IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 3, NO. 6, DECEMBER 2014 653

Low-Complexity Hybrid Precoding in Massive Multiuser MIMO SystemsLe Liang, Student Member, IEEE, Wei Xu, Member, IEEE, and Xiaodai Dong, Senior Member, IEEE

AbstractMassive multiple-input multiple-output (MIMO) isenvisioned to offer considerable capacity improvement, but atthe cost of high complexity of the hardware. In this paper, wepropose a low-complexity hybrid precoding scheme to approachthe performance of the traditional baseband zero-forcing (ZF)precoding (referred to as full-complexity ZF), which is considereda virtually optimal linear precoding scheme in massive MIMO sys-tems. The proposed hybrid precoding scheme, named phased-ZF(PZF), essentially applies phase-only control at the RF domain andthen performs a low-dimensional baseband ZF precoding basedon the effective channel seen from baseband. Heavily quantizedRF phase control up to 2 bits of precision is also considered andshown to incur very limited degradation. The proposed schemeis simulated in both ideal Rayleigh fading channels and sparselyscattered millimeter wave (mmWave) channels, both achievinghighly desirable performance.

Index TermsMassive MIMO, hybrid precoding, millimeterwave (mmWave) MIMO, RF chain limitations.

I. INTRODUCTION

MASSIVE multiple-input multiple-output (MIMO) isknown to achieve high capacity performance with sim-plified transmit precoding/receive combining design [1][3].Most notably, simple linear precoding schemes, such as zero-forcing (ZF), are virtually optimal and comparable to nonlinearprecoding like the capacity-achieving dirty paper coding (DPC)in massive MIMO systems [2]. However, to exploit multipleantennas, the convention is to modify the amplitudes and phasesof the complex symbols at the baseband and then upcovert theprocessed signal to around the carrier frequency after passingthrough digital-to-analog (D/A) converters, mixers, and poweramplifiers (often referred to as the radio frequency (RF) chain).Outputs of the RF chain are then coupled with the antennaelements. In other words, each antenna element needs to besupported by a dedicated RF chain. This is in fact too expensiveto be implemented in massive MIMO systems due to the largenumber of antenna elements.

On the other hand, cost-effective variable phase shifters arereadily available with current circuitry technology, making itpossible to apply high dimensional phase-only RF or analogprocessing [4][7]. Phase-only precoding is considered in [4],

Manuscript received August 30, 2014; accepted October 9, 2014. Date ofpublication October 17, 2014; date of current version December 17, 2014.This work was supported in part by the Natural Sciences and EngineeringResearch Council of Canada under Grant 261524 and by the NSFC underGrant 61471114 and by the 973 Program under 2013CB329204. The associateeditor coordinating the review of this paper and approving it for publication wasV. Raghavan.

L. Liang and X. Dong are with the Department of Electrical and ComputerEngineering, University of Victoria, Victoria, BC V8W 3P6, Canada (e-mail:liang@uvic.ca; xdong@ece.uvic.ca).

W. Xu is with the National Mobile Communications Research Laboratory,Southeast University, Nanjing 210096, China (e-mail: wxu@seu.edu.cn).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/LWC.2014.2363831

Fig. 1. System model of the hybrid mmWave precoding structure.

[5] to achieve full diversity order and near-optimal beam-forming performance through iterative algorithms. The limitedbaseband processing power can further be exploited to performmulti-stream signal processing as in [6], where both diversityand multiplexing transmissions of MIMO communications areaddressed with less RF chains than antennas. Reference [7]then takes into account more practical constraints such as onlyquantized phase control and finite-precision analog-to-digital(A/D) conversion. Works in [4][7], however, do not considerthe multiuser scenario and are not aimed to maximize thecapacity performance in the large array regime.

In this paper, we consider the practical constraints of RFchains and propose to design the RF precoder by extract-ing the phases of the conjugate transpose of the aggregatedownlink channel to harvest the large array gain in massiveMIMO systems, inspired by [6]. Low-dimensional baseband ZFprecoding is then performed based on the equivalent channelobtained from the product of the RF precoder and the actualchannel matrix. This hybrid precoding scheme, termed PZF, isshown to approach the performance of the virtually optimal yetpractically infeasible full-complexity ZF precoding in a mas-sive multiuser MIMO scenario. Furthermore, hybrid basebandand RF precoding has been considered for millimeter wave(mmWave) communications in works [8][10]. They share theidea of capturing dominant paths of mmWave channels usingRF phase control and the RF processing is constrained, more orless, to choose from array response vectors. We will also showin the simulation the desirable performance of our proposedPZF scheme in mmWave channels

II. SYSTEM MODEL

We consider the downlink communication of a massivemultiuser MIMO system as shown in Fig. 1, where the basestation (BS) is equipped with Nt transmit antennas, but drivenby a far smaller number of RF chains, namely, K. Thischain limitation restricts the maximum number of transmittedstreams to be K and we assume scheduling exactly K single-antenna users, each supporting single-stream transmission. Asdiscussed, the downlink precoding is divided among basebandand RF processing, denoted by W of dimension K K andF of dimension Nt K, respectively. Notably, both amplitudeand phase modifications are feasible for the baseband precoderW, but only phase changes can be made to the RF precoder F

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654 IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 3, NO. 6, DECEMBER 2014

with variable phase shifters and combiners [6]. Thus each entryof F is normalized to satisfy |Fi,j | = 1Nt where |Fi,j | denotesthe magnitude of the (i, j) th element of F.

We adopt a narrowband flat fading channel and obtain thesampled baseband signal received at the kth user

yk = hHk FWs+ nk (1)

where hHk is the downlink channel from the BS to the k thuser, and s CK1 denotes the signal vector for a total ofK users, satisfying E[ssH ] = PK IK where P is the transmitpower at the BS and E[] is the expectation operator. To meetthe total transmit power constraint, we further normalize W tosatisfy FW2F = K. nk denotes the additive noise, assumedto be circular symmetric Gaussian with unit variance, i.e., nk CN (0, 1). Then the received signal-to-interference-plus-noise-ratio (SINR) at the kth user is given by

SINRk =PK |hHk Fwk|

2

1 +

j =kPK |hHk Fwj |

2 (2)

where wj denotes the jth column of W. If Gaussian inputsare used, the system can achieve a long-term average (over thefading distribution) spectral efficiency

R =

Kk=1E [log2(1 + SINRk)] . (3)

III. HYBRID PRECODING IN MASSIVE MIMO SYSTEMS

In massive MIMO systems, ZF precoding is known as aprominent linear precoding scheme to achieve virtually optimalcapacity performance due to the asymptotic orthogonality ofuser channels in richly scattering environment [2]. It is typi-cally realized through baseband processing, requiring Nt RFchains performing RF-baseband frequency translation and A/Dconversion. This tremendous hardware requirement, however,restricts the array size from scaling large.

To alleviate the hardware constraints while realizing fullpotentials of massive multiuser MIMO systems, we propose toapply phase-only control to couple the K RF chain outputs withNt transmit antennas, using cost-effective RF phase shifters.Low-dimensional multi-stream processing is then performed atthe baseband to manage inter-user interference. The proposedlow-complexity hybrid precoding scheme, termed phased-ZF(PZF), can approach the performance of the full-complexity ZFprecoding, which is, as stated, practically infeasible due to therequirement of supporting each antenna with a dedicated RFchain. The spectral efficiency achieved by the proposed PZFscheme is then analyzed.

A. Hybrid Precoder Design

The structure shown in Fig. 1 is exploited to perform theproposed hybrid baseband and RF joint processing, where thebaseband precoder W modifies both the amplitudes and phasesof incoming complex symbols and the RF precoder F controlsphases of the upconverted RF signal. We propose to performphase-only control at the RF domain by extracting phases ofthe conjugate transpose of the aggregate downlink channel fromthe BS to multiple users. This is to align the phases of channelelements and can thus harvest the large array gain provided by

the excessive antennas in massive MIMO systems. To clarify,denote Fi,j as the (i, j)th element of F and we perform the RFprecoding according to

Fi,j =1Nt

eji,j (4)

where i,j is the phase of the (i, j)th element of the con-jugate transpose of the composite downlink channel, i.e.,[h1, ,hK ]. Here we implicitly assume perfect channelknowledge at the BS which can potentially be obtained, e.g.,through uplink channel estimation combined with channel reci-procity in time division duplex (TDD) systems [1]. We notethat efficient channel estimation techniques leveraging hybridstructures and rigorous treatment of frequency selectivity arean ongoing research topic of great practical interest.

Then at the baseband, we observe an equivalent chan-nel Heq = HF of a low dimension K K where H =[h1, ,hK ]H is the composite downlink channel. Hencemulti-stream baseband precoding can be applied to Heq , wheresimple low-dimensional ZF precoding is performed as

W = HHeq(HeqH

Heq

)1 (5)

where is a diagonal matrix, introduced for column powernormalization. With this PZF scheme, to support simultaneoustransmission of K streams, hardware complexity is substan-tially reduced, where only K RF chains are needed, as com-pared to Nt required by the full-complexity ZF precoding.

Quantized RF Phase Control: According to (4), each entryof the RF precoder F differs only in phases which assumecontinuous values. However, in practical implementation, thephase of each entry tends to be heavily quantized due to prac-tical constraints of variable phase shifters. Therefore, we needto investigate the performance of our proposed PZF precodingscheme in this realistic scenario, i.e., phases of the KNt entriesof F are quantized up to B bits of precision, each quantizedto its nearest neighbor based on closest Euclidean distance.The phase of each entry of F can thus be written as =(2n)/(2B) where n is chosen according to

n = arg minn{0,,2B1}

2n2B (6)

where is the unquantized phase obtained from (4). Then thebaseband precoder is computed by (5) with the quantized F.

B. Spectral Efficiency Analysis in Rayleigh Fading ChannelsIn this part, we analyze the spectral efficiency achieved by

our proposed PZF and full-complexity ZF precoding in thelimit of large transmit antenna size Nt assuming Rayleighfading. Closed-form expressions are derived, revealing the rolesdifferent parameters play in affecting system capacity.

Denoting the kth column of F by fk, we obtain

yk =[hHk f1, ,hHk fk, ,hHk fK

]Ws+ nk (7)

based on (1). As described in Section III, fk is designed byextracting the phases of hk, we thus have the diagonal term

hHk fk =1Nt

Nti=1

|hi,k| (8)

LIANG et al.: LOW-COMPLEXITY HYBRID PRECODING IN MASSIVE MULTIUSER MIMO SYSTEMS 655

where hi,k denotes the ith element of the vector hk. Underthe assumption that each element of hk is independent andidentically distributed (i.i.d.) complex Gaussian random vari-able with unit variance and zero mean, i.e., h CN (0, 1), weconclude that |h| follows Rayleigh distribution with mean

2

and variance 1 4 . When Nt tends to infinity, the central limittheorem indicates

hHk fk N(

Nt2

, 1 4

). (9)

For the off-diagonal term, i.e., j = k, we have hHk fj =1Nt

Nti=1

hi,keji,j

, where () gives the complex conjugation.Its distribution is characterized in the lemma below.

Lemma 1: In Rayleigh fading channels, the off-diagonalterm hHk fj is distributed according to hHk fj CN (0, 1).

Proof: The proof is achieved by analyzing the real andimaginary parts of hHk fj separately, followed by proving theirindependence. The proof is straightforward by definitions, andhence details are left out due to space limit.

Based on Lemma 1, we derive that the magnitude of theoff-diagonal term, i.e., |hHk fj | follows the Rayleigh distri-bution with mean

2 and variance 1 4 . Compared with

the diagonal term hHk fk given by (9), it is safe to say theoff-diagonal terms are negligible when the transmit antennanumber Nt is fairly large. This implies that the inter-userinterference is essentially negligible even without basebandprocessing at large Nt! However, we note when Nt assumessome medium high values, the residual interference may stilldeteriorate the system performance. Therefore we apply in ourproposed scheme ZF processing at the baseband to suppress itas in (5).

We reason that even with ZF processing at the baseband, thespectral efficiency achieved is still less than it would be if theoff-diagonal terms hHk fjs were precisely zero. In other words,the spectral efficiency achieved by PZF is upper bounded byKR with R = E[log2(1 + PK |hHk fk|

2)], which can be charac-

terized by the following theorem using the limit equivalencetype of argument [11].

Theorem: The spectral efficiency achieved by the pro-posed low-complexity PZF precoding scheme is tightly upperbounded by RPZF KR where

limNt

Rlog2

(1 + 4

PNtK

) = 1. (10)Proof: The per-user upper bound is derived as

R=E[log2

(1 +

P

K

(y +

Nt2

)2)]

= log2

(1+

4

PNtK

)+E

log2

1 + PK

(y+

Nt2

)21 + Nt4

PK

where y N (0, 2) with =1 4 . One may provelim

Nt = 0 by showing is both upper and lower bounded

by zero in the limit. An upper bound can be directly proved by

Fig. 2. Spectral efficiency achieved by different precoding schemes in largemultiuser MIMO systems with i.i.d. Rayleigh fading channels whereNt = 128and K = 4, obtained from averaging 1000 channel realizations.

applying the Jensens Inequality. Proof of the lower bound isinvolved. Briefly, by defining = PK and a

=

Nt2 , we have

limNt

limaE

[log2

(1 +

y

a

)2]+ lim

a log2a2

1 + a

2

= lima

2a log2 e2

+0

(lnx)ea2(x1)2

22

(1 + e

2a2x

2

)dx

(a)

lima

2a log2 e2

(1 + e

2a2

2

) 10

(lnx)ea2(x1)2

22 dx

= lima

2aea2

222 ln 2

(1 + e

2a2

2

) 10

(lnx)ea2x

22 dx

(b)

lima

2aea2

222 ln 2

(1 + e

2a2

2

) 22(1 e a222)2

= 0

where (a) holds by shortening the integral range and thenapplying the Mean Value Theorem for Integral with (0, 1).(b) is valid by using 10 (lnx)emxdx em+1m for m 0.

Remark 1: Considering that the off-diagonal terms hHk fjsare essentially negligible when Nt is large, we expect thederived closed-form upper bound to be very tight in the largeantenna regime. This is further verified in the simulation resultsas shown in Fig. 2. Thus the closed-form upper bound serves asa good approximation of the spectral efficiency achieved by theproposed PZF precoding scheme at large Nt.

The full-complexity ZF precoding vector (with unit norm)for the kth stream follows by projecting hk onto the nullspaceof Hk = [h1, ,hk1,hk+1, ,hK ]H . In the spectral effi-ciency analysis, we exploit the property that users channels areasymptotically orthogonal in massive multiuser MIMO systems[1]. It indicates full-complexity ZF precoding converges to con-jugate beamforming with inter-user interference forced to zero,achieving SINRk PK |hk|2, as Nt . Then according to(3), we obtain the spectral efficiency of full-complexity ZFprecoding in the limit of large Nt as [12]

RFCZF KE[log2

(1 +

P

K|hk|2

)]

=KeKP log2 e

Ntn=1

En

(K

P

)(11)

656 IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 3, NO. 6, DECEMBER 2014

by acknowledging that |hk|2 follows chi-squared distributionwith 2Nt degrees of freedom and En(x) is the exponentialintegral of order n.

IV. SIMULATION RESULTS

A. Large Rayleigh Fading Channels

We numerically compare our proposed PZF precodingscheme in Fig. 2 along with its quantized version against thefull-complexity ZF scheme, which is deemed virtually optimalin the large array regime but practically infeasible due to therequirement of Nt costly RF chains. It is observed that theproposed PZF precoding performs measurably close to the full-complexity ZF precoding, with less than 1 dB loss but substan-tially reduced complexity. As for the heavily quantized phasecontrol, we find that with B = 2 bits of precision, i.e., phasecontrol candidates of {0,2 , }, the proposed scheme suffersnegligible degradation, say less than 1 dB.

The derived analytical spectral efficiency expressions (10)and (11) are also plotted in Fig. 2. We observe that the derivedclosed-form expressions are quite accurate in characterizingspectral efficiencies achieved by the proposed PZF precod-ing and full-complexity ZF precoding schemes throughoutthe whole signal-to-noise (SNR)1 range, thus providing usefulguidelines in practical system designs.

B. Large mmWave Multiuser Channels

Apart from ideal i.i.d. Rayleigh fading channels, our pro-posed PZF scheme can also be applied to the mmWave com-munication which is known to have very limited multipathcomponents. To capture this poor scattering nature, in thesimulation, we adopt a geometric channel model [8][10]

hHk =

NtNp

Npl=1

kl aH(kl ,

kl

) (12)where each user is assumed to observe the same number ofpropagation paths, denoted by Np, the strength associated withthe lth path seen by the kth user is represented by kl (assumingkl CN (0, 1)), and kl (kl ) is the random azimuth (eleva-tion) angle of departure drawn independently from uniformdistributions over [0, 2]. a(kl , kl ) is the array response vectordepending only on array structures. Here we consider a uniformlinear array (ULA) whose array response vector admits a simpleexpression, given by [9, eq. (6)] where d is the normalizedantenna spacing.

We compare in Fig. 3 our proposed PZF scheme againstthe beamspace MIMO (B-MIMO) scheme proposed in [10],which essentially steers streams onto the approximate strongestpaths (using DFT matrix columns) at the RF domain andperforms low-dimensional baseband ZF precoding based on theequivalent channel. For fair comparison, the BS is also assumedto have a total of K chains. The B-MIMO scheme achievesdesirable performance in line-of-sight (LoS) channel but failsto capture sparse multipath components in non-LoS channels.

1Here SNR = P is the common average SNR received at each antenna withnoise variance normalized to unity.

Fig. 3. Spectral efficiency achieved by different precoding schemes in largemmWave multiuser systems with Nt = 128, K = 4, d = 12 , and Np = 10,obtained from averaging 1000 channel realizations.

V. CONCLUSION

In this paper, we have studied a large multiuser MIMOsystem under practical RF hardware constraints. We have pro-posed to approach the desirable yet infeasible full-complexityZF precoding with low-complexity hybrid PZF scheme. TheRF processing was designed to harvest the large power gainwith reasonable complexity, and the baseband precoder wasthen introduced to facilitate multi-stream processing. Its per-formance has been characterized in a closed form and furtherdemonstrated in both Rayleigh fading and poorly scatteredmmWave channels through computer simulations.

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