Energy-Efﬁcient Hybrid Analog and Digital Precoding for ...

998 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 34, NO. 4, APRIL 2016

Energy-Efficient Hybrid Analog and DigitalPrecoding for MmWave MIMO Systems

With Large Antenna ArraysXinyu Gao, Student Member, IEEE, Linglong Dai, Senior Member, IEEE, Shuangfeng Han, Member, IEEE,

Chih-Lin I, Senior Member, IEEE, and Robert W. Heath Jr., Fellow, IEEE

Abstract—Millimeter wave (mmWave) MIMO will likely usehybrid analog and digital precoding, which uses a small num-ber of RF chains to reduce the energy consumption associatedwith mixed signal components like analog-to-digital componentsnot to mention baseband processing complexity. However, mosthybrid precoding techniques consider a fully connected architec-ture requiring a large number of phase shifters, which is alsoenergy-intensive. In this paper, we focus on the more energy-efficient hybrid precoding with subconnected architecture, andpropose a successive interference cancelation (SIC)-based hybridprecoding with near-optimal performance and low complexity.Inspired by the idea of SIC for multiuser signal detection, wefirst propose to decompose the total achievable rate optimiza-tion problem with nonconvex constraints into a series of simplesubrate optimization problems, each of which only considers onesubantenna array. Then, we prove that maximizing the achievablesubrate of each subantenna array is equivalent to simply seekinga precoding vector sufficiently close (in terms of Euclidean dis-tance) to the unconstrained optimal solution. Finally, we proposea low-complexity algorithm to realize SIC-based hybrid precod-ing, which can avoid the need for the singular value decomposition(SVD) and matrix inversion. Complexity evaluation shows that thecomplexity of SIC-based hybrid precoding is only about 10% ascomplex as that of the recently proposed spatially sparse precodingin typical mmWave MIMO systems. Simulation results verify thatSIC-based hybrid precoding is near-optimal and enjoys higherenergy efficiency than the spatially sparse precoding and the fullydigital precoding.

Index Terms—MIMO, mmWave communications, hybrid pre-coding, energy-efficient, 5G.

Manuscript received July 27, 2015; accepted December 8, 2015. Date of pub-lication March 31, 2016; date of current version May 11, 2016. This work wassupported in part by the International Science and Technology CooperationProgram of China (Grant 2015DFG12760), in part by the National NaturalScience Foundation of China (Grant 61571270 and Grant 61271266), in partby the Beijing Natural Science Foundation (Grant 4142027), and in part bythe Foundation of Shenzhen Government. This work was presented in part atthe IEEE International Conference on Communications (ICC), London, U.K.,June 2015.

X. Gao and L. Dai are with the Tsinghua National Laboratory forInformation Science and Technology (TNList), Department of ElectronicEngineering, Beijing 100084, China (e-mail: [email protected]; [email protected]).

S. Han and C.-L. I are with the Green Communication ResearchCenter, China Mobile Research Institute, Beijing 100053, China (e-mail:[email protected]; [email protected]).

R. W. Heath is with the University of Texas at Austin, Austin, TX 78712USA (e-mail: [email protected]).

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

Digital Object Identifier 10.1109/JSAC.2016.2549418

I. INTRODUCTION

T HE integration of millimeter-wave (mmWave) and mas-sive multiple-input multiple-output (MIMO) technique

can achieve orders of magnitude increase in system through-put due to larger bandwidth [1] and higher spectral efficiency[2], which makes it promising for future 5G wireless com-munication systems [3]. On one hand, massive MIMO with avery large antenna array (e.g., 256 antennas) at the base station(BS) can simultaneously serve a set of users through the useof precoding [4]. It has been theoretically proved that massiveMIMO can achieve orders of magnitude increase in spectralefficiency, since it can provide more multi-user gain [2]. On theother hand, mmWave with high frequencies enables such largeantenna array in massive MIMO to be packed in small physicaldimension [5]. Furthermore, the large antenna array can alsoprovide sufficient array gain by precoding [6], [7] to overcomethe free-space pathloss of mmWave signals and establish linkswith satisfying signal-to-noise ratio (SNR) [8].

For MIMO in conventional cellular frequency band (e.g.,2–3 GHz), precoding is entirely realized in the digital domainto cancel interference between different data streams. For aconventional digital precoding, each antenna requires a dedi-cated energy-intensive radio frequency (RF) chain (includingdigital-to-analog converter, up converter, etc.), whose energyconsumption is a large part (about 250 mW per RF chain [9])of the total energy consumption at mmWave frequencies dueto the wide bandwidth. If the conventional digital precoding isapplied in mmWave massive MIMO system with a large num-ber of antennas, the corresponding large number of RF chainswill bring high energy consumption, e.g., 16 W is required bya mmWave massive MIMO system with 64 antennas. To solvethis problem, the hybrid analog and digital precoding has beenproposed [10]. The key idea is to divide the conventional digitalprecoder into a small-size digital precoder (realized by a smallnumber of RF chains) to cancel interference and a large-sizeanalog precoder (realized by a large number of analog phaseshifters (PSs)) to increase the antenna array gain. In this way,hybrid precoding can reduce the number of required RF chainswithout obvious performance loss, which makes it enjoy a muchhigher energy efficiency than digital precoding [10].

The existing hybrid precoding schemes can be divided intotwo categories. The first category of hybrid precoding basedon spatially sparse precoding was proposed in [11]–[13], whichformulated the achievable rate optimization problem as a sparse

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GAO et al.: ENERGY-EFFICIENT HYBRID ANALOG AND DIGITAL PRECODING FOR MmWAVE MIMO SYSTEMS 999

approximation problem and solved it by the orthogonal match-ing pursuit (OMP) algorithm [14] to achieve the near-optimalperformance. The second category of hybrid precoding basedon codebooks was proposed in [15]–[17], which involved aniterative searching procedure among the predefined codebooksto find the optimal hybrid precoding matrix. However, thesealgorithms are all designed for the hybrid precoding with thefully-connected architecture, where each RF chain is connectedto all BS antennas via PSs. As the number of BS antennas isvery large (e.g., 256 as considered in [11]), the fully-connectedarchitecture requires thousands of PSs, which may bring threeadditional limitations: 1) it consumes more energy for exci-tation like the giant phased array radar [18]; 2) it requiresmore energy to compensate for the insertion loss of PS [18];3) it involves higher computational complexity, which will alsobring more energy consumption [19]. In contrast, the hybridprecoding with the sub-connected architecture, where each RFchain is connected to only a subset of BS antennas, can sig-nificantly reduce the number of required PSs. Therefore, thesub-connected architecture is expected to be more energy-efficient and easier to be implemented for mmWave MIMOsystems. Unfortunately, designing the optimal hybrid precod-ing with the sub-connected architecture is still a challengingproblem [10], [20], since such architecture changes the con-straints on the original problem of hybrid precoding with thefully-connected architecture.

In this paper, we propose a successive interference cancela-tion (SIC)-based hybrid precoding with sub-connected archi-tecture. The contributions of this paper can be summarized asfollows.

1) Inspired by the idea of SIC derived for multi-user sig-nal detection [21], we propose to decompose the totalachievable rate optimization problem with non-convexconstraints into a series of simple sub-rate optimizationproblems, each of which only considers one sub-antennaarray. Then, we maximize the achievable sub-rate of eachsub-antenna array one by one until the last sub-antennaarray is considered.

2) We prove that maximizing the achievable sub-rate of eachsub-antenna array is equivalent to seeking a precodingvector which has the smallest Euclidean distance to theunconstrained optimal solution. Based on this fact, wecan easily obtain the optimal precoding vector for eachsub-antenna array.

3) We further propose a low-complexity algorithm to realizethe SIC-based precoding, which avoids the need for sin-gular value decomposition (SVD) and matrix inversion.Complexity evaluation shows that the complexity of SIC-based precoding is only about 10% as complex as thatof the spatially sparse precoding [11] in typical mmWaveMIMO systems. Simulation results verify that the pro-posed SIC-based hybrid precoding is near-optimal andenjoys higher energy efficiency than the spatially sparseprecoding [11] and the fully digital precoding.

It is worth pointing out that to the best of the authors’ knowl-edge, our work in this paper is the first one that considers thehybrid precoding design with sub-connected architecture.

The rest of the paper is organized as follows. Section IIbriefly introduces the system model of mmWave MIMO.

Fig. 1. Two typical architectures of the hybrid precoding in mmWave MIMOsystems: (a) Fully-connected architecture, where each RF chain is connectedto all BS antennas; (b) Sub-connected architecture, where each RF chain isconnected to only a subset of BS antennas.

Section III specifies the proposed SIC-based hybrid precod-ing, together with the complexity evaluation. The simulationresults of the achievable rate and energy efficiency are shownin Section IV. Finally, conclusions are drawn in Section V.

Notation: Lower-case and upper-case boldface letters denotevectors and matrices, respectively; (·)T , (·)H , (·)−1, and |·|denote the transpose, conjugate transpose, inversion, and deter-minant of a matrix, respectively; ‖·‖1 and ‖·‖2 denote thel1- and l2-norm of a vector, respectively; ‖·‖F denotes theFrobenius norm of a matrix; Re{·} and Im{·} denote the realpart and imaginary part of a complex number, respectively;E(·) denotes the expectation; Finally, IN is the N × N identitymatrix.

II. SYSTEM MODEL

Fig. 1 illustrates two typical architectures for hybrid pre-coding in mmWave MIMO systems, i.e., the fully-connectedarchitecture as shown in Fig. 1 (a) and the sub-connected archi-tecture as shown in Fig. 1 (b). In both cases the BS has N Mantennas but only N RF chains. From Fig. 1, we observe that thesub-connected architecture will likely be more energy-efficient,since it only requires N M PSs, while the fully-connectedarchitecture requires N 2 M PSs. To fully achieve the spatialmultiplexing gain, the BS usually transmits N independent datastreams to users employing K receive antennas [10].

In the sub-connected architecture as shown in Fig. 1 (b),N data streams in the baseband are precoded by the dig-ital precoder D. In cases where complexity is a concern,D can be further specialized to be a diagonal matrix asD = diag [d1, d2, . . . , dN ], where dn ∈ R for n = 1, 2, . . . , N[10]. Then the role of D essentially performs some power allo-cation. After passing through the corresponding RF chain, the


digital-domain signal from each RF chain is delivered to onlyM PSs [22] to perform the analog precoding, which can bedenoted by the analog weighting vector an ∈ C

M×1, whose ele-ments have the same amplitude 1/

√M but different phases

[22]. After the analog precoding, each data stream is finallytransmitted by a sub-antenna array with only M antennas asso-ciated with the corresponding RF chain. Then, the receivedsignal vector y = [y1, y2, . . . , yK ]T at the user in a narrowbandsystem1 can be presented as

y = √ρHADs + n = √

ρHPs + n, (1)

where ρ is the average received power; H ∈ CK×N M denotes

the channel matrix, A is the N M × N analog precoding matrixcomprising N analog weighting vectors {am}N

m=1 as

A =

⎡⎢⎢⎢⎣

a1 0 . . . 00 a2 0...

. . ....

0 0 . . . aN

⎤⎥⎥⎥⎦

N M×N

, (2)

s = [s1, s2, . . . , sN ]T represents the transmitted signal vectorin the baseband. In this paper, we assume the widely usedGaussian signals [10]–[13], [15]–[17] with normalized signalpower E(ssH ) = 1

N IN , while the practical system with finite-alphabet inputs [23], [24] will be also briefly discussed inSection IV. P = AD presents the hybrid precoding matrix ofsize N M × N , which satisfies ‖P‖F ≤ N to meet the totaltransmit power constraint [11]. Finally, n = [n1, n2, . . . , nN ]T

is an additive white Gaussian noise (AWGN) vector, whoseentries follow the independent and identical distribution (i.i.d.)CN(0, σ 2).

It is known that mmWave channel H will not likely fol-low the rich-scattering model assumed at low frequencies dueto the limited number of scatters in the mmWave proroga-tion environment [3]. In this paper, we adopt the geometricSaleh-Valenzuela channel model to embody the low rank andspatial correlation characteristics of mmWave communications[10]–[13], [15]–[17], [25] as

H = γ

L∑l=1

αl�r(φr

l , θrl

)�t(φt

l , θtl

)fr(φr

l , θrl

)fHt

(φt

l , θtl

),

(3)

where γ =√

N M KL is a normalization factor, L is the number

of effective channel paths corresponding to the limited numberof scatters, and we usually have L ≤ N for mmWave communi-cation systems. αl ∈ C is the gain of the lth path. φt

l (θ tl ) and φr

l(θr

l ) are the azimuth (elevation) angles of departure and arrival(AoDs/AoAs), respectively. �t

(φt

l , θtl

)and �r

(φr

l , θrl

)denote

the transmit and receive antenna array gain at a specific AoDand AoA, respectively. For simplicity but without loss of gen-erality, �t

(φt

l , θtl

)and �r

(φr

l , θrl

)can be set as one within the

range of AoDs/AoAs [26]. Finally, ft(φt

l , θtl

)and fr

(φr

l , θrl

)are the antenna array response vectors depending on the antenna

1While mmWave systems are expected to be broadband as in prior work [3],the narrowband system can be regarded as a reasonable first step. The extensionto broadband system is an interesting topic.

array structures at the BS and the user, respectively. For the uni-form linear array (ULA) with U elements, the array responsevector can be presented as [18]

fULA (φ) = 1√U

[1, e j 2π

λd sin(φ), . . . , e j (U−1) 2π

λd sin(φ)

]T, (4)

where λ denotes the wavelength of the signal, and d is theantenna spacing. Note that here we abandon the subscripts {t, r}in (3) and we also do not include θ since the ULA response vec-tor is independent of the elevation angle. Additionally, when weconsider the uniform planar array (UPA) with W1 and W2 ele-ments (W1W2 = U ) on horizon and vertical, respectively, thearray response vector can be given by [18]

fUPA (φ, θ) = 1√U

[1, . . . , e j 2π

λd(x sin(φ) sin(θ)+y cos(θ)), . . . ,

e j 2πλ

d((W1−1) sin(φ) sin(θ)+(W2−1) cos(θ))]T

, (5)

where 0 ≤ x ≤ (W1 − 1) and 0 ≤ y ≤ (W2 − 1).

III. SIC-BASED HYBRID PRECODING FOR MMWAVE

MIMO SYSTEMS

In this section, we propose a low-complexity SIC-basedhybrid precoding to achieve the near-optimal performance. Theevaluation of computational complexity is also provided toshow its advantages over current solutions.

A. Structure of SIC-based hybrid precoding

In this paper, we aim to maximize the total achievable rate Rof mmWave MIMO systems2, while other criteria such as themax-min fairness criterion [27] are also of interest. Specifically,R can be expressed as [11]

R = log2

(∣∣∣IK + ρ

Nσ 2HPPH HH

∣∣∣) . (6)

According to the system model (1) in Section II, sincethe hybrid precoding matrix P can be represented asP = AD = diag {a1, . . . , aN } · diag {d1, . . . , dN }, there arethree constraints for the design of P:

Constraint 1: P should be a block diagonal matrix similarto the form of A as shown in (2), i.e., P = diag {p1, . . . , pN },where pn = dn an is the M × 1 non-zero vector of the nthcolumn pn of P, i.e., pn = [01×M(n−1), pn, 01×M(N−n)

]T ;Constraint 2: The non-zero elements of each column of P

should have the same amplitude, since the digital precodingmatrix D is a diagonal matrix, and the amplitude of non-zeroelements of the analog precoding matrix A is fixed to 1/

√M ;

Constraint 3: The Frobenius norm of P should satisfy‖P‖F ≤ N to meet the total transmit power constraint, where Nis the number of RF chains equal to the number of transmitteddata streams.

Unfortunately, these non-convex constraints on P make max-imizing the total achievable rate (6) very difficult to be solved.

2The maximization sum-rate criterion can also suppress the interference asmuch as possible, and the mathematical quantification of such interference willbe an important topic for future work.


However, based on the special block diagonal structure of thehybrid precoding matrix P, we observe that the precoding ondifferent sub-antenna arrays are independent. This inspires usto decompose the total achievable rate (6) into a series of sub-rate optimization problems, each of which only considers onesub-antenna array.

In particular, we can divide the hybrid precoding matrix P asP = [PN−1 pN

], where pN is the N th column of P, and PN−1 is

an N M × (N − 1) matrix containing the first (N − 1) columnsof P. Then, the total achievable rate R in (6) can be rewritten as

R = log2

(∣∣∣IK + ρ

Nσ 2HPPH HH

∣∣∣)= log2

(∣∣∣IK + ρ

Nσ 2H[PN−1 pN

] [PN−1 pN

]H HH∣∣∣)

= log2

(∣∣∣IK + ρ

Nσ 2HPN−1PH

N−1HH

+ ρ

Nσ 2HpN pH

N HH∣∣∣)

(a)= log2 (|TN−1|) + log2

(∣∣∣IK + ρ

Nσ 2T−1

N−1HpN pHN HH

∣∣∣)(b)= log2 (|TN−1|) + log2

(1+ ρ

Nσ 2pH

N HH T−1N−1HpN

), (7)

where (a) is obtained by defining the auxiliary matrixTN−1 = IK + ρ

Nσ 2 HPN−1PHN−1HH , and (b) is true

due to the fact that |I + XY| = |I + YX| by definingX = T−1

N−1HpN and Y = pHN HH . Note that the second

term log2

(1 + ρ

Nσ 2 pHN HH T−1

N−1HpN

)on the right side of

(7) is the achievable sub-rate of the N th sub-antenna array,while the first term log2 (|TN−1|) shares the same form as(6). This observation implies that we can further decomposelog2 (|TN−1|) using the similar method in (7) as

log2 (|TN−2|)+log2

(1+ ρ

Nσ 2pH

N−1HH T−1N−2HpN−1

).

Then, after N such decompositions, the total achievable rate Rin (6) can be presented as

R =N∑

n=1

log2

(1 + ρ

Nσ 2pH

n HH T−1n−1Hpn

), (8)

where we have Tn = IK + ρ

Nσ 2 HPnPHn HH and T0 = IN .

From (8), we observe that the total achievable rate optimizationproblem can be transformed into a series of sub-rate optimiza-tion problems of sub-antenna arrays, which can be optimizedone by one3. After that, inspired by the idea of SIC for multi-user signal detection [21], we can optimize the achievablesub-rate of the first sub-antenna array and update the matrix T1.Then, the similar method can be utilized to optimize the achiev-able sub-rate of the second sub-antenna array. Such procedurewill be executed until the last sub-antenna array is considered.Fig. 2 shows the diagram of the proposed SIC-based hybrid pre-coding. Next, we will discuss how to optimize the achievablesub-rate of each sub-antenna array.

3Note that different precoding orders of sub-antenna arrays will lead to thesame performance, since the total achievable rate R in (8) can be exactly repre-sented by the summation of the sub-rate of each sub-antenna array without anyperformance loss.

Fig. 2. Diagram of the proposed SIC-based hybrid precoding.

B. Solution to the Subrate Optimization Problem

In this subsection, we focus on the sub-rate optimizationproblem of the nth sub-antenna array, which can be directlyapplied to other sub-antenna arrays. According to (8), the sub-rate optimization problem of the nth sub-antenna array bydesigning the nth precoding vector pn can be stated as

poptn = arg max

pn∈Flog2

(1 + ρ

Nσ 2pH

n Gn−1pn

), (9)

where Gn−1 is defined as Gn−1 = HH T−1n−1H, F is the set of

all feasible vectors satisfying the three constraints described inSection III-A. Note that the nth precoding vector pn only hasM non-zero elements from the (M(n − 1) + 1)th one to the(Mn)th one. Therefore, the sub-rate optimization problem (9)can be equivalently written as

poptn = arg max

pn∈Flog2

(1 + ρ

Nσ 2pH

n Gn−1pn

), (10)

where F includes all possible M × 1 vectors satisfyingConstraint 2 and Constraint 3, Gn−1 of size M × M is the cor-responding sub-matrix of Gn−1 by only keeping the rows andcolumns of Gn−1 from the (M(n − 1) + 1)th one to the (Mn)thone, which can be presented as

Gn−1 = RGn−1RH = RHH T−1n−1HRH , (11)

where R = [ 0M×M(n−1) IM 0M×M(N−n)

]is the correspond-

ing selection matrix.Define the singular value decomposition (SVD) of the

Hermitian matrix Gn−1 as Gn−1 = V�VH , where � is anM × M diagonal matrix containing the singular values of Gn−1in a decreasing order, and V is an M × M unitary matrix. It isknown that the optimal unconstrained precoding vector of (10)is the first column v1 of V, i.e., the first right singular vectorof Gn−1 [11]. However, according to the constraints mentionedin Section III-A, we cannot directly choose popt

n as v1 since theelements of v1 do not obey the constraint of same amplitude(i.e., Constraint 2). To find a feasible solution to the sub-rateoptimization problem (10), we need to further convert (10) intoanother form, which is given by the following Proposition 1.

Proposition 1: The optimization problem (10)

poptn = arg max

pn∈Flog2

(1 + ρ

Nσ 2pH

n Gn−1pn

)


is equivalent to the following problem

poptn = arg min

pn∈F‖v1 − pn‖2

2 , (12)

where v1 is the first right singular vector of Gn−1.

Proof: See Appendix A. �Proposition 1 indicates that we can find a feasible precod-

ing vector pn , which is sufficiently close (in terms of Euclideandistance) to the optimal but unpractical precoding vector v1, tomaximize the achievable sub-rate of the nth sub-antenna array.Since pn = dn an according to (1), the target ‖v1 − pn‖2

2 in (12)can be rewritten as

‖v1 − pn‖22

= (v1 − dn an)H (v1 − dn an)

= vH1 v1 + d2

n aHn an − 2dnRe

(vH

1 an

)(a)= 1 + d2

n − 2dnRe(

vH1 an

)=(

dn − Re(

vH1 an

))2 +(

1 −[Re(

vH1 an

)]2)

, (13)

where (a) is obtained based on the facts that vH1 v1 = 1 and

aHn an = 1, since v1 is the first column of the unitary matrix V

and each element of an has the same amplitude 1/√

M .From (13), we observe that the distance between pn and v1

consists of two parts. The first one is(dn − Re

(vH

1 an))2

, whichcan be minimized to zero by choosing dn = Re

(vH

1 an). The

second one is(

1 − [Re(vH

1 an)]2)

, which can be minimized

by maximizing∣∣Re

(vH

1 an)∣∣. Note that both an and v1 have a

fixed power of one, i.e., vH1 v1 = 1 and aH

n an = 1. Therefore,the optimal aopt

n to maximize∣∣Re

(vH

1 an)∣∣ is

aoptn = 1√

Me jangle(v1), (14)

where angle(v1) denotes the phase vector of v1, i.e., eachelement of aopt

n shares the same phase as the correspondingelement of v1

4. Accordingly, the optimal choice of doptn is

doptn =Re

(vH

1 an

)= 1√

MRe(

vH1 e jangle(v1)

)= ‖v1‖1√

M. (15)

Based on (14) and (15), the optimal solution poptn to the opti-

mization problem (12) (or equivalently (10)) can be obtained by

poptn = dopt

n aoptn = 1

M‖v1‖1e jangle(v1). (16)

It is worth pointing out that v1 is the first column of the uni-tary matrix V, each element vi of v1 (for i = 1, . . . , M) has

the amplitude less than one. Therefore, we have∥∥∥popt

n

∥∥∥2

2≤ 1.

4It is worth pointing out that the analog precoding vector an can be alsorestricted to a DFT vector to save the overhead of quantization for limited feed-back systems [11]. However, since more constraints are set on the design ofanalog precoding, such scheme may lead to some performance loss comparedto the proposed one (14).

Note that for all sub-antenna arrays, the optimal solution poptn

for n = 1, 2, . . . , N have a similar form. Thus, we can concludethat ∥∥Popt

∥∥2F =

∥∥∥diag{

popt1 , . . . , popt

N

}∥∥∥2

F≤ N , (17)

which demonstrates that the total transmit power constraint(Constraint 3) is satisfied.

After we have acquired poptn for the nth sub-antenna

array, the matrices Tn = IK + ρ

Nσ 2 HPnPHn HH (8) and

Gn = RHH T−1n HRH (11) can be updated. Then, the method

described above for the nth sub-antenna array can be reusedagain to optimize the achievable sub-rate of the (n + 1)th sub-antenna array. To sum up, solving the sub-rate optimizationproblem of the nth sub-antenna array consists of the followingthree steps.

Step 1: Execute the SVD of Gn−1 to obtain v1;Step 2: Let popt

n = 1M ‖v1‖1e jangle(v1) as the optimal solution

to the current nth sub-antenna array;Step 3: Update matrices Tn = IK + ρ

Nσ 2 HPnPHn HH and

Gn = RHH T−1n HRH for the next (n + 1)th sub-

antenna array.Note that although we can obtain the optimal solution popt

n bythe method above, we need to compute the SVD of Gn−1 (Step1) and the matrix Gn (Step 3) involving the matrix inversionof large size, which leads to high computational complexity aswell as high energy consumption for computation [19]. To thisend, next we will propose a low-complexity algorithm to obtainpopt

n without the complicated SVD and matrix inversion.

C. Low-Complexity Algorithm to Obtain the Optimal Solution

We start by considering how to avoid the SVD involving highcomputational complexity as well as a large number of divi-sions, which are difficult to be implemented in hardware. Weobserve from Step 1 that the SVD of Gn−1 does not need tobe computed to acquire � and V, as only the first column v1of V is enough to obtain popt

n . This observation inspires us toexploit the simple power iteration algorithm [28], which is usedto compute the largest eigenvalue and the corresponding eigen-vector of a diagonalizable matrix. Since Gn−1 is a Hermitianmatrix, it follows that: 1) Gn−1 is also a diagonalizable matrix;2) The singular values (right singular vectors) of Gn−1 are thesame as the eigenvalues (eigenvectors). Therefore, the poweriteration algorithm can be also utilized to compute v1 as well asthe largest singular value �1 of Gn−1 with low complexity.

More specifically, as shown by the pseudo-code inAlgorithm 1, the power iteration algorithm starts with an ini-tial solution u(0) ∈ C

M×1, which is usually set as [1, 1, . . . , 1]T

without loss of generality [28]. In each iteration, it first com-putes the auxiliary vector z(s) = Gn−1u(s−1) (s is the numberof iterations) and then extracts the element of z(s) havingthe largest amplitude as m(s). After that, u(s) is updated as

u(s) = z(s)

m(s) for the next iteration. The power iteration algorithmwill stop until the number of iterations reaches the predefinednumber S. Finally, m(S) and u(S)/

∥∥u(S)∥∥

2 will be output as thelargest singular value �1 and the first right singular vector v1of Gn−1, respectively.


Algorithm 1. Power iteration algorithm

Input: (1) Gn−1;(2) Initial solution u(0);(3) Maximum number of iteration S

for 1 ≤ s ≤ S1) z(s) = Gn−1u(s−1)

2) m(s) = arg maxz(s)

i

∣∣∣z(s)i

∣∣∣3) if 1 ≤ s ≤ 2

n(s) = m(s)

else

n(s) = m(s)m(s−2)−(m(s−1))2

m(s)−2m(s−1)+m(s−2)

end if4) u(s) = z(s)

n(s)

end forOutput: (1) The largest singular value �1 = n(S)

(2) The first singular vector v1 = u(S)

‖u(S)‖2

According to [28], we know that

m(s) = �1

[1 + O

((�2

�1

)s)], (18)

where �2 is the second largest singular value of Gn−1. From(18), we conclude that m(s) will converges to �1 as long as�1 �= �2. Similarly, when �1 �= �2, u(s)/

∥∥u(s)∥∥

2 will alsoconverge to v1, i.e.,

lims→∞ m(s) = �1, lim

s→∞u(s)∥∥u(s)∥∥

2

= v1. (19)

Although the power iteration algorithm is convergent, its con-vergence rate may be slow if �1 ≈ �2 based on (18). To solvethis problem, we propose to utilize the Aitken accelerationmethod [29] to further increase the convergence rate of thepower iteration algorithm. Specifically, we can compute{

n(s) = m(s), for 1 ≤ s ≤ 2,

n(s) = m(s)m(s−2)−(m(s−1))2

m(s)−2m(s−1)+m(s−2) , for 2 < s ≤ S.(20)

Then, u(s) and �1 will be correspondingly changed to

u(s) = z(s)

n(s) and �1 = n(S), respectively.Next, we will focus on how to reduce the complex-

ity to compute the matrices Tn = IK + ρ

Nσ 2 HPnPHn HH and

Gn = RHH T−1n HRH , which involve the complicated matrix-

to-matrix multiplication and matrix inversion of large size. Inparticular, with some standard mathematical manipulations, thecomputation of Gn can be significantly simplified as shown bythe following Proposition 2.

Proposition 2: The matrix Gn = RHH T−1n HRH , where

Tn = IK + ρ

Nσ 2 HPnPHn HH , can be simplified as

Gn ≈ Gn−1 −ρ

Nσ 2 �21v1vH

1

1 + ρ

Nσ 2 �1, (21)

where �1and v1 are the largest singular value and first rightsingular vector of Gn−1, respectively.

Proof: See Appendix B. �Proposition 2 implies that we can simply exploit �1 and v1

that have been obtained by Algorithm 1 as described above toupdate Gn , which only involves one vector-to-vector multipli-cation instead of the complicated matrix-to-matrix multiplica-tion and matrix inversion. Note that the evaluation of computa-tional complexity will be discussed in detail in Section III-E.

D. Summary of the Proposed SIC-Based Hybrid Precoding

Based on the discussion so far, the pseudo-code of theproposed SIC-based hybrid precoding can be summarized inAlgorithm 2, which can be explained as follows. The proposedSIC-based hybrid precoding starts by computing the largest sin-gular value �1 and first right singular vector v1 of Gn−1, whichis achieved by Algorithm 1. After that, according to Section III-B, the optimal precoding vector for the nth sub-antenna arraycan be obtained by utilizing v1. Finally, based on Proposition 2,Gn can be updated with low complexity for the next iteration.This procedure will be executed until the last (N th) sub-antennaarray is considered. Finally, after N iterations, the optimal dig-ital, analog, and hybrid precoding matrices D, A, and P can beobtained, respectively.

Algorithm 2. SIC-based hybrid precoding

Input: G0for 1 ≤ n ≤ N

1) Compute v1 and �1 of Gn−1 by Algorithm 12) aopt

n = 1√M

e jangle(v1), doptn = ‖v1‖1√

M,

poptn = 1

M ‖v1‖1e jangle(v1) (14)-(16)

3) Gn =Gn−1−ρ

Nσ2 �21 v1vH

1

1+ ρ

Nσ2 �1(Proposition 2)

end forOutput: (1) D = diag

{dopt

1 , . . . , doptN

}(2) A = diag

{aopt

1 , . . . , aoptN

}(3) P = AD

It is worth pointing out that the idea of SIC-based hybridprecoding can be also extended to the combining at the userfollowing the similar logic in [11]. When the number of RFchains at the BS is smaller than that at the user, we firstcompute the optimal hybrid precoding matrix P according toAlgorithm 2, where we assume that the combining matrixQ = I. Then, given the effective channel matrix HP, we cansimilarly obtain the optimal hybrid combining matrix Q byreferring to Algorithm 2, where the input G0 and the optimalunconstrained solution v1 should be correspondingly replaced.Conversely, when the number of RF chains at the BS is largerthan that at the user, we can assume P = I and obtain theoptimal hybrid combining matrix Q. After that, the optimal pre-coding matrix P can be acquired given the effective channelmatrix QH. Additionally, to further improve the performance,we can combine the above method with the “Ping-pong” algo-rithm [22], which involves an iteration procedure between theBS and the user, to jointly seek the optimal hybrid precodingand combining matrices pair. Further discussion about hybridcombining will be left for future work.


TABLE ICOMPLEXITY COMPARISON

E. Complexity Evaluation

In this subsection, we provide the complexity evaluationof the proposed SIC-based hybrid precoding in terms of therequired numbers of complex multiplications and divisions.From Algorithm 2, we observe that the complexity of SIC-based hybrid precoding comes from the following four parts:

1) The first one originates from the computation ofG0 = RHH HRH according to (11). Note that R is aselection matrix and H has the size K × N M . Therefore,this part involves K M2 times of multiplications withoutany division.

2) The second one is from executing Algorithm 1. It isobserved that in each iteration we need to computea matrix-to-vector multiplication z(s) = Gn−1u(s−1)

together with the Aitken acceleration method (20).Therefore, we totally require S

(M2 + 2

)− 4 and(2S − 2) times of multiplications and divisions,respectively.

3) The third one stems from acquiring the optimal solutionpopt

n in step 2 of Algorithm 2. We find that this part isquite simple, which only needs 2 times of multiplicationswithout any division, since v1 has been obtained and 1√

Mis a fixed constant.

4) The last one comes from the update of Gn . Accordingto Proposition 2, we know that this part mainly involvesa outer product v1vH

1 . Thus, it requires M2 times ofmultiplications with only one division.

To sum up, the proposed SIC-based hybrid precodingapproximately requires O

(M2 (N S+K )

)times of multiplica-

tions and O (2N S) times of divisions. Table I provides the com-plexity comparison between SIC-based hybrid precoding andthe recently proposed spatially sparse precoding [11], whichrequires O

(N 4 M + N 2L2 + N 2 M2L

)times of multiplications

and O(2N 3

)times of divisions. Here, L is the number of effec-

tive channel paths as defined in (3). Considering the typicalmmWave MIMO system with N = 8, M = 8, K = 16, L = 3[11], we observe that the complexity of SIC-based hybrid pre-coding is about 4 × 103 times of multiplications and 102 timesof divisions, where we set S = 5 (note that S ≥ 5 is usually suf-ficient to guarantee the performance, which is verified throughintensive simulations). By contrast, the complexity of the spa-tially sparse precoding is about 5 × 104 times of multiplicationsand 103 times of divisions. Therefore, the proposed SIC-basedhybrid precoding enjoys much lower complexity, which is onlyabout 10% as complex as that of the spatially sparse precoding.

Fig. 3. Achievable rate comparison for an N M × K = 64 × 16 (N = 8)mmWave MIMO system.

IV. SIMULATION RESULTS

In this section, we provide the simulation results ofthe achievable rate and energy efficiency to evaluate theperformance of the proposed SIC-based hybrid precoding. Wecompare the performance of SIC-based hybrid precoding withthe recently proposed spatially sparse precoding [11] and theoptimal unconstrained precoding based on the SVD of thechannel matrix, which are both with fully-connected archi-tecture. Additionally, we also include the conventional analogprecoding [30] and the optimal unconstrained precoding (i.e.,popt

n = v1) which are both with sub-connected architecture asbenchmarks for comparison.

The simulation parameters are described as follows. We gen-erate the channel matrix according to the channel model [31]described in Section II. The number of effective channel pathsis L = 3 [11]. The carrier frequency is set as 28GHz [15]. Boththe transmit and receive antenna arrays are ULAs with antennaspacing d = λ/2. Since the BS usually employs the directionalantennas to eliminate interference and increase antenna gain[3], the AoDs are assumed to follow the uniform distributionwithin

[−π6 , π

6

]. Meanwhile, due to the random position of

users, we assume that the AoAs follow the uniform distributionwithin [−π, π ], which means the omni-directional antennas areadopted by users. Furthermore, we set the maximum number ofiterations S = 5 to run Algorithm 2. Finally, SNR is definedas ρ

σ 2 .Firstly, we consider the perfect channel state information

(CSI) scenario. Fig. 3 shows the achievable rate comparison inmmWave MIMO system, where N M × K = 64 × 16 and thenumber of RF chains is N = 8. We observe from Fig. 3 thatthe proposed SIC-based hybrid precoding outperforms the con-ventional analog precoding with sub-connected architecture inwhole simulated SNR range. Meanwhile, Fig. 3 also verifies thenear-optimal performance of SIC-based hybrid precoding, sinceit can achieve about 99% of the rate achieved by the optimalunconstrained precoding with sub-connected architecture.


Fig. 4. Achievable rate comparison for an N M × K = 128 × 32 (N = 16)mmWave MIMO system.

Fig. 4 compares the achievable rate in mmWave MIMO sys-tem with N M × K = 128 × 32 and N = 16, where we observesimilar trends as those from Fig. 3. More importantly, Fig. 3and Fig. 4 show that the performance of SIC-based hybrid pre-coding is also close to the spatially sparse precoding and theoptimal unconstrained precoding with fully-connected archi-tecture. For example, when SNR = 0 dB, our method canachieve more than 90% of the rate achieved by the near-optimalspatially sparse precoding in both simulated mmWave MIMOconfigurations. Considering the low computational complex-ity of the proposed SIC-based hybrid precoding as analyzedbefore, we further conclude that SIC-based hybrid precodingcan achieve much better trade-off between the performance andcomputational complexity.

Fig. 5 provides a achievable rate comparison in mmWaveMIMO systems against the numbers of BS and user anten-nas, where N M = K , the number of RF chains is fixed toN = 8, and SNR = 0 dB. We find that the performance ofthe proposed SIC-based hybrid precoding can be improved byincreasing the number of BS and user antennas, which involvesmuch lower energy consumption than increasing the number ofenergy-intensive RF chains [18].

Fig. 6 shows the achievable rate comparison against the num-bers of user antennas K , where N M = 64, N = 8, and SNR =0 dB. We imply from Fig. 6 that the performance loss of SIC-based hybrid precoding due to the sub-connected architecturecan be compensated by increasing the number of user antennasK . For example, the achievable rate of SIC-based hybrid pre-coding when K = 30 is the same as that of the spatially sparseprecoding when K = 20. Note that in this case, the requirednumber of PSs of SIC-based hybrid precoding is N M = 64,while for the spatially sparse precoding, the number of requiredPSs is N 2 M = 512. That means much energy can saved bySIC-based hybrid precoding, which will be also verified bysimulation results later. In contrast, the cost of increasing thenumber of user antennas K will be negligible since the energyconsumption of user antenna is usually small [18].

Fig. 5. Achievable rate comparison against the numbers of BS and userantennas (N M = K ), where N = 8 and SNR = 0 dB.

Fig. 6. Achievable rate comparison against the number of user antennas K ,where N M = 64, N = 8, and SNR = 0 dB.

Next we evaluate the impact of imperfect CSI on the pro-posed SIC-based hybrid precoding. The estimated channelmatrix (imperfect CSI) H can be modeled as [4]

H = ξH+√

1 − ξ2E, (22)

where H is the actual channel matrix, ξ ∈ [0, 1] presents theCSI accuracy, and E is the error matrix with entries follow-ing the distribution i.i.d. CN (0, 1). Fig. 7 shows the achievablerate comparison for an N M × K = 64 × 16 (N = 8) mmWaveMIMO system, where the perfect CSI and the imperfect CSIwith different ξ scenarios are considered. We observe that theproposed SIC-based hybrid precoding is not sensitive to the CSIaccuracy. For example, the achievable rate of SIC-based hybridprecoding when ξ = 0.9 is quite close to that in the perfect CSIscenario, where the SNR gap is about 1 dB. Even when the CSIaccuracy is quite poor (i.e., ξ = 0.5), SIC-based hybrid pre-coding with imperfect CSI can still achieve more than 88% ofthe rate in the perfect CSI scenario. Additionally, Fig. 8 shows


Fig. 7. Impact of imperfect CSI on SIC-based hybrid precoding for anN M × K = 64 × 16 (N = 8) mmWave MIMO system.

Fig. 8. Impact of imperfect CSI on SIC-based hybrid precoding for anN M × K = 128 × 32 (N = 16) mmWave MIMO system.

the achievable rate comparison for an N M × K = 128 × 32(N = 16) mmWave MIMO system, where similar conclusionsas those from Fig. 7 can be derived.

After that, we will also evaluate the proposed SIC-basedhybrid precoding in practical systems with finite-alphabet sig-nals instead of ideal Gaussian signals. Here, we also aim tomaximize the achievable sum-rate, since it has been shown in[23, Section IV] that maximizing the achievable sum-rate is anexcellent criterion for precoding, and it also has direct impacton the coded bit error rate performance. For the finite-alphabetsignals s, whose values are taken from a practical constellationQ, the achievable rate R can be presented as [23]

R = N log2 |Q| − 1

|Q|N

|Q|N∑m=1

En

⎧⎨⎩log2

|Q|N∑k=1

e−um,k

⎫⎬⎭ , (23)

where N is the number of RF chains (also the number of trans-mitted data streams), |Q| is the cardinality of Q, En denotes the

Fig. 9. Achievable rate of SIC-based hybrid precoding with finite-alphabetinputs, where N =8, N M = K =64, and BSPK is adopted.

expectation with respect to the noise vector n, um,k is defined as

um,k =∥∥HP

(sm − sk

)+ n∥∥2

2 − ∥∥n∥∥2

2

σ 2, (24)

H and P are the channel matrix and precoding matrix, respec-tively, σ 2 is the power of noise, and sm is a possible signalvector with N elements taking values from Q.

From (23) we know that the achievable rate of practical sig-naling is quite different from ideal Gaussian signaling, wherethe upper bound is determined by N log2 |Q| [24], [32]. Fig. 9shows the achievable rate of SIC-based hybrid precoding withfinite-alphabet inputs, where N M × K = 64 × 16, N = 8, andthe simple BPSK modulation (|Q| = 2) is considered as anexample. We can observe that as the SNR becomes large, theproposed SIC-based hybrid precoding can also achieve theperformance quite close to the upper bound N log2 |Q| = 8bits.

In the end, we evaluate the energy efficiency of the proposedSIC-based hybrid precoding. Based on the energy consumptionmodel in [33], [34], the energy efficiency η can be defined as

η= R

Ptotal= R

Pt + NRF PRF + NPS PPS(bps/Hz/W) , (25)

where Ptotal�= Pt + NRF PRF + NPS PPS is the total energy con-

sumption, Pt is the transmitted energy, PRF is the energyconsumed by RF chain, PPS is the energy consumed by PS(including the energy for the excitation and the energy for thecompensation of insertion loss [18]), NRF and NPS are thenumbers of required RF chains and PSs, respectively.

In this paper, we use the practical values PRF =250 mW [9],PPS = 1 mW [18], and Pt = 1 W (about 30 dBm) in a smallcell transmission scenario [35], since mmWave is more likelyto be applied in small cells. Fig. 10 shows the energy effi-ciency comparison against the number of RF chains N , whereSNR = 0 dB, N M = K = 64 (N = 1, 2, 4, . . . , 64 to ensurethat M is an integer). We observe that both the conventionalspatially sparse precoding and the proposed SIC-based pre-coding can achieve higher energy efficiency than the optimal


Fig. 10. Energy efficiency comparison against the numbers of RF chains N ,where N M = K = 64.

unconstrained precoding (also known as the fully digital pre-coding), especially when the number of RF chains N is limited(e.g., N < 30). Besides, we also observe that the proposed SIC-based precoding is more energy efficient than the conventionalspatially sparse precoding.

V. CONCLUSIONS

In this paper, we proposed a SIC-based hybrid precodingwith sub-connected architecture for mmWave MIMO systems.We first showed that the total achievable rate optimizationproblem with non-convex constraints can be decomposed intoa series of sub-rate optimization problems, each of whichonly considers one sub-antenna array. Then, we proved thatthe sub-rate optimization problem of each sub-antenna arraycan be solved by simply seeking a precoding vector suffi-ciently close to the unconstrained optimal solution. Finally, alow-complexity algorithm was proposed to realize SIC-basedprecoding without the complicated SVD and matrix inver-sion. Complexity evaluation showed that the complexity of theproposed SIC-based hybrid precoding is only about 10% ofthat of the recently proposed spatially sparse precoding withfully-connected architecture in typical mmWave MIMO sys-tem. Simulation results verified the near-optimal performanceand high energy efficiency of the proposed SIC-based hybridprecoding. Our further work will focus on the limited feedbackscenario, where the angles of PSs are quantified.

APPENDIX APROOF OF PROPOSITION 1

Define the target of the optimization problem (10) as

Rn = log2

(1 + ρ

Nσ 2pH

n Gn−1pn

), (26)

and the SVD of Gn−1 as Gn−1 = V�VH . Then, by separatingthe matrices � and V into two parts:

� =[

�1 00 �2

], V = [v1 V2] , (27)

Rn in (26) can be rewritten as

Rn = log2

(1 + ρ

Nσ 2pH

n Gn−1pn

)= log2

(1 + ρ

Nσ 2pH

n V�VH pn

)= log2

(1 + ρ

Nσ 2

× pHn [v1 V2]

[�1 00 �2

][v1 V2]H pn

)

= log2

(1 + ρ

Nσ 2pH

n v1�1vH1 pn

+ ρ

Nσ 2pH

n V2�2VH2 pn

). (28)

Since we aim to find a vector pn sufficiently “close” to v1, it isreasonable to assume that pn is approximately orthogonal to thematrix V2, i.e., pH

n V2 ≈ 0 [11]. Then, (28) can be simplified as

Rn ≈ log2

(1 + ρ�1

Nσ 2pH

n v1vH1 pn

)(a)= log2

(1 + ρ�1

Nσ 2

)

+ log2

(1 −

(1 + ρ�1

Nσ 2

)−1ρ�1

Nσ 2

(1−pH

n v1vH1 pn

))

(b)≈ log2

(1 + ρ�1

Nσ 2

)+ log2

(pH

n v1vH1 pn

)(29)

where (a) is obtained by using the formulaI + XY = (I + X)

(I − (I + X)−1X (I − Y)

)[11], where

we define X = ρ�1Nσ 2 and Y = pH

n v1vH1 pn ; (b) is valid by

employing the high SNR approximation [36], i.e.,(1 + ρ�1

Nσ 2

)−1ρ�1

Nσ 2≈ 1. (30)

From (29), we observe that maximizing Rn is equivalent to

maximizing pHn v1vH

1 pn = ∥∥pHn v1

∥∥22, the square of inner prod-

uct between two vectors pn and v1. Note that v1 is a fixed vector.Therefore, exploring a vector pn , which has the largest projec-tion on v1, will lead to the smallest Euclidean distance to v1as well. Based on this fact, we conclude that the optimizationproblem (10) is equivalent to the following problem

poptn = arg min

pn∈F‖v1 − pn‖2

2 . (31)

APPENDIX BPROOF OF PROPOSITION 2

We first consider the matrix Tn = IK + ρ

Nσ 2 HPnPHn HH ,

which should be inversed to compute Gn (11). By partitioningPn as Pn = [Pn−1 pn

], Tn can be rewritten as

Tn = IK + ρ

Nσ 2HPnPH

n HH

= IK + ρ

Nσ 2H[Pn−1 pn

] [Pn−1 pn

]H HH

= IK + ρ

Nσ 2HPn−1PH

n−1HH + ρ

Nσ 2HpnpH

n HH

= Tn−1 + ρ

Nσ 2HpnpH

n HH . (32)


Then, by utilizing the Sherman-Morrison formula [[28],Eq 2.1.4]

(A + uvT

)−1 = A−1 − A−1uvT A−1

1 + vT A−1u, (33)

T−1n can be presented as

T−1n =

(Tn−1 + ρ

Nσ 2HpnpH

n HH)−1

= T−1n−1 −

ρ

Nσ 2 T−1n−1HpnpH

n HH T−1n−1

1 + ρ

Nσ 2 pHn HH T−1

n−1Hpn. (34)

Substituting (34) into Gn = HH T−1n H, we have

Gn = HH T−1n H

= HH

(T−1

n−1 −ρ

σ 2 T−1n−1HpnpH

n HH T−1n−1

1 + ρ

σ 2 pHn HH T−1

n−1Hpn

)H

= Gn−1 −ρ

σ 2 Gn−1pnpHn Gn−1

1 + ρ

σ 2 pHn Gn−1pn

. (35)

Then, according to (11), Gn can be obtained by

Gn = RGnRH

= R

(Gn−1 −

ρ

Nσ 2 Gn−1pnpHn Gn−1

1 + ρ

Nσ 2 pHn Gn−1pn

)RH

= Gn−1 −ρ

Nσ 2 Gn−1pn pHn Gn−1

1 + ρ

Nσ 2 pHn Gn−1pn

. (36)

Note that in Section III-B, we have obtained the precoding vec-tor pn sufficiently close to v1, i.e., pn ≈ v1. Thus, (36) can bewell approximated by replacing pn with v1 as

Gn = Gn−1 −ρ

Nσ 2 Gn−1pn pHn Gn−1

1 + ρ

Nσ 2 pHn Gn−1pn

≈ Gn−1 −ρ

Nσ 2 Gn−1v1vH1 Gn−1

1 + ρ

Nσ 2 vH1 Gn−1v1

(a)= Gn−1 −ρ

Nσ 2 �21v1vH

1

1 + ρ

Nσ 2 �1, (37)

where (a) is true due to fact that vH1 Gn−1 = �1vH

1 , since Gn−1is an Hermitian matrix.

REFERENCES

[1] T. Bai, A. Alkhateeb, and R. Heath, “Coverage and capacity ofmillimeter-wave cellular networks,” IEEE Commun. Mag., vol. 52, no. 9,pp. 70–77, Sep. 2014.

[2] T. L. Marzetta, “Noncooperative cellular wireless with unlimited numbersof base station antennas,” IEEE Trans. Wireless Commun., vol. 9, no. 11,pp. 3590–3600, Nov. 2010.

[3] Z. Pi and F. Khan, “An introduction to millimeter-wave mobile broadbandsystems,” IEEE Commun. Mag., vol. 49, no. 6, pp. 101–107, Jun. 2011.

[4] F. Rusek et al., “Scaling up MIMO: Opportunities and challenges withvery large arrays,” IEEE Signal Process. Mag., vol. 30, no. 1, pp. 40–60,Jan. 2013.

[5] L. Wei, R. Q. Hu, Y. Qian, and G. Wu, “Key elements to enable mil-limeter wave communications for 5G wireless systems,” IEEE WirelessCommun., vol. 21, no. 6, pp. 136–143, Dec. 2014.

[6] M. Samimi et al., “28 GHz angle of arrival and angle of departure analysisfor outdoor cellular communications using steerable beam antennas inNew York City,” in Proc. IEEE Veh. Technol. Conf. (VTC Spring’13),May 2013, pp. 1–6.

[7] A. Alkhateeb, J. Mo, N. González-Prelcic, and R. Heath, “MIMO pre-coding and combining solutions for millimeter-wave systems,” IEEECommun. Mag., vol. 52, no. 12, pp. 122–131, Dec. 2014.

[8] B. Yin et al., “High-throughput beamforming receiver for millimeterwave mobile communication,” in Proc. IEEE Global Commun. Conf.(GLOBECOM’13), Dec. 2013, pp. 3697–3702.

[9] P. Amadori and C. Masouros, “Low RF-complexity millimeter-wavebeamspace-MIMO systems by beam selection,” IEEE Trans. Commun.,vol. 63, no. 6, pp. 2212–2222, Jun. 2015.

[10] S. Han, C.-L. I, Z. Xu, and C. Rowell, “Large-scale antenna systems withhybrid precoding analog and digital beamforming for millimeter wave5G,” IEEE Commun. Mag., vol. 53, no. 1, pp. 186–194, Jan. 2015.

[11] O. El Ayach, S. Rajagopal, S. Abu-Surra, Z. Pi, and R. Heath, “Spatiallysparse precoding in millimeter wave MIMO systems,” IEEE Trans.Wireless Commun., vol. 13, no. 3, pp. 1499–1513, Mar. 2014.

[12] Y. Lee, C.-H. Wang, and Y.-H. Huang, “A hybrid RF/baseband precod-ing processor based on parallel-index-selection matrix-inversion-bypasssimultaneous orthogonal matching pursuit for millimeter wave MIMOsystems,” IEEE Trans. Signal Process., vol. 63, no. 2, pp. 305–317, Jan.2015.

[13] C.-E. Chen, “An iterative hybrid transceiver design algorithm for millime-ter wave MIMO systems,” IEEE Wireless Commun. Lett., vol. 4, no. 3,pp. 285–288, Jun. 2015.

[14] J. A. Tropp and A. C. Gilbert, “Signal recovery from random measure-ments via orthogonal matching pursuit,” IEEE Trans. Inf. Theory, vol. 53,no. 12, pp. 4655–4666, Dec. 2007.

[15] W. Roh et al., “Millimeter-wave beamforming as an enabling technol-ogy for 5G cellular communications: Theoretical feasibility and prototyperesults,” IEEE Commun. Mag., vol. 52, no. 2, pp. 106–113, Feb. 2014.

[16] T. Kim, J. Park, J.-Y. Seol, S. Jeong, J. Cho, and W. Roh, “Tens of Gbpssupport with mmWave beamforming systems for next generation com-munications,” in Proc. IEEE Global Commun. Conf. (GLOBECOM’13),Dec. 2013, pp. 3685–3690.

[17] C. Kim, J. S. Son, T. Kim, and J.-Y. Seol, “On the hybrid beamformingwith shared array antenna for mmWave MIMO-OFDM systems,” in Proc.IEEE WCNC’14), Apr. 2014.

[18] C. A. Balanis, Antenna Theory: Analysis and Design. Hoboken, NJ, USA:Wiley, 2012.

[19] D. Schneider, “Could supercomputing turn to signal processors (again)?”IEEE Spectr., vol. 49, no. 10, pp. 13–14, Oct. 2012.

[20] S. Han, C.-L. I, Z. Xu, and S. Wang, “Reference signals design for hybridanalog and digital beamforming,” IEEE Commun. Lett., vol. 18, no. 7,pp. 1191–1193, Jul. 2014.

[21] Y.-C. Liang, E. Y. Cheu, L. Bai, and G. Pan, “On the relationshipbetween MMSE-SIC and BI-GDFE receivers for large multiple-inputmultiple-output channels,” IEEE Trans. Signal Process., vol. 56, no. 8,pp. 3627–3637, Aug. 2008.

[22] S. Hur, T. Kim, D. Love, J. Krogmeier, T. Thomas, and A. Ghosh,“Millimeter wave beamforming for wireless backhaul and access in smallcell networks,” IEEE Trans. Commun., vol. 61, no. 10, pp. 4391–4403,Oct. 2013.

[23] C. Xiao, Y. R. Zheng, and Z. Ding, “Globally optimal linear precodersfor finite alphabet signals over complex vector Gaussian channels,” IEEETrans. Signal Process., vol. 59, no. 7, pp. 3301–3314, Apr. 2011.

[24] Y. Wu, C. Xiao, X. Gao, J. D. Matyjas, and Z. Ding, “Linear precoderdesign for MIMO interference channels with finite-alphabet signaling,”IEEE Trans. Commun., vol. 61, no. 9, pp. 3766–3780, Sep. 2013.

[25] F. Zhu, F. Gao, M. Yao, and H. Zou, “Joint information- and jamming-beamforming for physical layer security with full duplex base station,”IEEE Trans. Signal Process., vol. 62, no. 24, pp. 6391–6401, Dec. 2014.

[26] A. Alkhateeb, O. El Ayach, G. Leus, and R. W. Heath, “Hybrid precodingfor millimeter wave cellular systems with partial channel knowledge,” inProc. IEEE Inf. Theory Appl. Workshop (ITA’13), 2013, pp. 1–5.

[27] M. Tao and R. Wang, “Linear precoding for multi-pair two-way MIMOrelay systems with max-min fairness,” IEEE Trans. Signal Process.,vol. 60, no. 10, pp. 5361–5370, Oct. 2012.

[28] G. H. Golub and C. F. Van Loan, Matrix Computations. Baltimore, USA:JHU Press, 2012.

[29] Å. Björck, “Numerical methods in matrix computations,” Berlin,German: Springer, 2015.


[30] O. El Ayach, R. W. Heath, S. Rajagopal, and Z. Pi, “Multimode pre-coding in millimeter wave MIMO transmitters with multiple antennasub-arrays,” in Proc. IEEE Global Commun. Conf. (GLOBECOM’13),Dec. 2013, pp. 3476–3480.

[31] A. Alkhateeb, O. El Ayach, G. Leus, and R. Heath, “Channel estimationand hybrid precoding for millimeter wave cellular systems,” IEEE J. Sel.Topics Signal Process., vol. 8, no. 5, pp. 831–846, Oct. 2014.

[32] Y. Wu, C. Xiao, Z. Ding, X. Gao, and S. Jin, “Linear precoding for finite-alphabet signaling over MIMOME wiretap channels,” IEEE Trans. Veh.Technol., vol. 61, no. 6, pp. 2599–2612, Jul. 2012.

[33] S. Cui, A. J. Goldsmith, and A. Bahai, “Energy-constrained modulationoptimization,” IEEE Trans. Wireless Commun., vol. 4, no. 5, pp. 2349–2360, Sep. 2005.

[34] C. Masouros, M. Sellathurai, and T. Rantarajah, “Computationally effi-cient vector perturbation using thresholded optimization,” IEEE Trans.Commun., vol. 61, no. 5, pp. 1880–1890, May 2013.

[35] T. S. Rappaport, J. N. Murdock, and F. Gutierrez, “State of the art in 60-GHz integrated circuits and systems for wireless communications,” Proc.IEEE, vol. 99, no. 8, pp. 1390–1436, Aug. 2011.

[36] D. Tse and P. Viswanath, Fundamentals of Wireless Communication.Cambridge, U.K.: Cambridge Univ. Press, 2005.

Xinyu Gao (S’14) received the B.E. degree incommunication engineering from Harbin Instituteof Technology, Heilongjiang, China, in 2014. Heis currently pursuing the Ph.D. degree in elec-tronic engineering at Tsinghua University, Beijing,China. His research interests include massive MIMOand mmWave communications, with the empha-sis on signal detection and precoding. He hasauthored several journal and conference papers pub-lished in IEEE JOURNAL ON SELECTED AREAS

IN COMMUNICATIONS, IEEE TRANSACTIONS ON

VEHICULAR TECHNOLOGY, IEEE ICC, IEEE GLOBECOM, etc. He wasthe recipient of the National Scholarship in 2015.

Linglong Dai (M’11–SM’14) received the B.S.degree from Zhejiang University, Hangzhou, China,in 2003, the M.S. degree (with the highest Hons.)from the China Academy of TelecommunicationsTechnology (CATT), Beijing, China, in 2006, andthe Ph.D. degree (with the highest Hons.) fromTsinghua University, Beijing, China, in 2011. From2011 to 2013, he was a Postdoctoral Fellow withthe Department of Electronic Engineering, TsinghuaUniversity, where he has been an Assistant Professorsince July 2013 and now an Associate Professor. His

research interests include wireless communications, with a focus on multi-carrier techniques, multiantenna techniques, and multiuser techniques. He hasauthored more than 50 IEEE journal papers and more than 30 IEEE conferencepapers. He currently serves as a Co-Chair of the IEEE Special Interest Group(SIG) on Signal Processing Techniques in 5G Communication Systems. He wasthe recipient of the Outstanding Ph.D. Graduate of Tsinghua University Awardin 2011, the Excellent Doctoral Dissertation of Beijing Award in 2012, the IEEEICC Best Paper Award in 2013, the National Excellent Doctoral DissertationNomination Award in 2013, the IEEE ICC Best Paper Award in 2014, the URSIYoung Scientists Award in 2014, the IEEE Transactions on Broadcasting BestPaper Award in 2015, the IEEE RADIO Young Scientists Award in 2015.

Shuangfeng Han (M’06) received the M.S. andPh.D. degrees in electrical engineering fromTsinghua University, Beijing, China, in 2002 and2006, respectively. He joined Samsung Electronicsas a Senior Engineer, in 2006, working on MIMO,multi-BS MIMO, etc. Since 2012, he has been aSenior Project Manager in the Green CommunicationResearch Center, China Mobile Research Institute.His research interests include green 5G, mas-sive MIMO, full duplex, NOMA and EE-SEco-design. Currently, he is the Vice Chair of Wireless

Technology Work Group of China’s IMT-2020 (5G) promotion group.

Chih-Lin I (SM’03) received the Ph.D. degreein electrical engineering from Stanford University,Stanford, CA, USA. She has been working at mul-tiple world-class companies and research institutesleading the R&D, including AT&T Bell Labs; AT&THQ, ITRI of Taiwan, and ASTRI of Hong Kong.Currently, she is a China Mobile’s Chief Scientistof wireless technologies and has established theGreen Communications Research Center, spearhead-ing major initiatives including key 5G technologyR&D; high energy efficiency system architectures,

technologies and devices; green energy; and C-RAN and soft base stations.Her research interests include green communications, C-RAN, network con-vergence, bandwidth refarming, EE-SE co-design, massive MIMO, and activeantenna arrays. She was an elected Board Member of the IEEE ComSoc, theChair of the ComSoc Meetings and Conferences Board, and the FoundingChair of the IEEE WCNC Steering Committee. She is currently an ExecutiveBoard Member of GreenTouch and a Network Operator Council Memberof ETSI NFV. She was the recipient of the IEEE TRANSACTIONS ON

COMMUNICATIONS Stephen Rice Best Paper Award and is a winner of theCCCP National 1000 Talent program.

Robert W. Heath, Jr. (S’96–M’01–SM’06–F’11) received the B.S. and M.S. degrees from theUniversity of Virginia, Charlottesville, VA, USA, in1996 and 1997, respectively, and the Ph.D. degreefrom Stanford University, Stanford, CA, USA, in2002, all in electrical engineering. From 1998 to2001, he was a Senior Member of the Technical Staffthen a Senior Consultant with Iospan Wireless Inc.,San Jose, CA, USA, where he worked on the designand implementation of the physical and link layers ofthe first commercial MIMO-OFDM communication

system. Since January 2002, he has been with the Department of Electricaland Computer Engineering, University of Texas at Austin, Austin, TX, USA,where he is a Cullen Trust for Higher Education Endowed Professor, and is theDirector of the Wireless Networking and Communications Group. He is alsothe President and the CEO of MIMO Wireless Inc. and the Chief InnovationOfficer at Kuma Signals LLC. His research interests include several aspects ofwireless communication and signal processing: limited feedback techniques,multihop networking, multiuser and multicell MIMO, interference alignment,adaptive video transmission, manifold signal processing, and millimeter wavecommunication techniques.

Dr. Heath has been an Editor of the IEEE TRANSACTIONS ON

COMMUNICATION, an Associate Editor of the IEEE TRANSACTIONS ON

VEHICULAR TECHNOLOGY, a Lead Guest Editor of the IEEE JOURNAL

ON SELECTED AREAS IN COMMUNICATIONS special issue on LimitedFeedback Communication, and a Lead Guest Editor of the IEEE JOURNAL ON

SELECTED TOPICS IN SIGNAL PROCESSING special issue on HeterogenousNetworks. He currently serves on the steering committee for the IEEETRANSACTIONS ON WIRELESS COMMUNICATIONS. He was a member ofthe Signal Processing for Communications Technical Committee in the IEEESignal Processing Society and is a former Chair of the IEEE COMSOCCommunications Technical Theory Committee. He was a technical Co-Chairfor the 2007 Fall Vehicular Technology Conference, a General Chair of the2008 Communication Theory Workshop, a General Co-Chair, a TechnicalCo-Chair and a Co-Organizer of the 2009 IEEE Signal Processing forWireless Communications Workshop, a Local Co-Organizer for the 2009 IEEECAMSAP Conference, a Technical Co-Chair for the 2010 IEEE InternationalSymposium on Information Theory, the Technical Chair for the 2011 AsilomarConference on Signals, Systems, and Computers, the General Chair for the2013 Asilomar Conference on Signals, Systems, and Computers, a foundingGeneral Co-Chair for the 2013 IEEE GlobalSIP conference, and is a TechnicalCo-Chair for the 2014 IEEE GLOBECOM conference. He was a 2003 Frontiersin Education New Faculty Fellow. He is also a licensed amateur radio operatorand is a registered Professional Engineer in Texas. He was the recipient of theof Best Student Paper Awards at the IEEE VTC 2006 Spring, WPMC 2006,IEEE GLOBECOM 2006, IEEE VTC 2007 Spring, and IEEE RWS 2009, aswell as the corecipient of the Grand Prize in the 2008 WinTech WinCool DemoContest. He was also the corecipient of the 2010 and 2013 EURASIP Journalon Wireless Communications and Networking Best Paper Awards and the 2012Signal Processing Magazine Best Paper Award.

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