Receive Spatial Modulation in Correlated Massive MIMO With ...static.tongtianta.site/paper_pdf/12fef50e-891c-11e9-8d5e-00163e08bb86.pdfing on massive multiple-input multiple-output

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 67, NO. 5, MARCH 1, 2019 1237

Receive Spatial Modulation in Correlated MassiveMIMO With Partial CSI

Marjan Maleki , Kamal Mohamed-Pour, and Mojtaba Soltanalian , Member, IEEE

Abstract—Spatial modulation is emerging as a promising signaltransmission paradigm for future wireless communications rely-ing on massive multiple-input multiple-output (MIMO) technol-ogy. In this paper, the receive spatial modulation (RSM) is ap-plied to the downlink of multi-user MIMO systems with the basestation equipped with a massive array of antennas. Due to thesignificant overhead associated with acquiring channel state in-formation at the transmitter (CSIT) of frequency-division-duplexmassive MIMO systems, herein we propose a preprocessing schemefor correlated channels that reduces the channel estimation over-heads. The proposed preprocessing method relies on the partialCSIT and consists of two stages of beamforming and precod-ing. The beamforming is accomplished via block-diagonalizationto combat multi-user interference while the precoding matrix isdesigned by formulating optimization problems aiming to elimi-nate inter-antenna interference and enhancing RSM transmissionperformance. The design criteria of the problems include bit-error-rate minimization and discrete-input continuous-output (DCMC)capacity maximization with a transmit power constraint. The aris-ing non-convex problems are cast as semi-definite and second-ordercone programs that can be solved efficiently. The proposed schemesare readily extended to generalized RSM as the most comprehen-sive case in RSM-MIMO scenarios. Simulation results demonstratethat the proposed schemes outperform the imperfect-CSIT basedand two-stage zero-forcing procedures and offer similar or sig-nificant performance compared to conventional MIMO counter-parts at low to mid signal-to-noise ratio regimes, depending on theconfigurations.

Index Terms—Receive spatial modulation (RSM), massiveMIMO, partial channel state information at the transmitter(CSIT), two-stage preprocessing, optimization.

I. INTRODUCTION

MASSIVE MIMO has recently attracted a spurt of interestdue to its innate advantages over conventional MIMO,

including considerable power efficiency, reliability, security, aswell as higher data communication rates [1]–[4]. In such sys-tems, a base station (BS) equipped by hundreds of antennasserves to many user terminals simultaneously [2]. Although

Manuscript received January 15, 2018; revised August 25, 2018; acceptedDecember 5, 2018. Date of publication December 27, 2018; date of currentversion January 14, 2019. The associate editor coordinating the review of thismanuscript and approving it for publication was Prof. Kostas D. Berberidis.This work was supported by the National Science Foundation under GrantECCS-1809225. (Corresponding author: Marjan Maleki.)

M. Maleki and K. Mohamed-Pour are with the Department of Electricaland Computer Engineering, K. N. Toosi University of Technology, Tehran1969764499, Iran (e-mail:,[email protected]; [email protected]).

M. Soltanalian is with the Department of Electrical and Computer Engi-neering, University of Illinois at Chicago, Chicago, IL 60607 USA (e-mail:,[email protected]).

Digital Object Identifier 10.1109/TSP.2018.2890063

current research on multi-user massive MIMO systems typi-cally focuses on single antenna users, deployment of multipleantennas for user equipment (as a key assumption for the futurecommunication systems) offers potential for higher multiplex-ing and diversity gains [5].

Note that due to the radio frequency (RF)-chain connected toeach antenna element, employing multiple antennas for usersleads to an increased complexity, hardware cost and powerconsumption. Spatial modulation (SM) is a beneficial multi-antenna modulation paradigm introduced for single RF-chainmassive MIMO systems. This MIMO modulation scheme pro-vides higher throughput than single-antenna systems by usingthe index of active antennas as the third dimension of a spatialconstellation to convey extra information, in addition to ampli-tude phase modulated (APM) symbols [6].

While there are a wide range of research efforts focused onpoint-to-point architectures [7]–[10], several recent works haveinvestigated SM transmission scheme in the downlink of multi-user systems [11]–[13]. Such attempts rely on grouping thetransmit antennas and allocating each antenna set to a user.However, improving the overall system error performance inall of these schemes requires full knowledge of channel sateinformation at transmitter (CSIT) and imposes a higher com-plexity and implementation cost. A novel transmission scheme,called layered spatial modulation, is proposed in [14] whichassigns a set of antennas according to users’ symbols and at-tains higher data rates in comparison with existing SM-basedmultiuser schemes.

A reciprocal scheme for SM, named receive spatial modula-tion (RSM), also known as preprocessing aided spatial modu-lation (PSM), has been proposed in [15]. RSM primarily relieson receiving the information bits at one of the receive antennasand then employing the index of the intended antenna to forma spatial constellation. To overcome the limitation of havinga power of two number of antennas, the RSM scheme is ex-tended in [16]–[17] to its general framework, generalized RSM.Generalized RSM makes multi-stream transmission possible byselecting multiple antennas rather than one single antenna toreceive information bits at any given time. A more recent de-velopment on RSM is presented in [18], focusing on the designof minimum mean-square error (MMSE) and zero-forcing (ZF)precoders, particularly with per antenna power constraints inlieu of the common average total power constraint.

Recently, [19]–[21] have made use of RSM in the downlinkof multi-user systems. In [19] the authors consider performanceof RSM for multi-stream transmission in a broadcast channel

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1238 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 67, NO. 5, MARCH 1, 2019

and the bit error rate (BER) performance of ZF preprocessingscheme with the given full knowledge of CSIT is studied. In [20],the authors focus on the performance of RSM, using ZF precod-ing in the shadowing MIMO broadcast channel. As a practicalscenario, which the authors claim is suitable for the downlink,RSM has been used in a virtual MIMO dual-hop architecturein [21], where linear precoding methods with imperfect CSITare considered along with an analytical power analysis for eachprecoder.

Extending the concept of SM to massive MIMO systemsoffers higher system throughput and energy efficiency. It allowsusers to employ multiple antennas connected to only one singleRF-chain, and thus enjoy MIMO multiplexing and diversitygains without imposing RF hardware complexity and cost [22].It is demonstrated in [23] that generalized SM is a suitablecandidate for deployment in massive MIMO systems. In thecontext of massive MIMO systems, the literature on SM hasmostly focused on the uplink of multi-user systems [24]–[25].We refer the interested reader to the survey of proposed single-carrier schemes for massive MIMO systems in [26].

With this background, we summarize below the main contri-butions of this work, and particularly, elaborate on the differ-ences and points of departure from the existing literature:

� In this paper, the concept of RSM in downlink of a multi-user frequency division duplexing (FDD)-based massiveMIMO system is considered for the first time. Our goalis to design the preprocessing stage at the BS in order toovercome the problem of CSIT gathering.

� In our proposed framework, we use partial CSIT and as-sume that the second order statistics of user channels isavailable at the BS.

� In practical scenarios, it is assumed that the communicationantennas at the receive and transmit sides are surrounded bylocal scatterers resulting in correlated channels—where thecorrelation depends on antenna geometry, angular spreadand arrival angle of waves [27]. In this paper, we usethe spatially correlated Kronecker model [28] with theassumption that only the correlation matrices are availableat the BS.

� Using this partial information, we propose a novel two-stage preprocessing scheme in the downlink of multi-usermassive MIMO system. At the first stage, the ZF-basedmethod of block diagonalization (BD) is used—an effec-tive linear processing scheme described well as a gener-alization of channel inversion method for multi-antennasystems [29]–[31]. The second stage of preprocessing isdone through formulating two optimization problems. Inthe first problem, our goal is to minimize the union boundof the average BER, which is deemed difficult to deal withdirectly. This motivates further considering a maximizationof the minimum average Euclidean distance between thereceived noise-free codewords under a constraint on allow-able transmit power. The second problem is defined to de-sign precoding matrix in BS by maximizing discrete-inputcontinuous-output memoryless channel (DCMC) capacitywith a transmit power constraint. Using the semi-definiterelaxation (SDR) technique, we reformulate the arising

problems into a convex semi-definite programming (SDP)that can be solved efficiently. Moreover, the proposed pro-cedure is applicable to the generalized RSM scheme whichallows having more than one receiving antenna at once andmulti-stream transmission. Our simulation results show theeffectiveness of proposed partial CSIT based preprocess-ing techniques in multi-user RSM systems in compari-son with full CSIT preprocessing and conventional MIMOtransmissions.

Before this work, there have been some efforts on two-stageprecoding in massive MIMO systems mainly due to two prac-tical issues and considerations [32]–[36]. The first issue is thecost of acquiring CSIT in FDD systems in terms of downlinktraining and CSIT uplink feedback signaling. The correspond-ing overhead is proportional to the number of BS antennas andthe number of total active users antennas, respectively. Differ-ent from the existing works on signal training and precodingdesign in massive MIMO systems [37]–[38], the research workin [32] aims to resolve the CSIT issue in FDD mode by ex-ploiting transmit correlation through clustering users and im-plementing two-stage precoding. At the first and second stagesinter-cluster and intra-cluster interference is mitigated, respec-tively. The second issue associated to massive MIMO arrays isthe huge number of required RF chains which can be reduced bytwo-stage precoding, known as hybrid precoding. This schemewhich is designed as a cascade of a high dimensional RF stagefollowed by a low dimensional baseband stage, is found suitablefor mmWave communications to reduce hardware implementa-tion cost [34]–[36].

In light of the above, it is necessary to devise preprocessingtechniques for FDD systems that are less dependent on instanta-neous CSIT. In contrast to [34], we apply the idea of two-stagepreprocessing for RSM scheme in a way that the multi-userinterference (MUI) and signal leakage to unintended antennasusers could be eliminated at the same time and each user can de-tect its transmitted spatial symbols from the BS correctly. Com-bination of SM with mmWave communications due to sparsesignal nature in spatial domain has been studied in a few re-cent papers [39]–[41], which offers low-complexity and highdata-rate realization. Deployment of SM scheme with hybridprecoding technique can lead to twofold benefits in reducinghardware implementation cost which provides possibilities forfuture research.

In the context of SM, [42]–[44] use partial CSIT to de-sign adaptive SM at transmit side, for point-to-point scenar-ios. Given that full knowledge of CSIT is available at the BS,linear preprocessing methods such as ZF based ones proposedin [19]–[20], [45] can be used for RSM transmission in mul-tiuser systems. However, instantaneous CSI usually changesrapidly which makes tracking it difficult and imposes a largesignaling overhead feedback on the system. Although the pro-posed two-stage preprocessing technique in [45] mitigates therequired resources for gathering CSIT, it is not possible to havefull knowledge of the effective channel of all users. Therefore,different from [19]–[20], [45] we use partial CSIT, which com-pared to [21] that considers imperfect CSIT imposed by chan-nel estimation error, assumes only the knowledge of channel

MALEKI et al.: RECEIVE SPATIAL MODULATION IN CORRELATED MASSIVE MIMO WITH PARTIAL CSI 1239

covariance information is available at the BS. This is a practicalassumption, because the covariance information is more sta-ble and stationary across several coherence times which makespreprocessing design more tractable [46]–[47].

The rest of the paper is organized as follows: In Section II,we introduce a system model for downlink multi-user MIMOcommunication. The two-stage preprocessing scheme is pro-posed in Section III, where the inner preprocessing matrix andthe design criterion for the outer preprocessing matrix are pre-sented. The optimization problems based on BER minimizationand DCMC capacity maximization criterion are formulated andstudied in Sections IV and V, respectively. The proposed prepro-cessing scheme is extended to generalized RSM transmission inSection VI. In Section VII, the computation complexity of theproposed preprocessing schemes compared with other methodswhile their performance is illustrated by simulation results inSection VIII. Finally, Section IX concludes the paper.

Notation: Throughout this paper, the small and capital boldletters are used to denote vectors and matrices, respectively. (·)T

and (·)H represent the transpose and the hermitian transpose,respectively. Tr(·) denotes the trace of a square matrix argu-ment. E{·} and ‖ · ‖ denote the statistical expectation and theEuclidean norm of the vector arguments.

II. SYSTEM MODEL

Consider a downlink multiuser RSM-MIMO communica-tions system where at the transmitter, the BS employs Nt anten-nas serving K users and each user possesses Nr antennas. It isassumed that total number of transmit antennas at BS exceedsthe total number of receive antennas at the users, i.e. Nt > KNr .

The channel of the ith user is characterized by the widelyused spatially correlated Rayleigh fading and formulated viathe Kronecker model [28] as

Hi =√

βiR1/2i HiT

H/2i , (1)

where Ri = R1/2i RH/2

i and Ti = T1/2i TH/2

i denote Nr × Nr

receive and Nt × Nt transmit spatial correlation matrices ofith user, respectively. βi > 0 characterizes the large-scale fad-ing parameter between the ith user and the BS which accountspath loss and is determined by their distance. Elements of Hi

are independent and identically distributed (i.i.d.) Gaussian ran-dom variables with zero mean and unit variance. By assuminga block fading channel, Hi independently varies over differ-ent coherence time slots while remaining constant during eachtime slot.

Based on the local scattering model [48], in the downlink oftypical cellular networks, user terminals are surrounded by acluster of local scatterers such that the angle of arrival (AOA)spans an interval of [−Δ,Δ] where Δ is known as the angularspread. For a user located at azimuth angel θ and being servedby a uniform linear array (ULA) at the BS, the correlationcoefficient between the (m,n) pair of BS antennas is given by

[T]m,n =1

2Δ

∫ Δ

−Δej2πD (m−n)sin(α+θ)dα, (2)

for 1 ≤ m,n ≤ Nt and where D is the normalized distancebetween the BS antennas by the wavelength.

Let Ti = UiΛiUHi represent the singular value decomposi-

tion (SVD) of Ti . Using Karhunen-Loeve representation, thechannel matrix between ith user and the BS can be expressed as

Hi =√

βiR1/2i HiΛi

1/2UiH , (3)

where Hi ∈ CNr ×ti denotes a matrix composed of i.i.d.Gaussian entries with zero mean and unit variance, Λi is theti × ti diagonal matrix whose elements are non-zero eigenval-ues of Ti and Ui

H is the unitary matrix of eigenvectors ti × Nt

corresponding to non-zero eigenvalues.The RSM scheme is applied at the BS such that in a given

channel use, one of Nr antennas of each user is selected asthe receiving antenna and the information is coded as the indexof this target receive antenna. The remaining information bitsare each mapped onto a symbol from the conventional M -aryPSK/QAM alphabet (hereafter denoted by M) and transmitted.Thus, the total number of bits transmitted by the proposed RSMscheme is given by log2(MNr ) for each user. The flat-fadingbaseband signal received by ith user after applying preprocess-ing in transmitter is given by

yi = HiWs + ni , (4)

where� s ∈ CK Nr ×1 is the transmitted signal vector for a total

of K users. Namely, s =[sT1 , . . . , sT

i , . . . , sTK

]T, and

si = smi ek

i is the transmit super-symbol for ith user. smi ∈

M for m ∈ {1, 2, . . . ,M}, is the APM symbol with unitpower, ek

i for k ∈ {1, . . . , Nr}, is the kth column of INr,

representing kth spatial symbol;� W = [W1 , . . . , Wi , . . . , WK ] is the Nt × KNr pre-

processing matrix at the BS, in which Wi ∈ CNt ×Nr isthe preprocessing matrix of ith user; and,

� ni is the Nr × 1 vector of i.i.d. CN (0, N0INr) additive

complex Gaussian noise.The received signal of ith user in (4) can be rewritten as

yi = HiWisi +K∑

j=1,j �=i

HiWjsj + ni , (5)

where the first term is the desired signal of ith user and thesecond term accounts for the received interference from the otherusers. The goal is to design the preprocessing matrix W suchthat simultaneous downlink transmissions to the associated usersand error-free detection of transmitted spatial modulated symbolfor each user could be achieved. The procedure of designingpreprocessing matrix will be presented in the following.

III. TWO-STAGE PREPROCESSING DESIGN FOR MULTI-USER

RSM-MIMO

As mentioned above, to take advantage of the RSM techniquewhile avoiding MUI in cellular networks, preprocessing at trans-mitter is done with the two-stage structure W = BP whereB ∈ CNt ×K Ns is the outer preprocessing matrix designed toeliminate MUI and P ∈ CK Ns ×K Nr is the inner preprocessing


matrix considered to improve performance of RSM and decreas-ing the probability of error detection for each user.

We assume that each user only has knowledge of the instan-taneous CSI of its own channel. This can be easily achieved foreach user terminal by using periodic bursts of known trainingsymbols. The only CSI available at the BS from each user is thedistribution of {Hi} and the spatial correlation matrices {Ri}and {Ti} (i ∈ {1, . . . , K}) which are more stable than instan-taneous CSI and can be stationary over a given time frame. Inthis way, the amount of feedback overhead needed for collectingpartial CSI from users decreases. Based on the partial CSI avail-able at the BS, the proposed two-stage preprocessing structureis elaborated below.

A. Beamforming Matrix Design Based on Spatial Correlation

Assume the outer preprocessing matrix B is partitioned intosub-matrices in the form [B1 , . . . , BK ], where Bi ∈ CNt ×Ns

is the beamforming matrix associated to ith user for MUI miti-gation. From (5), we can see that this is equivalent to a designof Bi such that the second term is removed, leading to theconstraint:

HiWj ≈ 0 for i �= j. (6)

Considering the structure Wi = BiPi , this leads to employ-ing the outer preprocessing matrix in order to suppress MUI,converting the BD constraint to HiBj ≈ 0 for i �= j.

Since the partial CSI is available at the BS, the BD process-ing (which is a ZF-based method and exploits spatial corre-lation) can satisfy the above constraint. The outer preprocess-ing matrix B is a function of the matrices {Ui} in such away that Span(Bi) ⊆ Span⊥({Uj : j �= i}). At first, for useri ∈ {1, . . . , K}, we define

Ui =[U∗

1 , . . . , U∗i−1 , U∗

i+1 , . . . , U∗K

]T. (7)

The BD finds the subspace most orthogonal to subspacespanned by other users’ channels. Let qi = rank

(Ui

) ≤∑K

j=1,j �=i

tj , the BD is possible when max {q1 , . . . , qK } <

Nt [29]. Performing a SVD of the matrix Ui yields

Ui = UiΛi

[V(1)

i , V(0)i

]H, (8)

where V(1)i is the right singular vectors associated with domi-

nant eigenvalues of Ui . The preprocessing matrix Bi is sup-

posed to lie in the null space of Ui that is Span(V(0)

i

).

Following the water filling algorithm in order to improve thespectral efficiency, by projecting channel matrix of ith user onto

Span(V(0)

i

)which is orthogonal to the dominant eigen-space

spanned by other users’ channel, covariance matrix of HiV(0)i

is given as

Σi = Tr(R)(V(0)

i

)H

UiΛiUHi V(0)

i . (9)

It should be noted that the necessary condition for users to

transmit without interfering for each other is rank(HiV

(0)i

)≥

1. This condition is satisfied when the channel matrices of usersare not highly correlated and therefore this limits the number ofusers that can be served by the BS [29].

By decomposing the covariance matrix Σi into its SVD asEiΦiEH

i , we get Ei = [E′i ,E

′′i ] where E′

i contains right sin-gular vectors corresponding to Ns dominant singular values ofprojected channel of ith user. Eventually, the outer preprocess-ing matrix can be constructed by concatenating V(0)

i and E′i as

follows:

B =[V(0)

1 E′1 , . . . , V(0)

K E′K

]

Nt ×K Ns

. (10)

Let ta = min {t1 , . . . , tK }. Given that the needed conditionsfor BD feasibility are satisfied, we should have Nr ≤ Ns ≤min(Nt −

∑Kj=1 tj + ta , ta

)where Ns is a design parameter

that can be chosen properly to support data transmission to eachmulti-antenna user. In the following, we assume Ns = Nr .

B. Precoding Matrix Design Procedure

We assume that the MUI is eliminated at the first stage.From (5), the effective channel matrix in this case will have

the diagonal structure as H = diag(H1 , . . . , HK

), where

Hi � HiBi for i ∈ {1, . . . , K}. Accordingly, the inner pre-processing matrix also has a block diagonal structure given byP = diag(P1 , . . . ,PK ). The received signal of the ith user canthus be rewritten as

yi = HiPisi + ni . (11)

Note that due to MUI cancellation, our multi-user MIMO systemis effectively turned into parallel single user subsystems. There-fore, designing inner preprocessing matrix P leads to designingPi for i ∈ {1, . . . , K} and is equivalent to solving Nr indepen-dent sub-problems. In the following, we focus on designing Pi

following the same procedure for i ∈ {1, . . . , K}.For a fixed outer preprocessing matrix B, the inner prepro-

cessing matrix P will be designed with the aim of detectingtransmitted APM symbol and index of active receive antennacorrectly for each user. When full CSI of users is available atthe BS, the inner preprocessing matrix P can be designed usingeffective channel matrix, in order to eliminate interference leak-age to unintended receive antennas which is done based on theZF method. Because the BS has only partial CSI of transmit andreceive spatial correlation matrices, we in the sequel design theinner preprocessing matrix based on such information as wellas by using the following two criteria:

1) Error probability of detection: At ith user, the jointmaximum-likelihood (ML) detector of both conventionalsymbol sm

i and the spatial symbol k is formulated as

(k, m) = arg mink,m

∥∥∥yi − HiPieki sm

i

∥∥∥2. (12)

Note that each user estimates its channel coefficients andonly requires to be fed back its pre-processing matrixWi from the BS which due to designing based on longterm CSI remains unchanged or needs to be designedslowly with time. By defining xiε = sm

i eki , where ε =


(m − 1)Nr + k denotes the number of super-symbols forε ∈ {1, 2, . . . , Nc = MNr}, the average BER is boundedby pair-wise error probability (PEP) as [49]

Pe ≤Nc∑

ε=1

Nc∑

τ =ε+1

N(ε, τ)Nc log2Nc

P (ε → τ), (13)

where N(ε, τ) is the number of bits in error betweensuper-symbols ε and τ and P (ε → τ) is PEP betweenthem. Due to the fact that the noise component is complexGaussian, we can rewrite (13) as

Pe ≤Nc∑

ε=1

Nc∑

τ =1ε �=τ

N(ε, τ)Nc log2Nc

Q

(√1

2N0

∥∥∥HiPiΔi,ετ

∥∥∥)

,

(14)

where Δi,ετΔ= xiε − xiτ .

2) DCMC capacity: In contrast to Shannon’s channel ca-pacity which is defined for continuous-input continuous-output memoryless channels (CCMC) and maximized forGaussian distribution, DCMC capacity is defined whenemploying multi-dimensional signaling set for MIMOchannels. Considering the RSM scheme as a mixture ofclassical constellation and spatial constellation, with bear-ing in mind the received signal of ith user in (11), theDCMC capacity may be expressed as [50]

Ci,DC M C = log2(Nc)

− 1Nc

Nc∑

τ =1

EH i ,n i

{

log2

Nc∑

ε=1

P (yi |xiε)P (yi |xiτ )

}

,

(15)

where the conditional probability of receiving yi , giventhat super-symbol xiτ was transmitted, is given as

P {yi |xiτ } =1

(πN0)Nr

exp

⎛

⎜⎝−∥∥∥yi − HiPixiτ

∥∥∥

2

N0

⎞

⎟⎠ .

(16)

In the following, we explore two optimization problemscorresponding to the above performance criteria of SM-MIMO systems, in order to design the inner preprocessingmatrix Pi for i ∈ {1, . . . , K} based on partial CSIT.

IV. PROPOSED BER MINIMIZATION BASED

PRECODING DESIGN

In this section, we formulate an optimization problem to adap-tively design the inner preprocessing matrix Pi so to minimizethe BER upper bound based on partial CSIT. One can observefrom (14) that the receiver BER performance is affected byEuclidean distance among codewords; namely, BER increaseswith a decreasing Euclidean distance among codewords. How-ever, in order to minimize the BER upper bound, dealing withthe Q-function is a difficult task. Thus, instead of minimiz-ing the BER upper bound, we focus on designing Pi through

maximizing the minimum average Euclidean distance betweenthe noise-free received signal of the ith user

maxP i

minε,τ =1,...,Nc

ε �=τ

EH i

{∥∥∥HiPiΔi,ετ

∥∥∥

2}

s.t. ‖BiPi‖2 ≤ Pt, (17)

where Pt is the allowable transmit power of BS for ith user.Because the instantaneous CSI is not present at the BS, the av-erage Euclidean distance among received codewords is chosenas the objective function. Thus, by considering the expectationover channel statistics, the objective function can be simplifiedas

EH i

{∥∥∥HiPiΔi,ετ

∥∥∥

2}

= βiTr(Ri)∥∥∥TH/2

i BiPiΔi,ετ

∥∥∥

2.

(18)

Next, by introducing an auxiliary variable d, problem (17)can be equivalently formulated as follows:

maxP i ,d

d

s.t. βiTr(Ri)∥∥∥TH/2

i BiPiΔi,ετ

∥∥∥

2≥ d,∀ε �= τ ;

‖BiPi‖2 ≤ Pt. (19)

It is not difficult to verify that the above problem is non-convexwith respect to Pi . To deal with this problem, we recast (19)into a suitable form that can be handled efficiently. Also, itshould be noted that the last constraint of (19) for preserving thetransmit power, has to be satisfied with equality at the optimum.Otherwise, the optimal value of Pi can be scaled up to reachthe upper bound Pt , which leads to a larger value on the left-hand side of first group of constraints and therefore the objectivefunction will attain a larger amount that contradicts optimality.By using the identity [51]

Tr (ABCD) =(vec(DT )

)T (CT ⊗ A)vec(B), (20)

we transform (19) into a quadratic problem. As a result, (19)can be reformulated as

maxp i ,d

d (21a)

s.t. pHi Qi,ετ pi ≥ d,∀ε �= τ ; (21b)

pHi Bipi = Pt, (21c)

where pi = vec(Pi), Bi = INr⊗ (BH

i Bi) and Qi,ετ =βiTr(Ri)

((Δ∗

i,ετ ΔTi,ετ

)⊗ (BHi TiBi

)). The constraint (21b)

is not concave due to positive semi-definiteness of Qi,ετ andtherefore the optimization problem (21) is non-convex. By


introducing XiΔ= pipH

i , we further reformulate (21) as

maxX i ,d

d

s.t. Tr (Qi,ετ Xi) ≥ d,∀ε �= τ ;

Tr(BiXi

)= Pt ;

Xi � 0, rank(Xi) = 1. (22)

Note that by using semi-definite relaxation (SDR), we can dropthe non-convex rank constraint and the resulting semi-definiteprogram (SDP) can be efficiently tackled using interior pointmethods [52].

Particularly, the optimal solution of the relaxed problem, de-noted by Xopt

i , will yield a lower bound to (22). If the rankof Xopt

i is one, the optimal solution of (22) can be obtainedby its eigenvalue decomposition and will be the correspond-ing principal eigenvector. Otherwise, if the rank of Xopt

i ishigher than one we need to approximate the rank one solu-tion of (22) from the relaxed problem. Based on random-ization techniques [53], we design a stochastic programmingto achieve a good approximate solution of (22). For this pur-pose, we generate a random vector with covariance matrix Xopt

i

as xc = (Xopti )1/2υ where υ ∼ CN (0, IN 2

r) and the solution

of (22) will be as√

αxc . Inspired by problem (22), the scalar αis chosen as Pt/Tr(BixcxH

c ) to satisfy the power constraint.The optimal solution of (21) is find on average with generatinga number of candidate random vectors and their correspondingscaling factor. Finally, the best candidate will be selected asthe one which leads to the largest value of minimum Euclideandistance defined in (18).

V. PROPOSED DCMC CAPACITY MAXIMIZATION BASED

PRECODING DESIGN

As described in Section III-B, DCMC capacity defined formulti-dimensional signaling sets may be considered as a crite-rion for designing precoding matrix Pi , such that the achiev-able rate can approach DCMC capacity. From (15) and (16), theDCMC capacity may be formulated as

Ci,DC M C = log2(Nc) − 1Nc

Nc∑

τ =1

EH i ,n i

{

log2

(Nc∑

ε=1

1

(πN0)Nr

exp

(

− 1N0

∥∥∥HiPiΔi,ετ + ni

∥∥∥

2))}

+1

Nc

Nc∑

τ =1

EH i ,n i

{

log2

(1

(πN0)Nr

exp

(

−‖ni‖2

N0

))}

. (23)

The above expression is rather complicated to be used in de-signing the inner preprocessing matrix. To proceed further, weneed a more tractable form for DCMC capacity. The followinglemma paves the way for achieving this goal.

Lemma 1: The DCMC capacity for the ith user in (23) islower bounded as

Ci,DC M C ≥ log2(Nc) − log2

(1

Nc

Nc∑

τ =1

Nc∑

ε=1

Nr∏

j=1

(

1+

βiλji,R

2N0

∥∥∥TH/2

i BiPiΔi,ετ

∥∥∥

2)−1)

+ Nr

(1 − 1

ln(2)

).(24)

Proof: See Appendix A. �In the sequel, in lieu of the DCMC capacity, we will design

the inner preprocessing matrix based on the lower bound foundin (24). Under a constraint on the allocated transmit power ofthe BS for ith user, the design problem can be expressed as

minP i

Nc∑

τ =1

Nc∑

ε=1

Nr∏

j=1

(

1 +βiλ

ji,R

2N0

∥∥∥TH/2i BiPiΔi,ετ

∥∥∥2)−1

s.t. ‖BiPi‖2 ≤ Pt, (25)

where the log(.) function of (24) is dropped due to monotonicity.Note that the above problem is non-convex and its optimummay not be easy to find. Therefore, we propose an approachto transform this problem into an equivalent form which leadsto a tractable problem. By introducing slack variable rk for

k ∈ {1, . . . , N 2c }, denoted in vector form as r =

[r1 , . . . , rN 2

c

],

we can rewrite (25) as

minP i ,r

N 2c∑

l=1

rl (26a)

s.t.Nr∏

j=1

(

1 +βiλ

ji,R

2N0

∥∥∥TH/2

i BiPiΔi,k

∥∥∥

2)−1

≤ rk ,∀k;

(26b)

‖BiPi‖2 ≤ Pt, (26c)

where for simplicity of notation, we have used k = ετ . Thefirst N 2

c constraints in (26b) have fractional form and this stillcauses the problem to remain non-convex. We develop an al-ternate way to solve this problem and try to transform (26b) toconvex constraints using second-order cones [54]. Therefore,we first introduce slack variables {bk} for k ∈ {1, . . . , N 2

c }in correspondence with the denominator of kth multiplicativefractional expression in (26b) which can be denoted by the

vector b ∈ RN 2c

+ . Then by introducing a vector of new vari-

ables y ∈ RN 2c ×Nr

+ , for k ∈ {1, . . . , N 2c } and j ∈ {1, . . . , Nr}

we form an augmented system to represent (26b) leading to:

minP i ,yb,r

N 2c∑

l=1

rl (27a)

s.t.1bk

≤ rk ,∀k; (27b)


bk ≤⎛

⎝Nr∏

j=1

ykj

⎞

⎠

1/Nr

,∀k; (27c)

1 ≤ ykj ≤ 1 +βiλ

ji,R

2N0

∥∥∥TH/2i BiPiΔi,k

∥∥∥2,∀j, k;

(27d)

‖BiPi‖2 ≤ Pt, (27e)

in which we have used the geometric mean inequality

Nr∏

j=1

(ykj )1/Nr ≤

Nr∏

j=1

ykj

to form a lower bound, since all multiplicative factors {ykj} aregreater than one. It should be noted that although (26b) is castas strict constraints (27b), (27c) and (27d), this step guaranteessatisfaction of (27c) and makes the problem tractable. The ge-ometric mean on the right-hand side (RHS) of constraint (27c)is concave and increasing, which can be expressed as a systemof second-order cone (SOC) constraints [55] — to be discussedshortly.

The constraint in (27d) is non-convex due to the normstructure on the RHS of the inequality. Thus, we recast itinto a suitable form to proceed further. Using the defini-tions in the previous section for (20), an equivalent transfor-

mation of (27d) is∥∥∥TH/2

i BiPiΔi,k

∥∥∥

2= pH

i Qi,kpi , where

Qi,k =(Δ∗

i,kΔTi,k

)⊗ (BH

i TiBi

). Similar to our approach in

the previous section, we can transform (27) into the following

problem by using the change of variable XiΔ= pipi

H :

minX i ,yb,r

N 2c∑

l=1

rl (28a)

s.t.1bk

≤ rk ,∀k; (28b)

bk ≤⎛

⎝Nr∏

j=1

ykj

⎞

⎠

1/Nr

,∀k; (28c)

1 ≤ ykj ≤ 1 +βiλ

ji,R

2N0Tr(XiQi,k

),∀j, k; (28d)

Tr(BiXi

)≤ Pt ; (28e)

Xi � 0, rank(Xi) = 1. (28f)

The rank constraint in (28f) is non-convex. Therefore, we re-sort to eliminating it by employing SDR. Now we look intothe constraint (28c) and reformulate it as a set of SOC con-straints, so that the required computation cost is reduced [54].Since ykj ≥ 1 for j ∈ {1, . . . , Nr} and k ∈ {1, . . . , Nc

2}, one

can modify (28c) by adding Nr Nc2

2 new variables cast in

w ∈ RN 2c ×Nr /2

+ , leading to the problem:

minX i ,yb,r,w

N 2c∑

l=1

rl (29a)

s.t.1bk

≤ rk ,∀k; (29b)

(bk )N r2 ≤

N r2∏

j=1

wkj ,∀k; (29c)

w2kj ≤ yk(2m )yk(2m−1) ,∀j, k,m; (29d)

1 ≤ ykj ≤ 1 +βiλ

ji,R

2N0Tr(XiQi,k

),∀j, k; (29e)

Tr

(BiXi

)≤ Pt ; (29f)

Xi � 0. (29g)

The new constraints in (29d) are convex and ad-mit SOC representation, i.e. yk(2m ) + yk(2m−1) ≥√

4w2kj + (yk(2m ) − yk(2m−1))

2 for k ∈ {1, . . . , N 2c } and

m ∈ {1, . . . , Nr

2 }. Note for Nr > 4, we have Nr

2 variables inthe product. Therefore, the process of adding new variablesand second order cone constraints is repeated log2(

Nr

2 )times. It can be shown that there exists w∗ such that w∗,y∗ and b∗ satisfy (29c) and (29d), if and only if, y∗ and b∗

satisfy (28c) [54]. This leads to a convex reformulation of theproblem which is a combination of semi-definite and SOCprograms and can be handled efficiently form a computationalviewpoint [52].

Let Xiop denote the optimum of the latter problem. The ob-

tained Xiop is optimal only when its rank is equal to one; then

the solution to the original problem will be its normalized prin-cipal eigenvector. If Xi

op has ranks higher than one, we can findan optimal rank-one candidate using the randomization proce-dure [53] detailed in the following. At first, we generate randomvectors as zc = (Xop

i )1/2ω with ω ∼ CN (0, IN 2r). The solution

of (28) will be constructed as√

αzc where the scalar α shouldsatisfy the constraints of (28), and hence will be the solution tothe following optimization problem:

minα,y

b,r,w

N 2c∑

l=1

rl

s.t. (29b) − (29d);

1 ≤ ykj ≤ 1 +βiαλj

i,R

2N0Tr(zczH

c Qi,k

),∀j, k;

αTr(BizczH

c

)≤ Pt ;

α ≥ 0. (30)

Finally, after generating a candidate set of random vec-tors with sufficient number of samples, one can pick the best


candidate, i.e. the candidate resulting in the minimum objectivevalue for problem (30).

VI. EXTENSION TO GENERALIZED RECEIVE

SPATIAL MODULATION

In this section, we generalize the proposed preprocessingscheme to the case where each user can have more than oneactive receive antenna. This configuration enables simultaneoustransmission of multiple streams and comprises the most gen-eral spatial symbol to convey additional information. In gen-eralized RSM transmission mode, Na antennas from amongthe Nr receive antennas for each user are selected which en-able the user to transmit Na parallel data streams and thereforeconvey Na log2(M) bits. Let C denote the set of legitimate ac-tivation patterns, the number of bits conveyed by spatial con-stellation will be Nb = �log2(

Nr

Na)� and therefore the total num-

ber of bits by generalized RSM for each is Nb + Na log2(M).If Ω = MNa × C is the set of all possible states of transmit-ted super-symbols of BS, with the assumption of transmittingsuper-symbol corresponding to mth combination of Na constel-lation points for m ∈ {1, . . . , MNa } and kth legitimate antennacombination for k ∈ {1, . . . , Nb}, the intended data vector forith user is given by smk

i =∑Na

j=1 smj

i eC(k,j )i , where C(k, j) and

smj

i denote the jth activated receive antenna of the kth legiti-mate antenna combination and jth symbol of mth combinationof constellation points, respectively. The received signal of ithuser for generalized RSM transmission is expressed as

yi = HiWi smki +

K∑

j=1,j �=i

HiWj snlj + ni . (31)

The preprocessing matrix W has a two-stage design structure(i.e. the same as Section III) where at the first stage we deal withthe MUI and then the outer preprocessing matrix B is designedusing BD based on partial CSIT knowledge of spatial receiveand transmit correlation matrices.

The purpose of designing the inner preprocessing matrix P isminimizing the interference leakage to unintended receive an-tennas and maximizing the probability of detecting transmittedsuper-symbol for each user. At the ith user side, the optimaldetector for joint detection of spatial constellation symbol andAPM symbols is a ML one which is expressed as

(k, m1 , . . . , mNa) = arg min

sm ki ∈Ω

∥∥∥yi − HiPi smki

∥∥∥2. (32)

This can be used for designing inner preprocessing matrix Pwith the aim of minimizing the average BER. By defining xiε �smki for ε ∈ {1, . . . , Nc

′ = NbMNa }, we can achieve an upper

bound as (14) by replacing xiε with xiε and consequently theoptimization problem in accordance with (17) is derived as

maxP i

minε,τ =1,...,Nc

′ε �=τ

EH i

{∥∥∥HiPiΔ′i,ετ

∥∥∥2}

s.t. ‖BiPi‖2 ≤ Pt, (33)

where equivalently Δ′i,ετ � xiε − xiτ . Another criterion for

designing P is maximizing DCMC capacity considered in (25)for RSM transmission which can be generalized and formulatedas

maxP i

Nc′

∑

τ =1

Nc′

∑

ε=1

Nr∏

j=1

(

1 +βiλ

ji,R

2N0

∥∥∥TH/2

i BiPiΔ′i,ετ

∥∥∥

2)−1

s.t. ‖BiPi‖2 ≤ Pt. (34)

It can be seen that the above generalized optimization problemsare the same as previously considered RSM problems and canbe solved with similar procedures proposed in previous sections.In fact, RSM is a special case of generalized RSM with Na = 1.

VII. COMPUTATIONAL COMPLEXITY

In this section, we compare the computational complexity ofthe proposed partial-CSIT based RSM schemes with the full-CSIT based ZF precoding scheme in [19] (denoted as 1-stageZF) and the two-stage precoding method in [45] (denoted as2-stage ZF). Also, we apply the ZF-like precoding schemesbased on Stochastic Robust Approximation (SRA) and Worst-Case Robust Approximation (WCRA) in [21] (denoted as SRAand WCRA, respectively) at the second stage of our proposedscheme which both take channel imperfections into account.Moreover, the MSE-based precoding design under per-antennapower constraint (PAPC) in [18] (denoted as MSE-PAPC) thatexploits full-CSIT, is investigated at the second stage of our pre-coding scheme to produce more insights. For fair comparison,

the scaling factor for ith user is considered as√

Pt

‖B i P i ‖2 , to meet

the total power constraint at the same time which leads to a non-convex optimization problem in MSE-PAPC precoding method.The solution can be found through the heuristic convex-concaveprocedure [56], which should be initialized with a feasible point,reducing to a SOC problem that is to be solved in each iteration.

The computational cost of one-stage ZF precoding schemebased on full-CSIT in [19] can be approximated asO(2(KNr )2Nt + (KNr )3). Furthermore, the computationalcomplexity of the first stage of the proposed preprocessingschemes and two-stage ZF method [45] is dominated by theSVD computation of the matrix Ui and is approximately equal

to O(Nt(∑K

i=1 ti)2) [57]. For the sake of comparison, the com-

putational cost of applying the different full-CSIT and partial-CSIT precoding techniques at the second stage of preprocessorare shown in Table I, where υ is the accuracy parameter [58].

The computational burdens of two-stage procedures and one-stage ZF similarly grow linearly with the number of BS anten-nas. From Table I, one can infer that the MSE-PAPC precodingdesign algorithm, which converges in a few ten iterations, hasthe worst case complexity cost, growing as a function of Nt

per iteration in addition to the imposed cost of finding the initialfeasible point. Among the remaining methods, the two proposedoptimized methods have a higher complexity. However, due tothe fact that each user is only equipped with limited number ofantennas the associated complexity cost is negligble in practice.Also, it should be noted that the proposed optimized methods in


TABLE ICOMPUTATIONAL COST OF VARIOUS RSM PRECODING

METHODS AT THE SECOND STAGE

this paper have a clear advantage in utilizing partial-CSIT whichreduces the overhead of needed feedback in FDD systems.

VIII. SIMULATION RESULTS

To evaluate the proposed multi-user preprocessing methodsfor RSM scheme we provide the numerical results. The perfor-mance of our proposed preprocessing techniques is consideredfor different channel parameters and system setups in compari-son with full-CSIT and imperfect-CSIT based ZF methods. Weconsider Rayleigh block fading channels with a coherence timelarger than the symbol transmission time and a ULA at theBS with antenna elements placed along the y-axis with spacing0.5λ. We set Pt = Nr

Nato restrict the transmit power of BS for

each user’s symbol to 1. The azimuth AoA of ith user is as-sumed θi = −π + Δ + (i − 1) 2π

K for i ∈ {1, . . . , K}. Unlessotherwise stated, we consider the angular spread of all usersΔ = 15◦ and there is no channel attenuation between users andthe BS, implying βi = 1 for i ∈ {1, . . . , K}. Without loss ofgenerality and for simplicity, we assume the receive correla-tion matrices at all users are identical (R1 = · · · = RK ) andtheir elements are generated according to [42] with ρ = 0.4.Since the transmit signal of BS for each user has unity power,SNR = 1/N0 is the transmitted SNR of BS for each user inall figures (except Fig. 5 where SNR varies). For the imperfect-CSIT ZF-like SRA/WCRA methods (comparison benchmarks),we set the variance of channel uncertainty for 0.1.

In Fig. 1, the BER performance of the first proposed prepro-cessing method based on BER minimization criterion is shownfor obtained simulation results and the analytical upper boundin (14) for different spatial constellation sizes. We can see thegood match between the simulation results and the theoreticalbound in mid values of SNR. Also, the success of ML detectorin detection of symbols with larger spatial dimension despite thedeclining of performance with enlarging the APM constellationsize.

Figs. 2 and 3 compare the BER and DCMC capacity perfor-mance of the proposed preprocessing techniques with ZF-basedpreprocessing methods, both in presence of and without thelarge-scale attenuation effect. Since the path loss effect is deter-mined by the distance between the BS and the user, we providethe results by averaging over different users locations. It can benoticed that the proposed two partial CSIT based criteria havevery close performance to each other, presumably due to thefact that the DCMC capacity implicitly relies on the capabil-ity of ML detector in correctly detecting the mixed continuous

Fig. 1. BER performance of the proposed preprocessing scheme based onthe BER minimization criterion and theoretical upper bound (dashed line) fordifferent spatial constellation sizes for different spatial constellation sizes. Weset Nt = 100, K = 4.

Fig. 2. BER performance of the proposed preprocessing schemes in com-parison with full-CSI and imperfect-CSIT ZF-based preprocessing schemes,without (solid line) and in presence of (dashed lines) the large-scale fadingeffect. We set Nt = 100, Nr = 4, K = 4, and use 4-QAM.

and discrete symbols of the RSM scheme [17]. Apart from one-stage ZF method which is based on full-CSIT (and performssignificantly better than other procedures), one can observe thatthe proposed partial-CSIT based preprocessing schemes signifi-cantly outperform the two-stage ZF and MSE-PAPC preprocess-ing procedures with the full knowledge of CSIT and imperfect-CSIT based ZF preprocessing schemes. This clearly shows thatthe considered optimization criteria can design codewords withgreater Euclidean distances and achieve better performance thanZF based method independent of instantaneous channel statis-tics. As a further consequence, it can be noticed from Fig. 3 thatin presence of large-scale channel attenuation, the performancereduction is about 2.5 dB for our optimized preprocessors andimperfect-CSIT ZF-based SRA/WCRA schemes for the same


Fig. 3. DCMC capacity of the proposed preprocessing schemes in comparisonwith full-CSIT and imperfect-CSIT ZF-based preprocessing schemes, without(solid line) and in presence of (dashed line) the large-scale fading effect. We setNt = 100, Nr = 4, K = 4, and use 4-QAM.

Fig. 4. DCMC capacity of proposed RSM scheme based on DCMC capacitymaximization criterion (solid lines) and the related lower bound obtained withrandomization (dashed lines) and without randomization (solid shapes). We setNt = 100 and K = 4.

DCMC capacity, while this is almost 4.5 dB for one-stage/two-stage ZF methods.

Fig. 4 plots the DCMC capacity of the DCMC maximizationproblem versus SNR for different spatial constellation sizes. Asexpected, by increasing the size of APM constellation and/orthe number of receive antennas for each user, the user can attainhigher achievable capacity. In Fig. 4, the lower bound of DCMCcapacity in (24) has been depicted for the solution of SDP-SOCproblem of (29) and the obtained approximated rank one solu-tion using randomization technique in (30). It can be observedthat the approximated lower bound using randomization tech-nique is close to the result obtained by SDR. Theoretically, theobtained solution using SDP and the approximated solution us-ing randomization approach are upper and lower bounds on the

Fig. 5. BER performance of the proposed RSM scheme based on the BERminimization procedure, compared with other ZF-based methods for differentnumber of users, transmit antennas and angular spreads. We set Nr = 4 anduse 4-QAM.

optimal DCMC capacity lower bound, respectively. This impliesthe loss of performance due to the randomization is negligible.

Fig. 5 investigates the efficiency of BD technique in elim-inating MUI and the resulting average BER performance ofproposed preprocessing methods versus the received SNR. Incase of Δ = 15◦, the required conditions to guarantee zero-interference for serving K = 4 users are satisfied. As the num-ber of antennas in the BS increases to Nt = 200, the interfer-ence elimination can be performed more efficiently with theBD method at the first stage of preprocessing scheme and theBER performance becomes better. By increasing the number ofusers to K = 6, the BER performance degrades because MUIincreases and the needed constraint for applying BD is vio-lated. For the angular spread of Δ = 10◦, the channels are lesscorrelated. Therefore, as it can be seen from Fig. 5, the BER per-formance improves and the MUI can be suppressed. We also seethat by increasing the number of users to six, the proposed opti-mized preprocessor has a clearly superior performance, despiterobust performance of one-stage full-CSIT based ZF preproces-sor under all cases. The proposed preprocessor exhibits betterperformance for increased number of BS antennas and less cor-related channels while the two-stage ZF and imperfect-CSIT ZFbased methods degrade significantly by increasing the numberof served users. More specifically, at high SNR regimes ouroptimized procedure performs better while imperfect-CSIT andtwo-stage ZF based methods experience bit error floor.

A typical factor that influences the system functionality isthe correlation between receive antennas of each user. As it canbe seen from Figs. 6 and 7, the preprocessing schemes obtainbetter performance in the lower correlation case than highly cor-related receive antennas. For example, in case of ρ = 0.4, thereis a gain of about 2 dB over ρ = 0.8 for BER minimizationprocedure and imperfect-CSIT based SRA and WCRA prepro-cessors, to achieve BER performance of about 10−3 while thisis about 4.5 dB for perfect-CSIT based one-stage and two-stage


Fig. 6. BER performance of the proposed RSM scheme with the BERminimization criterion for different receive correlation coefficients ρ. We setNt = 100, Nr = 4, K = 4 and use 4-QAM.

Fig. 7. DCMC capacity of the proposed RSM scheme with the DCMC capac-ity maximization criterion for different receive correlation coefficients ρ. Weset Nt = 100, Nr = 4, K = 4 and use 4-QAM.

ZF preprocessors. For low SNR values in Fig. 7, decreasing thecorrelation coefficient at a constant transmission rate providesa performance improvement of about 1 dB for DCMC capacitymaximization, SRA/WCRA ZF and MSE-based preprocessingtechniques, compared with 4.5 dB gain for one-stage/two-stageZF preprocessors. Based on these results, compared to the full-CSIT one-stage ZF method, other preprocessing techniques aremore robust to correlation among receive antennas.

In Figs. 8 and 9, the BER and DCMC capacity performance ofthe proposed preprocessing procedures are shown for the gener-alized RSM and conventional MIMO scheme (corresponding tothe case Na = Nr ). In Fig. 8, the simulation results follow theunion upper bound (note that all upper bounds are not depicted toavoid a crowded figure, but they follow similar trends). For a faircomparison, we consider cases with the same transmission rate.It is observed that using BPSK modulation exhibits better BERperformance than 4-QAM modulation and at low and mid range

Fig. 8. BER performance of our generalized RSM scheme with the BERminimization criterion and the corresponding theoretical upper bound (dashedlines), in comparison with conventional MIMO. We set Nt = 100 and K = 4.

Fig. 9. DCMC capacity of our generalized RSM scheme with the DCMCcapacity maximization criterion in comparison with conventional MIMO. Weset Nt = 100 and K = 4.

SNR. For the proposed generalized RSM scheme, by increasingthe number of intended receive antennas, more transmit poweris needed to overcome inter-antenna interference and attain acertain BER, while the higher DCMC capacity can be achievedat a constant SNR. Inspecting Fig. 9 reveals that in differentconfigurations the same or even better and worse throughputthan conventional MIMO benchmark can be obtained beforereaching maximum value of the throughput. It can be observedthat both cases of (Nr ,Na,M) ∈ {(4, 2, 4), (5, 3, 2)} providenearly 2 dB performance improvement compared with conven-tional MIMO case of Nr = 3 and 4-QAM modulation, at lowSNR regimes. On the other hand, for Nr = 5 and BPSK, con-ventional MIMO achieves the same DCMC capacity as gen-eralized RSM with Na = 2 intended receive antennas, whichis more reasonable and efficient. At the same time, it offers


better throughput performance than generalized RSM scenarios(Nr ,Na,M) ∈ {(4, 3, 2), (3, 2, 4)} at low SNR regions. More-over, the generalized RSM scheme with Na = 2 intended re-ceiving antennas out of Nr = 4 available antennas and BPSKmodulation can attain significantly better performance than con-ventional MIMO under the setting of Nr = 2 and 4-QAM mod-ulation while providing the same DCMC capacity at high SNRregimes.

IX. CONCLUSION

Optimal preprocessing design for RSM in the downlink ofmulti-user massive-MIMO systems was investigated. Due to thehigh cost of CSIT feedback in FDD massive MIMO systems,the practical assumption was made that only the partial CSI oftransmit and receive correlation matrices of spatially correlatedusers’ channels is available at the BS. We proposed a two-stage preprocessing scheme partitioned into outer preprocessorand inner preprocessor. The outer preprocessor was designedto suppress MUI based on the BD method using partial CSIT.The design of inner preprocessor was formulated as two opti-mization problems with the aim to eliminate the transmit powerleakage to undesired receive antennas. The considered problemswere reformulated via their convex alternatives as a mixture ofsemi-definite and second-order cone programs. Furthermore, itwas shown that the proposed preprocessing procedures are alsoapplicable to the generalized RSM scheme. The simulation re-sults demonstrated improving the performance of BD methodin MUI suppression with increasing the number of BS anten-nas, especially in high correlated environments. Also, it wasshow that the two optimal partial-CSIT based RSM schemesoffer an enhanced performance in comparison with full-CSITand imperfect-CSIT ZF-based RSM transmission methods andeven conventional MIMO under some specified conditions.

APPENDIX ADERIVATION OF (24)

By applying Jensen’s inequality to the second term of RHSof (23), denoted as ci , and based on the chain rule for computingexpected value of random variables, the following upper boundcan be established:

ci ≤ 1Nc

Nc∑

τ =1

log2

(

EH i

{

En i |H i

{Nc∑

ε=1

1

(πN0)Nr

exp(− 1

N0


∥∥∥2)}})

. (35)

Moreover, it can be shown that the conditional expected valueof the second term of this upper bound, is evaluated for givenHi as

En i |H i

{Nc∑

ε=1

1

(πN0)Nr

exp(− 1

N0


∥∥∥2)}

=Nc∑

ε=1

1

(2πN0)Nr

exp(− 1

2N0


∥∥∥2)

. (36)

Let Ri = Ui,RΛi,RUHi,R represent the SVD of Ri , and define

δi,ετΔ=√

βiHiTH/2i BiPiΔi,ετ . Since Ui,R is a unitary ma-

trix, we can write


∥∥∥2

=Nr∑

j=1

βiλji,R

∣∣∣δji,ετ

∣∣∣2, (37)

where λji,R is the jth diagonal element of Λi,R and δj

i,ετ is thejth element of vector δi,ετ which consists of i.i.d. zero-meanGaussian random elements with the following variance

var{

δji,ετ

}=∥∥∥√

βiTH/2i BiPiΔ

ji,ετ

∥∥∥2, (38)

with Δji,ετ being the jth element of δi,ετ . Note that δj

i,ετ isa random variable with exponential distribution and parameter

1/var{

δji,ετ

}. Therefore, the expected value of RHS of (36)

with respect to Hi can be simplified as

EH i

{Nc∑

ε=1

1

(2πN0)Nr

exp(− 1

2N0


∥∥∥2)}

=Nc∑

ε=1

1

(2πN0)Nr

Nr∏

j=1

(

1 +βiλ

ji,R

2N0

∥∥∥TH/2i BiPiΔ

ji,ετ

∥∥∥2)−1

.

(39)

Now we substitute the RHS of (39) into (35) and make use ofthe fact that logarithm is a concave function. As a result, usingJensen’s inequality for the outer summation we have that

ci ≤ 1Ns

Ns∑

τ =1

log2

(1

(2πN0)Nr

Nc∑

ε=1

Nr∏

j=1

(

1 +λj

i,R

2N0

∥∥∥TH/2

i BiPiΔji,ετ

∥∥∥

2)−1)

≤ log2

(1

(2πN0)Nr Nc

Nc∑

τ =1

Nc∑

ε=1

Nr∏

j=1

(

1 +βiλ

ji,R

2N0

∥∥∥TH/2

i BiPiΔji,ετ

∥∥∥

2)−1)

. (40)

On the other hand, the third term on the RHS of (23) can beobtained as

1Nc

Nc∑

τ =1

En i ,H i

{

log2

(1

(πN0)Nr

exp

(

−‖ni‖2

N0

))}

= log2

(1

(πN0)Nr

)

+Nr

ln(2). (41)

Considering the upper bound of ci in (35), the lower boundof the DCMC capacity in (23) is derived in a straightforwardmanner by substituting (41).

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Marjan Maleki received the M.Sc. degree from IranUniversity of Technology, Tehran, Iran, in 2013, theB.Sc. degree in electrical engineering from IsfahanUniversity of Technology, Isfahan, Iran, in 2011. Sheis currently working toward the Ph.D. degree withthe Department of Electrical and Computer Engineer-ing, K. N. Toosi University of Technology, Tehran,Iran. From May 2017 to December 2017, she wasa Visiting Researcher with the Electrical and Com-puter Engineering Department, University of Illinoisat Chicago, Chicago, IL, USA. Her current area of

research includes signal processing, spatial modulation, massive MIMO, multi-user communications, and cooperative communications. She has served as aReviewer of the IEEE TRANSACTIONS ON COMMUNICATIONS, IEEE TRANS-ACTIONS ON VEHICULAR TECHNOLOGY, IEEE TRANSACTIONS ON WIRELESS

COMMUNICATIONS, and as a Guest Reviewer of IEEE conferences.

Kamal Mohamed-Pour was born in 1954. He re-ceived the B.Sc. degree from the Telecommunica-tion College, Tehran, Iran, in 1979, and the M.Sc.degree from the University of Tarbiat Modares,Tehran, Iran, in 1987, both in electrical engineering-telecommunication. He received the Ph.D. degreefrom the University of Manchester, Manchester,U.K., in 1995. He was a Lecturer with the ElectricalEngineering Department, K. N. Toosi University ofTechnology, Tehran, Iran. He is currently a Full Pro-fessor with the Department of Electrical and Com-

puter Engineering, K. N. Toosi University of Technology, and has authoredand coauthored several books in digital signal processing and digital commu-nications. His main research interests are broadband wireless communications,MIMO-OFDM, and communication networks.

Mojtaba Soltanalian (S’08–M’14) received thePh.D. degree in electrical engineering (with spe-cialization in signal processing) from the Depart-ment of Information Technology, Uppsala University,Uppsala, Sweden, in 2014.

He is currently with the Faculty of the Electricaland Computer Engineering Department, Universityof Illinois at Chicago (UIC), Chicago, IL, USA. Be-fore joining UIC, he held Research Positions withthe Interdisciplinary Centre for Security, Reliabil-ity and Trust (SnT, University of Luxembourg), and

California Institute of Technology (Caltech). His research interests lie in the in-terplay of signal processing, learning, and optimization theory, and specificallydifferent ways the optimization theory can facilitate a better processing anddesign of signals for collecting information, communication, as well as to forma more profound understanding of data, whether it is in everyday applicationsor in large-scale, complex scenarios.

Dr. Soltanalian serves as a member of the editorial board of Signal Process-ing, and as the Vice Chair of the IEEE Signal Processing Society Chapter inChicago. He has been a recipient of the 2017 IEEE Signal Processing SocietyYoung Author Best Paper Award, as well as the 2018 European Signal Process-ing Association (EURASIP) Best Ph.D. Award.

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