 GHz conventional MIMO systems [10] due to the power consumption and the high cost of mmWave mixed-signal components. Another issue is the mmWave propagation characteristics, which are quite different from sub-6 GHz because the mmWave channels are not so rich in multipath propagation effects [11, 12], which should be taken into consideration in the techniques design for these systems. To overcome the limitation of the RFs chains number, purely analog beamforming can use phase shifters [8], with some schemes proposed in [13, 14], where statistical channel knowledge is used through phase shifters, to optimally adjust the arrays response in space, applying a beam steering solution.The performance of the pure-analog techniques is limited by constraints on the amplitudes of phase shifters and due to the phases of the ones quantized. Therefore, analog beamforming is usually limited to single-stream transmission [15]. These limitations are overcome by doing some signal processing at an analog level and the rest at the digital level. These architectures are called hybrid analog/digital architectures and have been addressed in [15, 16]. Precoding and/or combining/equalization designs for single-user systems have been addressed for fully connected hybrid architectures in [17–19]. In these architectures, each of the RF chains is connected to all receive-and-transmit antennas. In [17], a hybrid spatially sparse precoding/combining approach was designed for mmWave mMIMO systems. The spatial structure of mmWave channels was used to transform the single-user multistream precoding and combining scheme into a sparse reconstruction problem. In [18], joint turbo-like beamforming was designed to compute transmit/receiving analog beamforming coefficients; however the digital processing part was not considered. In [19], codebook design approaches were addressed for single-stream transmissions through an analog beamforming structure. For multiuser systems, some beamforming approaches have been proposed for fully connected hybrid architectures [20–22]. The authors of [20] proposed uplink receiving beamforming where they assume only single antenna user-terminals (UTs), and at both stages analog and digital ones dealt with multiuser interference. Reference [21] proposed for the downlink a limited feedback analog/digital two-stage precoding and combining algorithm. Transmit/receiving analog beamforming are jointly computed in the first stage to maximize the power of the desired signal, and then the interference is explicitly mitigated using conventional linear zero-forcing (ZF) precoding in the second stage, that is, in the digital domain. An efficient hybrid iterative block space-time multiuser equalizer was proposed in [22]. This equalizer was designed based on the iterative block decision feedback equalization (IB-DFE) principle [23]. IB-DFE was originally proposed in [24]. It does not need the feedback loop of the channel decoder output, and it can be considered as a low complexity turbo equalizer. IB-DFE has been extended to several scenarios, like diversity scenarios and conventional and cooperative MIMO systems, among many others [25–30].In addition to the fully connected architectures, there are subconnected architectures that allow us to reduce the number of phase shifters fromNNRF to N, when compared with fully connected counterparts, where N and NRFare the number of antennas and the number of RF chains [31]. Thus, the power consumption used to excite and to compensate the insertion loss of phase shifters is reduced, and the computational complexity is also lower [31]. There are two types of subconnected architectures, dynamic and fixed [32]. In the dynamic subconnected case, each RF chain can dynamically connect to a different set of antennas, and in the fixed subconnected one, each RF chain is always physically connected to the same set of antennas. Precoding schemes for dynamic subconnected hybrid architectures have been proposed in [32, 33]. Reference [32] uses a relaxation of the mutual information maximization problem to design a technique that adapts the subarray structure according to the channel covariance matrix for frequency selective channels. The authors of [33] proposed a two-step algorithm for single-user narrowband systems that iteratively optimizes the hybrid precoder for spectral efficiency maximization, obtaining an extra data stream via the index of the active antenna set without any extra RF chain. Fixed subconnected hybrid architectures were addressed in [34–37]. In [34], precoder and combiner schemes for narrowband single-user systems are proposed, where a two-layer optimization method jointly exploiting the interference alignment and fractional programming was employed. First, the analog precoder and combiner are optimized via the alternating-direction optimization method, and then the digital precoder and combiner are optimized based on an effective channel matrix. The authors of [35] designed a precoder and a combiner for a wideband single-user system where the overall spectral efficiency is maximized considering a power budget constraint for each subcarrier. The works in [36, 37] are focused on multiuser downlink systems. In [36], the total achievable rate optimization problem with nonconvex constraints is decomposed to a series of subrate optimization problems for each subantenna array, and then an algorithm is implemented to perform a successive interference cancelation-based hybrid precoding. Precoding and combining schemes are performed in [37] for downlink, where virtual path selection maximizes the channel gain of the analog effective channel, and then a zero-forcing precoding in the digital domain is applied to mitigate the interference.In this manuscript, we propose an efficient hybrid iterative block multiuser equalizer for subconnected mmWave mMIMO systems. The limitation that each RF chain is only physically connected to a subset of antennas makes the design of the proposed subconnected hybrid iterative multiuser equalizer harder than for the fully connected based approaches. To the best of our knowledge, iterative block detection designed for subconnected mmWave mMIMO architectures has not been addressed in the literature. We propose low complexity UTs without access to the Channel State Information (CSI), using a single RF chain and pure-analog random precoding. A time encoder/precoder is applied to guarantee that the transmit signal and thus the noise plus interference at the receiver side are Gaussian distributed. This not only simplifies the receiver optimization problem but also increases the diversity effects on the mmWave mMIMO system. The designed subconnected hybrid equalizer is optimized by using the average bit-error-rate (BER) as a metric. We compute the analog part of the equalizer sequentially over the RF chains using a dictionary built by the array response vectors. Finally, we show that the BER performance of the proposed subconnected hybrid iterative multiuser equalizer tends to the BER performance of a fully connected hybrid equalizer as the number of iterations increases.This paper is organized as follows. Section2 presents the subconnected hybrid multiuser mmWave mMIMO system model. Section 3 begins with the description of a channel model for mmWave mMIMO systems, followed by the description of user-terminals (UT) and finally the design of the proposed subconnected hybrid iterative multiuser equalizer. Section 4 shows some BER performance results, and in Section 5, major conclusions are presented.Notations. Matrices are denoted in boldface capital letters and column vectors in boldface lowercase letters. The operations tr(·), (·)∗, (·)T, and (·)H represent the trace, the conjugate, the transpose, and the Hermitian transpose of a matrix. sign(a) is the operator that represents the sign of a real number a and signc=signRc+jsign(I(c)) if c is a complex number. It can also be employed elementwise to matrices. The functions R(c) and I(c) represent the real part and imaginary part of c. The functions diag(a) and diag(A), where a is a vector and A is a square matrix, correspond to a diagonal matrix where the entries of diagonal are equal to a and to a vector equal to the diagonal entries of A. The element of row j and column l of matrix B is denoted by B(j,l). The identity matrix N×N is IN.


## 2. System Characterization

In this section, we describe the channel model, the UTs, and the receiver signals for the mmWave mMIMO system. We consider a multiuser system withU users, each one with Ntx transmit antennas, that sends one data stream per time slot to the base station with Nrx receiving antennas.


### 2.1. Channel Model

We consider aT-sized block fading channel, that is, a channel that remains constant during a block but varies independently between blocks. The channel follows the clustered sparse mmWave channel model discussed in [17] where Hu∈CNrx×Ntx is the channel matrix, which contributes the sum of Ncl clusters, each one with the contribution of Nray propagation paths. The channel matrix may be expressed to the uth user as (1)Hu=γ∑j,lαjl,uΛrx,uϕjlrx,u,θjlrx,uΛtx,uϕjltx,u,θjltx,u×arx,uϕjlrx,u,θjlrx,uatx,uϕjltx,u,θjltx,uH.γ=NtxNrx/NclNray is a normalization factor, and αjl,u is the complex gain of the lth ray in the jth scattering cluster. The functions Λrx,uϕjlrx,u,θjlrx,u and Λtx,uϕjltx,u,θjltx,u represent transmit and receiving antenna element gain for ϕjlrx,uθjlrx,u and ϕjltx,uθjltx,u, that is, the azimuth (elevation) angles of arrival and departure. The vectors arx,uϕjlrx,u,θjlrx,u and atx,uϕjltx,u,θjltx,u represent the normalized receiving and transmit array response vectors for the corresponding angles. Reference [17] addressed the random distributions used to generate the path gains and the angles of channel, such that E[HuF2]=NrxNtx.


### 2.2. User-Terminal Model Description

We assume that each user has only a single RF chain and sends only one data stream per time slot over theNtx transmit antennas, as shown in Figure 1. We also consider that the UTs have no access to CSI, simplifying the overall system design. The analog precoder of uth UT at the instant t is mathematically modelled by fa,u,t∈CNtx and is physically realized using a vector of analog phase shifters, where all elements of vector fa,u,t have equal norm (fa,u,tl2=Ntx-1). Therefore, the analog precoder vector of the uth UT at the instant t is generated randomly according to(2)fa,u,t=ej2πϕnu,t1≤n≤Ntx,1≤t≤T,where ϕnu,t with n∈{1,…,Ntx},t∈{1,…,T} and u∈{1,…,U} are i.i.d. uniform random variables such that ϕnu,t∈[0,1].Figure 1
User-terminalu block diagram.To guarantee that the transmit signal and then the noise plus interference are Gaussian distributed at the receiver side, the transmit signalXu=[xu,1,…,xu,T] is built by using a space-time block code (STBC). A Discrete Fourier Transform (DFT) performs the time and the analog precoder performs the space encoding. The STBC simplifies receiver optimization and increases the diversity of the mmWave mMIMO system. Mathematically, the operation is expressed by(3)xu,t=fa,u,tct,u,(4)cuT=suTWT,where WT∈CT×T denotes a T-point DFT matrix and cu=[ct,u]1≤t≤T is the time encoded version of su=[st,u]1≤t≤T∈CT. st,u,t∈{1,…,T} designates a complex data symbol selected from a QAM constellation with E[|st,u|2]=σu2, where ∑u=1Uσu2=U. The transmitter total power constraint is XuF2=T. For the sake of simplicity, and without loss of generality, in this manuscript, only QPSK constellations are considered.


### 2.3. Receiver Model Description

For a givenT-sized block, the received signal is given by(5)yt=∑u=1UHuxu,t+nt,where yt∈CNrx denotes the received signal vector and nt∈CNrx is the zero mean Gaussian noise vector with variance σn2. This signal is processed at the receiver by an iterative block decoder based in subconnected hybrid architecture, as seen in Figure 2. First, the received signal yt is processed through the phase shifters in the analog part, modelled by the vector ga,p,t∈CR, where p∈1,…,NrxRF represents the index of the RF chain connected to a set of R=Nrx/NrxRF antennas. The global analog matrix Ga,t∈CNrxRF×Nrx has a block-diagonal structure: (6)Ga,t=ga,1,tT00000⋱00000ga,p,tT00000⋱00000ga,NRF,tT.Figure 2
Hybrid iterative multiuser equalizer for subconnected architecture.Baseband processing follows that containsNrxRF chains, each one connected to a subset of R antennas, and a closed-loop comprising a digital forward and feedback path. All elements of the vectors ga,p,t have equal norms (|ga,p,t(l)|2=Nrx-1). For the digital forward path, the signal first passes by a linear filter Gd,t∈CU×NrxRF and then follows a time equalizer and decoding. In the digital feedback path, the recovered data from the forward path first passes through the time precoder and then the feedback matrix Bd,t=bd,1,t,…,bd,U,t∈CU×U. The time encoding and decoding of the data symbols obey (4) and(7)S~=C~WTH,where the code-word matrix C~=[c~1,…,c~T],c~t=[c~u,t]1≤u≤U is a soft estimate of transmitted symbols. Both feedback and feedforward paths are combined; the signal output of the feedback path is subtracted from the filtered received signal Gd,tGa,tyt.This proposed receiver structure is a subconnected hybrid iterative feedback multiuser equalizer, where the main difference from the conventional iterative block decision feedback based equalizers is the analog front-end of constant amplitude phase shifters. The limitation that all elements of each analog vector must have the same norm makes the design of the proposed equalizer harder than for the conventional fully digital one. On the other hand, it assumes a subconnected hybrid architecture, where only a set of antennas is connected to each RF chain, which simplifies the physical implementation when compared with the fully connected hybrid architecture. In the following sections the analog and digital forward matrices,Ga,t and Gd,t, are designed, as well as the digital feedback matrix, Bd,t.


## 2.1. Channel Model

We consider aT-sized block fading channel, that is, a channel that remains constant during a block but varies independently between blocks. The channel follows the clustered sparse mmWave channel model discussed in [17] where Hu∈CNrx×Ntx is the channel matrix, which contributes the sum of Ncl clusters, each one with the contribution of Nray propagation paths. The channel matrix may be expressed to the uth user as (1)Hu=γ∑j,lαjl,uΛrx,uϕjlrx,u,θjlrx,uΛtx,uϕjltx,u,θjltx,u×arx,uϕjlrx,u,θjlrx,uatx,uϕjltx,u,θjltx,uH.γ=NtxNrx/NclNray is a normalization factor, and αjl,u is the complex gain of the lth ray in the jth scattering cluster. The functions Λrx,uϕjlrx,u,θjlrx,u and Λtx,uϕjltx,u,θjltx,u represent transmit and receiving antenna element gain for ϕjlrx,uθjlrx,u and ϕjltx,uθjltx,u, that is, the azimuth (elevation) angles of arrival and departure. The vectors arx,uϕjlrx,u,θjlrx,u and atx,uϕjltx,u,θjltx,u represent the normalized receiving and transmit array response vectors for the corresponding angles. Reference [17] addressed the random distributions used to generate the path gains and the angles of channel, such that E[HuF2]=NrxNtx.


## 2.2. User-Terminal Model Description

We assume that each user has only a single RF chain and sends only one data stream per time slot over theNtx transmit antennas, as shown in Figure 1. We also consider that the UTs have no access to CSI, simplifying the overall system design. The analog precoder of uth UT at the instant t is mathematically modelled by fa,u,t∈CNtx and is physically realized using a vector of analog phase shifters, where all elements of vector fa,u,t have equal norm (fa,u,tl2=Ntx-1). Therefore, the analog precoder vector of the uth UT at the instant t is generated randomly according to(2)fa,u,t=ej2πϕnu,t1≤n≤Ntx,1≤t≤T,where ϕnu,t with n∈{1,…,Ntx},t∈{1,…,T} and u∈{1,…,U} are i.i.d. uniform random variables such that ϕnu,t∈[0,1].Figure 1
User-terminalu block diagram.To guarantee that the transmit signal and then the noise plus interference are Gaussian distributed at the receiver side, the transmit signalXu=[xu,1,…,xu,T] is built by using a space-time block code (STBC). A Discrete Fourier Transform (DFT) performs the time and the analog precoder performs the space encoding. The STBC simplifies receiver optimization and increases the diversity of the mmWave mMIMO system. Mathematically, the operation is expressed by(3)xu,t=fa,u,tct,u,(4)cuT=suTWT,where WT∈CT×T denotes a T-point DFT matrix and cu=[ct,u]1≤t≤T is the time encoded version of su=[st,u]1≤t≤T∈CT. st,u,t∈{1,…,T} designates a complex data symbol selected from a QAM constellation with E[|st,u|2]=σu2, where ∑u=1Uσu2=U. The transmitter total power constraint is XuF2=T. For the sake of simplicity, and without loss of generality, in this manuscript, only QPSK constellations are considered.


## 2.3. Receiver Model Description

For a givenT-sized block, the received signal is given by(5)yt=∑u=1UHuxu,t+nt,where yt∈CNrx denotes the received signal vector and nt∈CNrx is the zero mean Gaussian noise vector with variance σn2. This signal is processed at the receiver by an iterative block decoder based in subconnected hybrid architecture, as seen in Figure 2. First, the received signal yt is processed through the phase shifters in the analog part, modelled by the vector ga,p,t∈CR, where p∈1,…,NrxRF represents the index of the RF chain connected to a set of R=Nrx/NrxRF antennas. The global analog matrix Ga,t∈CNrxRF×Nrx has a block-diagonal structure: (6)Ga,t=ga,1,tT00000⋱00000ga,p,tT00000⋱00000ga,NRF,tT.Figure 2
Hybrid iterative multiuser equalizer for subconnected architecture.Baseband processing follows that containsNrxRF chains, each one connected to a subset of R antennas, and a closed-loop comprising a digital forward and feedback path. All elements of the vectors ga,p,t have equal norms (|ga,p,t(l)|2=Nrx-1). For the digital forward path, the signal first passes by a linear filter Gd,t∈CU×NrxRF and then follows a time equalizer and decoding. In the digital feedback path, the recovered data from the forward path first passes through the time precoder and then the feedback matrix Bd,t=bd,1,t,…,bd,U,t∈CU×U. The time encoding and decoding of the data symbols obey (4) and(7)S~=C~WTH,where the code-word matrix C~=[c~1,…,c~T],c~t=[c~u,t]1≤u≤U is a soft estimate of transmitted symbols. Both feedback and feedforward paths are combined; the signal output of the feedback path is subtracted from the filtered received signal Gd,tGa,tyt.This proposed receiver structure is a subconnected hybrid iterative feedback multiuser equalizer, where the main difference from the conventional iterative block decision feedback based equalizers is the analog front-end of constant amplitude phase shifters. The limitation that all elements of each analog vector must have the same norm makes the design of the proposed equalizer harder than for the conventional fully digital one. On the other hand, it assumes a subconnected hybrid architecture, where only a set of antennas is connected to each RF chain, which simplifies the physical implementation when compared with the fully connected hybrid architecture. In the following sections the analog and digital forward matrices,Ga,t and Gd,t, are designed, as well as the digital feedback matrix, Bd,t.


## 3. Iterative Receiver Design

In this section, the proposed hybrid iterative multiuser equalizer is derived for subconnected mmWave mMIMO based systems, discussed in the previous section. We assume a decoupled joint transmitter-receiver optimization problem, with the focus on the design of the subconnected hybrid multiuser equalizer.


### 3.1. MSE Calculation

A block diagram of the proposed iterative multiuser equalizer is shown in Figure2. The received signal corresponding to user u at the tth time slot for the ith iteration is given by (8)c~u,ti=∑p=1NrxRFgd,up,tiga,p,tiTyp,t-bd,u,tiTc^ti-1,where gd,up,t(i)=Gd,t(i)u,p is the element of uth row and pth column of Gd,t(i), ga,p,t(i)∈CR denotes the analog part of the feedforward vector of user p, yp,t=yt(j)Rp-1+1≤j≤Rp∈CR, bd,u,t(i)∈CU is the feedback vector of user u, and c^t=[c^u,t]1≤u≤U∈CU is the estimate of the transmitted symbols. The matrix C^(i-1)=[c^1(i-1),…,c^T(i-1)]∈CU×T is the DFT of the detector output S^(i-1):(9)C^i-1=S^i-1WT,where S^(i)=[s^1,…,s^t,…,s^T] is the hard decision related to the QPSK data symbols S=[s1,…,st,…,sT]. At iteration i(10)s^i=signs~i.From the central limit theorem,ct=[cu,t]1≤u≤U, u∈{1,…,U}, t∈{1,…,T} approximately are Gaussian distributed. In addition, since the input-output relationship between ct and c^t(i), u∈{1,…,U}, t∈{1,…,T} is memoryless, by the Bussgang theorem [38], we have(11)c^ti=Ψict+ϵ^ti,t∈1,…,T,where Ψ(i) is a diagonal matrix given by (12)Ψi=diagψ1i,…,ψui,…,ψUi,(13)ψui=Ec^tiuct∗uEctu2,u∈1,…,U,and ϵ^t(i) is an error vector with zero mean and uncorrelated with ct,t∈{1,…,T}, with (14)Eϵ^tiϵ^tiH=IU-Ψi2σu2.For practical systems, the transmitted signals ct(u) are not known a priori and definition (13) cannot be used to compute matrix Ψ(i), which must be estimated using other methods. Please refer to work [39] where several methods are presented to estimate matrix Ψ(i).DefineHt=H1fa,1,t,…,HUfa,U,t∈CNrx×U, H-p,t=Ht(j,l)Rp-1+1≤j≤Rp,1≤l≤U∈CR×U; therefore, from (8) and (11), it follows that(15)c~u,ti=∑p=1NrxRFgd,up,tiga,p,tiTH-p,tct+np,t-bd,u,tiTΨi-1ct+ϵ^ti-1=∑p=1NrxRFgd,up,tiga,p,tiTH-p,tct+∑p=1NrxRFgd,up,tiga,p,tiTnp,t-bd,u,tiTΨi-1ct-bd,u,tiTϵ^ti-1,where nu,t=nt(j)Ru-1+1≤j≤Ru∈CR. Define gad,up,t(i)=gd,up,t(i)ga,p,t(i)∈CR and ϵ~u,t(i)=c~u,t(i)-cu,t which denotes the overall error. Then, by (15), the overall error is(16)ϵ~u,ti=∑p=1NrxRFgad,up,tiTH-p,t-bd,u,tiTΨi-1ct-cu,t︸ResidualISI-bd,u,tiTϵ^ti-1︸Errorfromestimatec^ti+∑p=1NrxRFgad,up,tiTnp,t︸ChannelNoise.As we can see in (16), the error has three terms: (1) the residual ISI term; (2) the error stemming from the estimate made by c^t(i) of cu,t; and (3) the term corresponding to the channel noise. Let us assume that error ϵ~u,t(i) is complex Gaussian distributed. From the Bussgang [38] theorem, the zero mean error variable ϵ^u,t(i) is uncorrelated with cu,t, and then, from (16), the average error (ϵ~u,t(i)) power, that is, the MSE at time slot t, is given by(17)MSEu,ti=Eϵ~u,ti2=∑p=1NrxRFgad,up,tiTH-p,t-vuT-bd,u,tiTΨi-1F2σu2+bd,u,tiTIU-Ψi-121/2F2σu2+∑p=1NrxRFgad,up,tiTF2σn2,where vu=0,0,…,1,…,0,0T∈CU, with the only nonzero value of vu in the uth position, and then cu,t=vuTct.


### 3.2. Design of Subconnected Hybrid Iterative Multiuser Equalizer

We start by considering the optimal case, that is, a full digital linear feedforward filter, without the sparsity constraint. We design the iterative multiuser equalizer based on the IB-DFE principles, and we use the average BER as the metric. The average BER minimization, according to [22], is equivalent to the MSE minimization:(18)gad,up,tiopt,bd,u,tiopt=argminMSEu,tis.t.∑t=1T∑p=1NrxRFgad,up,tiTH-p,t=TzUT,where gad,up,t(i) is the linear feedforward filter, bd,u,t(i) the feedback filter, and zU=1,1,…,1T∈CU a vector with U ones. The solution to optimization problem (18) is (see the Appendix)(19)gad,up,tioptT=ωuH-p,tHRp,ti-1-1,bd,u,tioptT=∑p=1NrxRFgad,up,tiTH-p,t-vuTΨi-1H,ωu=TzUT∑t=1T∑p=1NrxRFH-p,tHRp,ti-1-1H-p,t-1,Rp,ti-1=H-p,tIU-Ψi-12H-p,tH+σn2σu2IR.The previous optimization problem of (18) does not reflect the analog domain constraints. Designate Ga as the set of vectors with constant-magnitude entries, that is, the set of feasible RF equalizers. Thus, the reformulated optimization problem for the subconnected hybrid iterative equalizer is as follows:(20)ga,u,tiopt,gd,up,tiopt,bd,u,tiopt=argminMSEu,tis.t.∑t=1T∑p=1NrxRFgd,up,tiga,p,tiTH-p,t=TzUT,ga,p,ti∈Ga,p∈1,…,NrxRF.The feedback equalizer for the subconnected hybrid iterative equalizer is analogous to the fully digital iterative equalizer referred to in the previous section, because the analog domain constraints do not impose any restriction on the vectorbd,u,t(i) and thus are given by(21)bd,u,tioptT=∑p=1NrxRFgd,up,tioptga,p,tioptTH-p,t-vuTΨi-1H.From (17) and (21), the MSE expression is(22)MSEu,ti=∑p=1NrxRFgd,up,tiga,p,tiTH-p,t-vuT×IU-Ψi-121/2F2σu2+∑p=1NrxRFgd,up,tiga,p,tiTF2σn2.According to the constraint of (18) ∑p=1NrxRF(gad,up,t(i))optTH-p,t=vuT and therefore (22) can be rewritten as (23)MSEu,ti=∑p=1NrxRFgd,up,tiga,p,tiT-gad,up,tioptTH-p,t×IU-Ψi-121/2F2σu2+∑p=1NrxRFgd,up,tiga,p,tiTF2σn2.Because the feasible setGa has a nonconvex nature, an analytical solution to problem (20) is very hard to obtain. However, an approximate solution is possible by assuming that the vector ga,p,t(i) is a version of vector arx,p(θj,lrx,p). Let Arx=[Arx,1,…,Arx,U]∈CNrx×NclNrayU, with Arx,u=[arx,u(θ1,1rx,u),…,arx,u(θNcl,Nrayrx,u)]∈CNrx×NclNray, forming the matrix with the array response vectors of the receiver, corresponding to user u. As each RF chain is not connected to all antennas, in the selection process of the vector for the pth RF chain, the entries corresponding to the unconnected antennas are removed. For the uth RF chain we use the dictionary Arxu=Arx(j,l)Ru-1+1≤j≤Ru,1≤l≤NclNrayU∈CR×NclNrayU, which is a submatrix of Arx. Then, the vector ga,u,t(i) is selected from dictionary Arxu. Therefore, optimization problem (20) can be approximated as follows:(24)g⃛d,up,tiopt=argminMSE¯u,tig⃛d,up,ti,Arxps.t.∑t=1T∑p=1NrxRFg⃛d,up,tiArxpHH-p,t=TzUTg⃛d,up,tig⃛d,up,tiH0=1,where g⃛d,up,t(i)=[gd,up,t(i),0]T∈CNclNrayU and(25)MSE¯u,tig⃛d,up,ti,Arxp=∑p=1NrxRFg⃛d,up,tiArxpH-gad,up,tioptH-p,t×IU-Ψi-121/2F2σu2+∑p=1NrxRFg⃛d,up,tiArxpHF2σn2.The constraint g⃛d,up,t(i)(g⃛d,up,t(i))H0=1 enforces that only one element of vector g⃛d,up,t(i) is nonzero, representing the sparsity constraint. The optimum digital feedforward vector (gd,up,t(i))opt is computed from the solution (g⃛d,up,t(i))opt of the optimization problem in (24), by removing the zero elements. The optimum analog feedforward vector (ga,u,t(i))opt is given by the selected column from (Arxu)H, corresponding to the nonzero element of g⃛d,up,t(i). Consider optimization problem (24), without the sparsity constraint: (26)g⃛d,u,tiopt=argminMSE¯u,tig⃛d,up,ti,Arxps.t.∑t=1T∑p=1NrxRFg⃛d,up,tiArxpHH-p,t=TzUT.The associated Lagrangian to problem (26) is(27)Lμu,g⃛d,up,ti=MSE¯u,tig⃛d,up,ti,Arxp+μu∑t=1T∑p=1NrxRFg⃛d,up,tiArxpHH-p,t-TzUT,where μu, u∈{1,…,U}, are the Lagrange multipliers. Taking the derivate in order to obtain g⃛d,up,t(i), we obtain the optimality condition(28)∂Lμu,g⃛d,up,ti∂g⃛d,up,ti∗=gres,up,tiArxp=0,where (28) is the orthogonality principle relative to MSE estimator (24) and gres,up,t(i) is the residue vector given by(29)gres,up,ti=g⃛d,up,tiArxpH-gad,up,tioptH-p,tIU-Ψi-12H-p,tH+g⃛d,up,tiArxpHσn2σu2+μu2σu2H-p,tH.An iterative greedy method is used to select the best column of the dictionary Arxu allowing enforcement of the sparsity constraint. Due to the mmWave channel nature, the number of the paths is small and, consequently, the complexity of this selection process is low.Denotegd,up,t(i) and (ga,u,t(i))opt as the digital and analog terms of the feedforward vector, corresponding to the selected indices from g⃛d,up,t(i). For example, if the selected indices are in the first positions, then g⃛d,up,t(i)=[gd,up,t(i),0]. Thus, from (29) the orthogonality condition simplifies to(30)gd,up,tiga,p,tiT-gad,up,tiTH-p,tIU-Ψi-12H-p,tH+gd,up,tiga,p,tiTσn2σu2+μu2σu2H-p,tHga,u,ti∗=0.The solution to (30) is(31)gd,up,tiopt=ωdH-p,tHga,p,ti∗×ga,p,tiTRp,ti-1ga,p,ti∗-1,with ωd, and(32)ωd=gad,up,tiTH-p,tIU-Ψi-12+μu2σu2.The Lagrangian multiplier μu is computed by the power constraint of problem (24), given by(33)μu2σu2=TzUT∑t=1T∑p=1NrxRFH-p,tHga,p,ti∗×ga,p,tiTRp,ti-1ga,p,ti∗-1ga,p,tiTH-p,t-1-gad,up,tiHH-p,tIU-Ψi-12.After obtaininggd,up,t(i), the optimum value of the digital feedforward part, the previous steps are repeated until u=NrxRF. All steps are synthesized in Algorithm 1, pseudocode of the proposed subconnected hybrid iterative block multiuser equalizer.Algorithm 1:The proposed multiuser equalizer for subconnected mmWave massive MIMO architecture.
Input: (gad,up,t(i))opt
(1)foru≤NrxRFdo
(2)  gres,uu,t(i)=gd,uu,tiga,u,ti-gad,uu,tioptH-u,tIU-Ψi-12H-u,tH+gd,uu,tiga,u,tiσn2σu2+μu2σu2H-u,tH
(3)  k=argmaxl=1,…,NclNray⁡ArxuHgres,uu,tiHgres,uu,tiArxul,l
(4) ga,u,tiopt=ga,u,tioptH∣ArxukH
(5)end for
(6)gd,up,tiopt=ωdH-p,tHga,p,tiHga,p,t(i)Rp,ti-1ga,p,tiH-1
(7)return(ga,u,t(i))opt,(gd,up,t(i))opt


## 3.1. MSE Calculation

A block diagram of the proposed iterative multiuser equalizer is shown in Figure2. The received signal corresponding to user u at the tth time slot for the ith iteration is given by (8)c~u,ti=∑p=1NrxRFgd,up,tiga,p,tiTyp,t-bd,u,tiTc^ti-1,where gd,up,t(i)=Gd,t(i)u,p is the element of uth row and pth column of Gd,t(i), ga,p,t(i)∈CR denotes the analog part of the feedforward vector of user p, yp,t=yt(j)Rp-1+1≤j≤Rp∈CR, bd,u,t(i)∈CU is the feedback vector of user u, and c^t=[c^u,t]1≤u≤U∈CU is the estimate of the transmitted symbols. The matrix C^(i-1)=[c^1(i-1),…,c^T(i-1)]∈CU×T is the DFT of the detector output S^(i-1):(9)C^i-1=S^i-1WT,where S^(i)=[s^1,…,s^t,…,s^T] is the hard decision related to the QPSK data symbols S=[s1,…,st,…,sT]. At iteration i(10)s^i=signs~i.From the central limit theorem,ct=[cu,t]1≤u≤U, u∈{1,…,U}, t∈{1,…,T} approximately are Gaussian distributed. In addition, since the input-output relationship between ct and c^t(i), u∈{1,…,U}, t∈{1,…,T} is memoryless, by the Bussgang theorem [38], we have(11)c^ti=Ψict+ϵ^ti,t∈1,…,T,where Ψ(i) is a diagonal matrix given by (12)Ψi=diagψ1i,…,ψui,…,ψUi,(13)ψui=Ec^tiuct∗uEctu2,u∈1,…,U,and ϵ^t(i) is an error vector with zero mean and uncorrelated with ct,t∈{1,…,T}, with (14)Eϵ^tiϵ^tiH=IU-Ψi2σu2.For practical systems, the transmitted signals ct(u) are not known a priori and definition (13) cannot be used to compute matrix Ψ(i), which must be estimated using other methods. Please refer to work [39] where several methods are presented to estimate matrix Ψ(i).DefineHt=H1fa,1,t,…,HUfa,U,t∈CNrx×U, H-p,t=Ht(j,l)Rp-1+1≤j≤Rp,1≤l≤U∈CR×U; therefore, from (8) and (11), it follows that(15)c~u,ti=∑p=1NrxRFgd,up,tiga,p,tiTH-p,tct+np,t-bd,u,tiTΨi-1ct+ϵ^ti-1=∑p=1NrxRFgd,up,tiga,p,tiTH-p,tct+∑p=1NrxRFgd,up,tiga,p,tiTnp,t-bd,u,tiTΨi-1ct-bd,u,tiTϵ^ti-1,where nu,t=nt(j)Ru-1+1≤j≤Ru∈CR. Define gad,up,t(i)=gd,up,t(i)ga,p,t(i)∈CR and ϵ~u,t(i)=c~u,t(i)-cu,t which denotes the overall error. Then, by (15), the overall error is(16)ϵ~u,ti=∑p=1NrxRFgad,up,tiTH-p,t-bd,u,tiTΨi-1ct-cu,t︸ResidualISI-bd,u,tiTϵ^ti-1︸Errorfromestimatec^ti+∑p=1NrxRFgad,up,tiTnp,t︸ChannelNoise.As we can see in (16), the error has three terms: (1) the residual ISI term; (2) the error stemming from the estimate made by c^t(i) of cu,t; and (3) the term corresponding to the channel noise. Let us assume that error ϵ~u,t(i) is complex Gaussian distributed. From the Bussgang [38] theorem, the zero mean error variable ϵ^u,t(i) is uncorrelated with cu,t, and then, from (16), the average error (ϵ~u,t(i)) power, that is, the MSE at time slot t, is given by(17)MSEu,ti=Eϵ~u,ti2=∑p=1NrxRFgad,up,tiTH-p,t-vuT-bd,u,tiTΨi-1F2σu2+bd,u,tiTIU-Ψi-121/2F2σu2+∑p=1NrxRFgad,up,tiTF2σn2,where vu=0,0,…,1,…,0,0T∈CU, with the only nonzero value of vu in the uth position, and then cu,t=vuTct.


## 3.2. Design of Subconnected Hybrid Iterative Multiuser Equalizer

We start by considering the optimal case, that is, a full digital linear feedforward filter, without the sparsity constraint. We design the iterative multiuser equalizer based on the IB-DFE principles, and we use the average BER as the metric. The average BER minimization, according to [22], is equivalent to the MSE minimization:(18)gad,up,tiopt,bd,u,tiopt=argminMSEu,tis.t.∑t=1T∑p=1NrxRFgad,up,tiTH-p,t=TzUT,where gad,up,t(i) is the linear feedforward filter, bd,u,t(i) the feedback filter, and zU=1,1,…,1T∈CU a vector with U ones. The solution to optimization problem (18) is (see the Appendix)(19)gad,up,tioptT=ωuH-p,tHRp,ti-1-1,bd,u,tioptT=∑p=1NrxRFgad,up,tiTH-p,t-vuTΨi-1H,ωu=TzUT∑t=1T∑p=1NrxRFH-p,tHRp,ti-1-1H-p,t-1,Rp,ti-1=H-p,tIU-Ψi-12H-p,tH+σn2σu2IR.The previous optimization problem of (18) does not reflect the analog domain constraints. Designate Ga as the set of vectors with constant-magnitude entries, that is, the set of feasible RF equalizers. Thus, the reformulated optimization problem for the subconnected hybrid iterative equalizer is as follows:(20)ga,u,tiopt,gd,up,tiopt,bd,u,tiopt=argminMSEu,tis.t.∑t=1T∑p=1NrxRFgd,up,tiga,p,tiTH-p,t=TzUT,ga,p,ti∈Ga,p∈1,…,NrxRF.The feedback equalizer for the subconnected hybrid iterative equalizer is analogous to the fully digital iterative equalizer referred to in the previous section, because the analog domain constraints do not impose any restriction on the vectorbd,u,t(i) and thus are given by(21)bd,u,tioptT=∑p=1NrxRFgd,up,tioptga,p,tioptTH-p,t-vuTΨi-1H.From (17) and (21), the MSE expression is(22)MSEu,ti=∑p=1NrxRFgd,up,tiga,p,tiTH-p,t-vuT×IU-Ψi-121/2F2σu2+∑p=1NrxRFgd,up,tiga,p,tiTF2σn2.According to the constraint of (18) ∑p=1NrxRF(gad,up,t(i))optTH-p,t=vuT and therefore (22) can be rewritten as (23)MSEu,ti=∑p=1NrxRFgd,up,tiga,p,tiT-gad,up,tioptTH-p,t×IU-Ψi-121/2F2σu2+∑p=1NrxRFgd,up,tiga,p,tiTF2σn2.Because the feasible setGa has a nonconvex nature, an analytical solution to problem (20) is very hard to obtain. However, an approximate solution is possible by assuming that the vector ga,p,t(i) is a version of vector arx,p(θj,lrx,p). Let Arx=[Arx,1,…,Arx,U]∈CNrx×NclNrayU, with Arx,u=[arx,u(θ1,1rx,u),…,arx,u(θNcl,Nrayrx,u)]∈CNrx×NclNray, forming the matrix with the array response vectors of the receiver, corresponding to user u. As each RF chain is not connected to all antennas, in the selection process of the vector for the pth RF chain, the entries corresponding to the unconnected antennas are removed. For the uth RF chain we use the dictionary Arxu=Arx(j,l)Ru-1+1≤j≤Ru,1≤l≤NclNrayU∈CR×NclNrayU, which is a submatrix of Arx. Then, the vector ga,u,t(i) is selected from dictionary Arxu. Therefore, optimization problem (20) can be approximated as follows:(24)g⃛d,up,tiopt=argminMSE¯u,tig⃛d,up,ti,Arxps.t.∑t=1T∑p=1NrxRFg⃛d,up,tiArxpHH-p,t=TzUTg⃛d,up,tig⃛d,up,tiH0=1,where g⃛d,up,t(i)=[gd,up,t(i),0]T∈CNclNrayU and(25)MSE¯u,tig⃛d,up,ti,Arxp=∑p=1NrxRFg⃛d,up,tiArxpH-gad,up,tioptH-p,t×IU-Ψi-121/2F2σu2+∑p=1NrxRFg⃛d,up,tiArxpHF2σn2.The constraint g⃛d,up,t(i)(g⃛d,up,t(i))H0=1 enforces that only one element of vector g⃛d,up,t(i) is nonzero, representing the sparsity constraint. The optimum digital feedforward vector (gd,up,t(i))opt is computed from the solution (g⃛d,up,t(i))opt of the optimization problem in (24), by removing the zero elements. The optimum analog feedforward vector (ga,u,t(i))opt is given by the selected column from (Arxu)H, corresponding to the nonzero element of g⃛d,up,t(i). Consider optimization problem (24), without the sparsity constraint: (26)g⃛d,u,tiopt=argminMSE¯u,tig⃛d,up,ti,Arxps.t.∑t=1T∑p=1NrxRFg⃛d,up,tiArxpHH-p,t=TzUT.The associated Lagrangian to problem (26) is(27)Lμu,g⃛d,up,ti=MSE¯u,tig⃛d,up,ti,Arxp+μu∑t=1T∑p=1NrxRFg⃛d,up,tiArxpHH-p,t-TzUT,where μu, u∈{1,…,U}, are the Lagrange multipliers. Taking the derivate in order to obtain g⃛d,up,t(i), we obtain the optimality condition(28)∂Lμu,g⃛d,up,ti∂g⃛d,up,ti∗=gres,up,tiArxp=0,where (28) is the orthogonality principle relative to MSE estimator (24) and gres,up,t(i) is the residue vector given by(29)gres,up,ti=g⃛d,up,tiArxpH-gad,up,tioptH-p,tIU-Ψi-12H-p,tH+g⃛d,up,tiArxpHσn2σu2+μu2σu2H-p,tH.An iterative greedy method is used to select the best column of the dictionary Arxu allowing enforcement of the sparsity constraint. Due to the mmWave channel nature, the number of the paths is small and, consequently, the complexity of this selection process is low.Denotegd,up,t(i) and (ga,u,t(i))opt as the digital and analog terms of the feedforward vector, corresponding to the selected indices from g⃛d,up,t(i). For example, if the selected indices are in the first positions, then g⃛d,up,t(i)=[gd,up,t(i),0]. Thus, from (29) the orthogonality condition simplifies to(30)gd,up,tiga,p,tiT-gad,up,tiTH-p,tIU-Ψi-12H-p,tH+gd,up,tiga,p,tiTσn2σu2+μu2σu2H-p,tHga,u,ti∗=0.The solution to (30) is(31)gd,up,tiopt=ωdH-p,tHga,p,ti∗×ga,p,tiTRp,ti-1ga,p,ti∗-1,with ωd, and(32)ωd=gad,up,tiTH-p,tIU-Ψi-12+μu2σu2.The Lagrangian multiplier μu is computed by the power constraint of problem (24), given by(33)μu2σu2=TzUT∑t=1T∑p=1NrxRFH-p,tHga,p,ti∗×ga,p,tiTRp,ti-1ga,p,ti∗-1ga,p,tiTH-p,t-1-gad,up,tiHH-p,tIU-Ψi-12.After obtaininggd,up,t(i), the optimum value of the digital feedforward part, the previous steps are repeated until u=NrxRF. All steps are synthesized in Algorithm 1, pseudocode of the proposed subconnected hybrid iterative block multiuser equalizer.Algorithm 1:The proposed multiuser equalizer for subconnected mmWave massive MIMO architecture.
Input: (gad,up,t(i))opt
(1)foru≤NrxRFdo
(2)  gres,uu,t(i)=gd,uu,tiga,u,ti-gad,uu,tioptH-u,tIU-Ψi-12H-u,tH+gd,uu,tiga,u,tiσn2σu2+μu2σu2H-u,tH
(3)  k=argmaxl=1,…,NclNray⁡ArxuHgres,uu,tiHgres,uu,tiArxul,l
(4) ga,u,tiopt=ga,u,tioptH∣ArxukH
(5)end for
(6)gd,up,tiopt=ωdH-p,tHga,p,tiHga,p,t(i)Rp,ti-1ga,p,tiH-1
(7)return(ga,u,t(i))opt,(gd,up,t(i))opt


## 4. Performance Results

In this section, we show the performance of the proposed hybrid iterative block multiuser equalizer for subconnected millimeter wave massive MIMO systems. The BER is considered the performance metric, presented as a function ofEb/N0, where Eb is the average bit energy and N0 is the one-sided noise power spectral density. The power of each UT is normalized to σ12=⋯=σU2=1 and the channel matrix, as previously mentioned, is normalized, such that E[HuF2]=NrxNtx. Then, the average Eb/N0 for all users u∈{1,…,U} is identical and given by Eb/N0=σu2/(2σn2)=σn-2/2.The carrier frequency was set to 72 GHz and was considered a clustered narrowband channel model [40] for each user with Ncl=8 clusters, all with the same average power, and Nray=4 rays per cluster. The azimuth angles of arrival and departure of the channel model are Laplacian distributed as in [21]. We assumed an angle spread equal to 8° at both the transmitter and receiver and uniform linear arrays (ULAs) with antenna element spacing equal to half-wavelength; however, the subconnected hybrid equalizer developed in this paper can be applied to any antenna arrays. The channel remains constant during a block with size T but varies independently between them. We assume perfect synchronization and CSI knowledge at the receiver side. All results were obtained for a QPSK modulation. We present results for two main scenarios, all with Ntx=8. The other parameters are as follows.Scenario 1 (U=NrxRF=8).
(1.a)
N
r
x
=
16
R
=
2.
(1.b)
N
r
x
=
32
R
=
4.
(1.c)
N
r
x
=
48
R
=
6.Scenario 2 (U=NrxRF=4).
(2.a)
N
r
x
=
8
R
=
2.
(2.b)
N
r
x
=
16
R
=
4.
(2.c)
N
r
x
=
24
R
=
6.HereR=Nrx/NrxRF represents the number of antennas per RF chain. In both scenarios, we assume the full load case (worst case), where the number of users and RF chains is equal. To compute the proposed subconnected hybrid equalizer, we consider that fa,u,t, the analog precoder for the uth user, is generated according to (2). As mentioned before, the motivation is to keep the UTs with very low complexity without the knowledge of CSI before transmission. The proposed subconnected hybrid iterative multiuser equalizer is referred to here assubconnected. The results are compared with both fully digital, referred to here asdigital, and fully connected, referred to asfully connected, which was recently proposed in [22], approaches.Start with the results obtained for scenario (1.b) presented in Figures3 and 4. In Figure 3, we present results for iterations 1, 2, 3, and 4 of the proposed subconnected hybrid iterative multiuser equalizer. It can be seen that the performance improves as the number of iterations increases, as expected. Additionally, the gaps between the 1st and 2nd iterations are much higher than between the 3rd and 4th iterations. This occurs because most of the residual intersymbol and multiuser interferences are removed from the 1st to the 2nd iteration. For this scenario, the results with the fully digital and fully connected hybrid multiuser equalizer are presented in Figure 4. In this figure, we verify that thesubconnected performance is worse than bothdigital andfully connected approaches, mainly for the first iteration. However, for the 4th iteration we can see that the performance of the proposed subconnected hybrid multiuser equalizer is close to thedigital andfully connected counterparts. Therefore, we can argue that the dictionary approximation and the sequential optimization are quite precise.Figure 3
Performance of the proposed subconnected hybrid multiuser equalizer for scenario (1.b).Figure 4
Performance comparison of the proposed subconnected hybrid multiuser equalizer with a fully connected approach for scenario (1.b).Compare scenario (1.b) with scenario (2.b) presented in Figure5, where the number of users and RF chains (U=NrxRF) changes, but the number of antennas per RF chain is the same; that is, R=4. In scenario (1.b), at a target BER of 10-3, the penalties of the proposed subconnected equalizer for thefully connected anddigital approaches are 2.3 dB and 3.8 dB, respectively, at the 4th iteration. In scenario (2.b), for the same target BER and iteration, the penalties forfully connected anddigital are approximately 1.6 dB and 3.2 dB, slightly decreasing for lower U=NrxRF. We can also see that for scenario (1.b) a lower Eb/N0 is needed to achieve a BER of 10-5 compared with scenario (2.b). This is because scenario (1.b) can achieve higher diversity and array gain given by the larger dimension of the receiving antenna array. When we reduce the number of antennas per RF chain by half, that is, for R=2, which corresponds to scenario (1.a) presented in Figure 6, we verify that the penalty forfully connected anddigital approaches is approximately 1.2 dB and 1.9 dB, thus decreasing for lower R. This occurs because when we reduce R, the subconnected architecture tends toward the fully digital case. In the extreme case where R=1, we have one antenna per RF chain, that is, the digital architecture. As expected, by reducing R the degrees of freedom of the subconnected architecture increase, and the gap with thefully connected anddigital approaches decreases. For the digital equalizer the curve for iteration 1 is not presented in Figures 4, 5, and 6 to improve the intelligibility since it almost overlaps with the curve for iteration 4 of the proposed scheme.Figure 5
Performance comparison of the proposed subconnected hybrid multiuser equalizer with a fully connected approach for scenario (2.b).Figure 6
Performance comparison of the proposed subconnected hybrid multiuser equalizer with a fully connected approach for scenario (1.a).In Figures7 and 8, we compare the performance of thesubconnected approach for Scenarios 1 and 2 by considering R=2,4, and 6 antennas per RF chain. Although the penalty of the proposed subconnected multiuser equalizer relative to thefully connected anddigital increases with R, as previously verified, the BER performance of the subconnected multiuser equalizer improves with R because the diversity and antenna gain are higher due to a larger number of receiving antennas.Figure 7
Performance comparison of the proposed subconnected hybrid multiuser equalizer forR=2,4, and 6, for Scenario 1.Figure 8
Performance comparison of the proposed subconnected hybrid multiuser equalizer forR=2,4, and 6, for Scenario 2.Notably, the above was considered the worst case, whereU=NrxRF. Improved BER performance would be achieved for U<NrxRF, that is, for a number of users (streams) lower than the number of RF chains. For that case, the performance gaps between the proposed subconnected hybrid multiuser equalizer and thedigital/fully connected approaches would be even smaller.


## 5. Conclusions

We designed a new hybrid iterative multiuser equalizer for a subconnected mmWave massive MIMO architecture. We considered low complexity UTs employing randomly analog-only precoding and a single RF chain. First, it was verified that the proposed hybrid iterative multiuser equalizer converges and that it achieves a performance close to thedigital andfully connectedcounterparts, requiring very few iterations. This structure efficiently manages multiuser interference; the dictionary approximation and the sequential optimization are quite precise. We observed that when we reduce the number of receiving antennas per RF chain, that is, R=Nrx/NrxRF, the BER performance gets worse due to the reduction of both the antenna gain and diversity. We also observed that the gap between the proposed subconnectedscheme and thedigital/fully connected approaches decreases when R decreases, because the architecture becomes closer to thedigital one.The performance of the proposed receiver structure shows that it is interesting for practical mmWave massive MIMO based systems, that is, the systems where hardware constraints are considered. This means that the number of transmit/receiving RF chains is much lower than the number of transmit/receiving antennas, and each RF chain is only connected to a reduced number of antennas, ensuring good performance at a low cost.



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*Source: 9171068-2017-12-20.xml*